Test
Scenario
Interpretation
Used when dealing with large sample sizes or when the population standard deviation is known.
A small p-value (smaller than 0.05) indicates strong evidence against the null hypothesis, leading to its rejection.
Appropriate for small sample sizes or when the population standard deviation is unknown.
Similar to the Z-test
Used for tests of independence or goodness-of-fit.
A small p-value indicates that there is a significant association between the categorical variables, leading to the rejection of the null hypothesis.
Commonly used in Analysis of Variance (ANOVA) to compare variances between groups.
A small p-value suggests that at least one group mean is different from the others, leading to the rejection of the null hypothesis.
Measures the strength and direction of a linear relationship between two continuous variables.
A small p-value indicates that there is a significant linear relationship between the variables, leading to rejection of the null hypothesis that there is no correlation.
In general, a small p-value indicates that the observed data is unlikely to have occurred by random chance alone, which leads to the rejection of the null hypothesis. However, it’s crucial to choose the appropriate test based on the nature of the data and the research question, as well as to interpret the p-value in the context of the specific test being used.
The table given below shows the importance of p-value and shows the various kinds of errors that occur during hypothesis testing.
|
|
|
| Correct decision based | Type I error |
| Type II error | Incorrect decision based |
Type I error: Incorrect rejection of the null hypothesis. It is denoted by α (significance level). Type II error: Incorrect acceptance of the null hypothesis. It is denoted by β (power level)
A researcher wants to investigate whether there is a significant difference in mean height between males and females in a population of university students.
Suppose we have the following data:
Starting with interpreting the process of calculating p-value
H0: There is no significant difference in mean height between males and females.
H1: There is a significant difference in mean height between males and females.
The appropriate test statistic for this scenario is the two-sample t-test, which compares the means of two independent groups.
The t-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.
So, the calculated two-sample t-test statistic (t) is approximately 5.13.
The t-distribution is used for the two-sample t-test . The degrees of freedom for the t-distribution are determined by the sample sizes of the two groups.
The t-distribution is a probability distribution with tails that are thicker than those of the normal distribution.
The degrees of freedom (63) represent the variability available in the data to estimate the population parameters. In the context of the two-sample t-test, higher degrees of freedom provide a more precise estimate of the population variance, influencing the shape and characteristics of the t-distribution.
T-Statistic
The t-distribution is symmetric and bell-shaped, similar to the normal distribution. As the degrees of freedom increase, the t-distribution approaches the shape of the standard normal distribution. Practically, it affects the critical values used to determine statistical significance and confidence intervals.
Step 5 : Calculate Critical Value.
To find the critical t-value with a t-statistic of 5.13 and 63 degrees of freedom, we can either consult a t-table or use statistical software.
We can use scipy.stats module in Python to find the critical t-value using below code.
Comparing with T-Statistic:
The larger t-statistic suggests that the observed difference between the sample means is unlikely to have occurred by random chance alone. Therefore, we reject the null hypothesis.
In case the significance level is not specified, consider the below general inferences while interpreting your results.
Graphically, the p-value is located at the tails of any confidence interval. [As shown in fig 1]
Fig 1: Graphical Representation
The p-value in hypothesis testing is influenced by several factors:
Understanding these factors is crucial for interpreting p-values accurately and making informed decisions in hypothesis testing.
The p-value is a crucial concept in statistical hypothesis testing, serving as a guide for making decisions about the significance of the observed relationship or effect between variables.
Let’s consider a scenario where a tutor believes that the average exam score of their students is equal to the national average (85). The tutor collects a sample of exam scores from their students and performs a one-sample t-test to compare it to the population mean (85).
Since, 0.7059>0.05 , we would conclude to fail to reject the null hypothesis. This means that, based on the sample data, there isn’t enough evidence to claim a significant difference in the exam scores of the tutor’s students compared to the national average. The tutor would accept the null hypothesis, suggesting that the average exam score of their students is statistically consistent with the national average.
The p-value is a crucial concept in statistical hypothesis testing, providing a quantitative measure of the strength of evidence against the null hypothesis. It guides decision-making by comparing the p-value to a chosen significance level, typically 0.05. A small p-value indicates strong evidence against the null hypothesis, suggesting a statistically significant relationship or effect. However, the p-value is influenced by various factors and should be interpreted alongside other considerations, such as effect size and context.
Why is p-value greater than 1.
A p-value is a probability, and probabilities must be between 0 and 1. Therefore, a p-value greater than 1 is not possible.
It means that the observed test statistic is unlikely to occur by chance if the null hypothesis is true. It represents a 1% chance of observing the test statistic or a more extreme one under the null hypothesis.
A good p-value is typically less than or equal to 0.05, indicating that the null hypothesis is likely false and the observed relationship or effect is statistically significant.
It is a measure of the statistical significance of a parameter in the model. It represents the probability of obtaining the observed value of the parameter or a more extreme one, assuming the null hypothesis is true.
A low p-value means that the observed test statistic is unlikely to occur by chance if the null hypothesis is true. It suggests that the observed relationship or effect is statistically significant and not due to random sampling variation.
Compare p-values: Lower p-value indicates stronger evidence against null hypothesis, favoring results with smaller p-values in hypothesis testing.
Similar reads, improve your coding skills with practice.
Understanding p-value.
Yarilet Perez is an experienced multimedia journalist and fact-checker with a Master of Science in Journalism. She has worked in multiple cities covering breaking news, politics, education, and more. Her expertise is in personal finance and investing, and real estate.
In statistics, a p-value is defined as a number that indicates how likely you are to obtain a value that is at least equal to or more than the actual observation if the null hypothesis is correct.
The p-value serves as an alternative to rejection points to provide the smallest level of significance at which the null hypothesis would be rejected. A smaller p-value means stronger evidence in favor of the alternative hypothesis.
P-value is often used to promote credibility for studies or reports by government agencies. For example, the U.S. Census Bureau stipulates that any analysis with a p-value greater than 0.10 must be accompanied by a statement that the difference is not statistically different from zero. The Census Bureau also has standards in place stipulating which p-values are acceptable for various publications.
Jessica Olah / Investopedia
P-values are usually found using p-value tables or spreadsheets/statistical software. These calculations are based on the assumed or known probability distribution of the specific statistic tested. The sample size, which determines the reliability of the observed data, directly influences the accuracy of the p-value calculation. he p-value approach to hypothesis testing uses the calculated he p-value approach to hypothesis testing uses the calculated P-values are calculated from the deviation between the observed value and a chosen reference value, given the probability distribution of the statistic, with a greater difference between the two values corresponding to a lower p-value.
Mathematically, the p-value is calculated using integral calculus from the area under the probability distribution curve for all values of statistics that are at least as far from the reference value as the observed value is, relative to the total area under the probability distribution curve. Standard deviations, which quantify the dispersion of data points from the mean, are instrumental in this calculation.
The calculation for a p-value varies based on the type of test performed. The three test types describe the location on the probability distribution curve: lower-tailed test, upper-tailed test, or two-tailed test . In each case, the degrees of freedom play a crucial role in determining the shape of the distribution and thus, the calculation of the p-value.
In a nutshell, the greater the difference between two observed values, the less likely it is that the difference is due to simple random chance, and this is reflected by a lower p-value.
The p-value approach to hypothesis testing uses the calculated probability to determine whether there is evidence to reject the null hypothesis. This determination relies heavily on the test statistic, which summarizes the information from the sample relevant to the hypothesis being tested. The null hypothesis, also known as the conjecture, is the initial claim about a population (or data-generating process). The alternative hypothesis states whether the population parameter differs from the value of the population parameter stated in the conjecture.
In practice, the significance level is stated in advance to determine how small the p-value must be to reject the null hypothesis. Because different researchers use different levels of significance when examining a question, a reader may sometimes have difficulty comparing results from two different tests. P-values provide a solution to this problem.
Even a low p-value is not necessarily proof of statistical significance, since there is still a possibility that the observed data are the result of chance. Only repeated experiments or studies can confirm if a relationship is statistically significant.
For example, suppose a study comparing returns from two particular assets was undertaken by different researchers who used the same data but different significance levels. The researchers might come to opposite conclusions regarding whether the assets differ.
If one researcher used a confidence level of 90% and the other required a confidence level of 95% to reject the null hypothesis, and if the p-value of the observed difference between the two returns was 0.08 (corresponding to a confidence level of 92%), then the first researcher would find that the two assets have a difference that is statistically significant , while the second would find no statistically significant difference between the returns.
To avoid this problem, the researchers could report the p-value of the hypothesis test and allow readers to interpret the statistical significance themselves. This is called a p-value approach to hypothesis testing. Independent observers could note the p-value and decide for themselves whether that represents a statistically significant difference or not.
An investor claims that their investment portfolio’s performance is equivalent to that of the Standard & Poor’s (S&P) 500 Index . To determine this, the investor conducts a two-tailed test.
The null hypothesis states that the portfolio’s returns are equivalent to the S&P 500’s returns over a specified period, while the alternative hypothesis states that the portfolio’s returns and the S&P 500’s returns are not equivalent—if the investor conducted a one-tailed test , the alternative hypothesis would state that the portfolio’s returns are either less than or greater than the S&P 500’s returns.
The p-value hypothesis test does not necessarily make use of a preselected confidence level at which the investor should reset the null hypothesis that the returns are equivalent. Instead, it provides a measure of how much evidence there is to reject the null hypothesis. The smaller the p-value, the greater the evidence against the null hypothesis.
Thus, if the investor finds that the p-value is 0.001, there is strong evidence against the null hypothesis, and the investor can confidently conclude that the portfolio’s returns and the S&P 500’s returns are not equivalent.
Although this does not provide an exact threshold as to when the investor should accept or reject the null hypothesis, it does have another very practical advantage. P-value hypothesis testing offers a direct way to compare the relative confidence that the investor can have when choosing among multiple different types of investments or portfolios relative to a benchmark such as the S&P 500.
For example, for two portfolios, A and B, whose performance differs from the S&P 500 with p-values of 0.10 and 0.01, respectively, the investor can be much more confident that portfolio B, with a lower p-value, will actually show consistently different results.
A p-value less than 0.05 is typically considered to be statistically significant, in which case the null hypothesis should be rejected. A p-value greater than 0.05 means that deviation from the null hypothesis is not statistically significant, and the null hypothesis is not rejected.
A p-value of 0.001 indicates that if the null hypothesis tested were indeed true, then there would be a one-in-1,000 chance of observing results at least as extreme. This leads the observer to reject the null hypothesis because either a highly rare data result has been observed or the null hypothesis is incorrect.
If you have two different results, one with a p-value of 0.04 and one with a p-value of 0.06, the result with a p-value of 0.04 will be considered more statistically significant than the p-value of 0.06. Beyond this simplified example, you could compare a 0.04 p-value to a 0.001 p-value. Both are statistically significant, but the 0.001 example provides an even stronger case against the null hypothesis than the 0.04.
The p-value is used to measure the significance of observational data. When researchers identify an apparent relationship between two variables, there is always a possibility that this correlation might be a coincidence. A p-value calculation helps determine if the observed relationship could arise as a result of chance.
U.S. Census Bureau. “ Statistical Quality Standard E1: Analyzing Data .”
$H_o$: | |||
$H_a$: | μ | ≠ | μ₀ |
$n$ | = | $\bar{x}$ | = | = |
$\text{Test Statistic: }$ | = |
$\text{Degrees of Freedom: } $ | $df$ | = |
$ \text{Level of Significance: } $ | $\alpha$ | = |
$H_o$: | $\mu$ | ||
$H_a$: | $\mu$ | ≠ | $\mu_0$ |
$n$ | = | σ | = | $\mu$ | = |
$\text{Level of Significance: }$ | $\alpha$ | = |
The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.
$\sigma$ Known | $\sigma$ Unknown | |
Test Statistic | $ z = \dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{{\color{Black} n}}} $ | $ t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}} $ |
Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.
Lower Tail Test | Upper Tail Test | Two-Tailed Test |
$H_0 \colon \mu \geq \mu_0$ | $H_0 \colon \mu \leq \mu_0$ | $H_0 \colon \mu = \mu_0$ |
$H_a \colon \mu | $H_a \colon \mu \neq \mu_0$ |
In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.
To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.
In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.
To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.
Lower Tail Test | Upper Tail Test | Two-Tailed Test |
If $z \leq -z_\alpha$, reject $H_0$. | If $z \geq z_\alpha$, reject $H_0$. | If $z \leq -z_{\alpha/2}$ or $z \geq z_{\alpha/2}$, reject $H_0$. |
If $t \leq -t_\alpha$, reject $H_0$. | If $t \geq t_\alpha$, reject $H_0$. | If $t \leq -t_{\alpha/2}$ or $t \geq t_{\alpha/2}$, reject $H_0$. |
When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.
Condition | ||||
$H_0$ True | $H_a$ True | |||
Conclusion | Accept $H_0$ | Correct | Type II Error | |
Reject $H_0$ | Type I Error | Correct |
Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.
Hypothesis testing is a critical part of statistical analysis and is often the endpoint where conclusions are drawn about larger populations based on a sample or experimental dataset. Central to this process is the p-value. Broadly, the p-value quantifies the strength of evidence against the null hypothesis. Given the importance of the p-value, it is essential to ensure its interpretation is correct. Here are five essential tips for ensuring the p-value from a hypothesis test is understood correctly.
First, it is essential to understand what a p-value is. In hypothesis testing, the p-value is defined as the probability of observing your data, or data more extreme, if the null hypothesis is true. As a reminder, the null hypothesis states no difference between your data and the expected population.
For example, in a hypothesis test to see if changing a company’s logo drives more traffic to the website, a null hypothesis would state that the new traffic numbers are equal to the old traffic numbers. In this context, the p-value would be the probability that the data you observed, or data more extreme, would occur if this null hypothesis were true.
Therefore, a smaller p-value indicates that what you observed is unlikely to have occurred if the null were true, offering evidence to reject the null hypothesis. Typically, a cut-off value of 0.05 is used where any p-value below this is considered significant evidence against the null.
Based on the research question under exploration, there are two types of hypotheses: one-sided and two-sided. A one-sided test specifies a particular direction of effect, such as traffic to a website increasing after a design change. On the other hand, a two-sided test allows the change to be in either direction and is effective when the researcher wants to see any effect of the change.
Either way, determining the statistical significance of a p-value is the same: if the p-value is below a threshold value, it is statistically significant. However, when calculating the p-value, it is important to ensure the correct sided calculations have been completed.
Additionally, the interpretation of the meaning of a p-value will differ based on the directionality of the hypothesis. If a one-sided test is significant, the researchers can use the p-value to support a statistically significant increase or decrease based on the direction of the test. If a two-sided test is significant, the p-value can only be used to say that the two groups are different, but not that one is necessarily greater.
A common pitfall in interpreting p-values is falling into the threshold thinking trap. The most commonly used cut-off value for whether a calculated p-value is statistically significant is 0.05. Typically, a p-value of less than 0.05 is considered statistically significant evidence against the null hypothesis.
However, this is just an arbitrary value. Rigid adherence to this or any other predefined cut-off value can obscure business-relevant effect sizes. For example, a hypothesis test looking at changes in traffic after a website design may find that an increase of 10,000 views is not statistically significant with a p-value of 0.055 since that value is above 0.05. However, the actual increase of 10,000 may be important to the growth of the business.
Therefore, a p-value can be practically significant while not being statistically significant. Both types of significance and the broader context of the hypothesis test should be considered when making a final interpretation.
Similarly, some study conditions can result in a non-significant p-value even if practical significance exists. Statistical power is the ability of a study to detect an effect when it truly exists. In other words, it is the probability that the null hypothesis will be rejected when it is false.
Power is impacted by a lot of factors. These include sample size, the effect size you are looking for, and variability within the data. In the example of website traffic after a design change, if the number of visits overall is too small, there may not be enough views to have enough power to detect a difference.
Simple ways to increase the power of a hypothesis test and increase the chances of detecting an effect are increasing the sample size, looking for a smaller effect size, changing the experiment design to control for variables that can increase variability, or adjusting the type of statistical test being run.
Whenever multiple p-values are calculated in a single study due to multiple comparisons, there is an increased risk of false positives. This is because each individual comparison introduces random fluctuations, and each additional comparison compounds these fluctuations.
For example, in a hypothesis test looking at traffic before and after a website redesign, the team may be interested in making more than one comparison. This can include total visits, page views, and average time spent on the website. Since multiple comparisons are being made, there must be a correction made when interpreting the p-value.
The Bonferroni correction is one of the most commonly used methods to account for this increased probability of false positives. In this method, the significance cut-off value, typically 0.05, is divided by the number of comparisons made. The result is used as the new significance cut-off value. Applying this correction mitigates the risk of false positives and improves the reliability of findings from a hypothesis test.
In conclusion, interpreting p-values requires a nuanced understanding of many statistical concepts and careful consideration of the hypothesis test’s context. By following these five tips, the interpretation of the p-value from a hypothesis test can be more accurate and reliable, leading to better data-driven decision-making.
Mehrnaz holds a Masters in Data Analytics and is a full time biostatistician working on complex machine learning development and statistical analysis in healthcare. She has experience with AI and has taught university courses in biostatistics and machine learning at University of the People.
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Arcu felis bibendum ut tristique et egestas quis:
Hypothesis testing.
Key Topics:
sampled from a with unknown mean μ and known variance σ . : μ = μ H : μ ≤ μ H : μ ≥ μ | : μ ≠ μ H : μ > μ H : μ < μ |
It is either likely or unlikely that we would collect the evidence we did given the initial assumption. (Note: “likely” or “unlikely” is measured by calculating a probability!)
If it is likely , then we “ do not reject ” our initial assumption. There is not enough evidence to do otherwise.
If it is unlikely , then:
In statistics, if it is unlikely, we decide to “ reject ” our initial assumption.
First, state 2 hypotheses, the null hypothesis (“H 0 ”) and the alternative hypothesis (“H A ”)
Usually the H 0 is a statement of “no effect”, or “no change”, or “chance only” about a population parameter.
While the H A , depending on the situation, is that there is a difference, trend, effect, or a relationship with respect to a population parameter.
Then, collect evidence, such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, handwriting samples, etc. (In statistics, the data are the evidence.)
Next, you make your initial assumption.
In statistics, we always assume the null hypothesis is true .
Then, make a decision based on the available evidence.
If the observed outcome, e.g., a sample statistic, is surprising under the assumption that the null hypothesis is true, but more probable if the alternative is true, then this outcome is evidence against H 0 and in favor of H A .
An observed effect so large that it would rarely occur by chance is called statistically significant (i.e., not likely to happen by chance).
The p -value represents how likely we would be to observe such an extreme sample if the null hypothesis were true. The p -value is a probability computed assuming the null hypothesis is true, that the test statistic would take a value as extreme or more extreme than that actually observed. Since it's a probability, it is a number between 0 and 1. The closer the number is to 0 means the event is “unlikely.” So if p -value is “small,” (typically, less than 0.05), we can then reject the null hypothesis.
Significance level, α, is a decisive value for p -value. In this context, significant does not mean “important”, but it means “not likely to happened just by chance”.
α is the maximum probability of rejecting the null hypothesis when the null hypothesis is true. If α = 1 we always reject the null, if α = 0 we never reject the null hypothesis. In articles, journals, etc… you may read: “The results were significant ( p <0.05).” So if p =0.03, it's significant at the level of α = 0.05 but not at the level of α = 0.01. If we reject the H 0 at the level of α = 0.05 (which corresponds to 95% CI), we are saying that if H 0 is true, the observed phenomenon would happen no more than 5% of the time (that is 1 in 20). If we choose to compare the p -value to α = 0.01, we are insisting on a stronger evidence!
Neither decision of rejecting or not rejecting the H entails proving the null hypothesis or the alternative hypothesis. We merely state there is enough evidence to behave one way or the other. This is also always true in statistics! |
So, what kind of error could we make? No matter what decision we make, there is always a chance we made an error.
Errors in Criminal Trial:
Errors in Hypothesis Testing
Type I error (False positive): The null hypothesis is rejected when it is true.
Type II error (False negative): The null hypothesis is not rejected when it is false.
There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!
The power of a statistical test is its probability of rejecting the null hypothesis if the null hypothesis is false. That is, power is the ability to correctly reject H 0 and detect a significant effect. In other words, power is one minus the type II error risk.
\(\text{Power }=1-\beta = P\left(\text{reject} H_0 | H_0 \text{is false } \right)\)
Which error is worse?
Type I = you are innocent, yet accused of cheating on the test. Type II = you cheated on the test, but you are found innocent.
This depends on the context of the problem too. But in most cases scientists are trying to be “conservative”; it's worse to make a spurious discovery than to fail to make a good one. Our goal it to increase the power of the test that is to minimize the length of the CI.
We need to keep in mind:
(see the handout). To study the tradeoffs between the sample size, α, and Type II error we can use power and operating characteristic curves.
Assume data are independently sampled from a normal distribution with unknown mean μ and known variance σ = 9. Make an initial assumption that μ = 65. Specify the hypothesis: H : μ = 65 H : μ ≠ 65 z-statistic: 3.58 z-statistic follow N(0,1) distribution
The -value, < 0.0001, indicates that, if the average height in the population is 65 inches, it is unlikely that a sample of 54 students would have an average height of 66.4630. Alpha = 0.05. Decision: -value < alpha, thus Conclude that the average height is not equal to 65. |
What type of error might we have made?
Type I error is claiming that average student height is not 65 inches, when it really is. Type II error is failing to claim that the average student height is not 65in when it is.
We rejected the null hypothesis, i.e., claimed that the height is not 65, thus making potentially a Type I error. But sometimes the p -value is too low because of the large sample size, and we may have statistical significance but not really practical significance! That's why most statisticians are much more comfortable with using CI than tests.
Based on the CI only, how do you know that you should reject the null hypothesis? The 95% CI is (65.6628,67.2631) ... What about practical and statistical significance now? Is there another reason to suspect this test, and the -value calculations? |
There is a need for a further generalization. What if we can't assume that σ is known? In this case we would use s (the sample standard deviation) to estimate σ.
If the sample is very large, we can treat σ as known by assuming that σ = s . According to the law of large numbers, this is not too bad a thing to do. But if the sample is small, the fact that we have to estimate both the standard deviation and the mean adds extra uncertainty to our inference. In practice this means that we need a larger multiplier for the standard error.
We need one-sample t -test.
: μ = μ H : μ ≤ μ H : μ ≥ μ | : μ ≠ μ H : μ > μ H : μ < μ |
Let's go back to our CNN poll. Assume we have a SRS of 1,017 adults.
We are interested in testing the following hypothesis: H 0 : p = 0.50 vs. p > 0.50
What is the test statistic?
If alpha = 0.05, what do we conclude?
We will see more details in the next lesson on proportions, then distributions, and possible tests.
In Statistics, the researcher checks the significance of the observed result, which is known as test static . For this test, a hypothesis test is also utilized. The P-value or probability value concept is used everywhere in statistical analysis. It determines the statistical significance and the measure of significance testing. In this article, let us discuss its definition, formula, table, interpretation and how to use P-value to find the significance level etc. in detail.
Table of Contents:
The P-value is known as the probability value. It is defined as the probability of getting a result that is either the same or more extreme than the actual observations. The P-value is known as the level of marginal significance within the hypothesis testing that represents the probability of occurrence of the given event. The P-value is used as an alternative to the rejection point to provide the least significance at which the null hypothesis would be rejected. If the P-value is small, then there is stronger evidence in favour of the alternative hypothesis.
The P-value table shows the hypothesis interpretations:
| |
P-value > 0.05 | The result is not statistically significant and hence don’t reject the null hypothesis. |
P-value < 0.05 | The result is statistically significant. Generally, reject the null hypothesis in favour of the alternative hypothesis. |
P-value < 0.01 | The result is highly statistically significant, and thus rejects the null hypothesis in favour of the alternative hypothesis. |
Generally, the level of statistical significance is often expressed in p-value and the range between 0 and 1. The smaller the p-value, the stronger the evidence and hence, the result should be statistically significant. Hence, the rejection of the null hypothesis is highly possible, as the p-value becomes smaller.
Let us look at an example to better comprehend the concept of P-value.
Let’s say a researcher flips a coin ten times with the null hypothesis that it is fair. The total number of heads is the test statistic, which is two-tailed. Assume the researcher notices alternating heads and tails on each flip (HTHTHTHTHT). As this is the predicted number of heads, the test statistic is 5 and the p-value is 1 (totally unexceptional).
Assume that the test statistic for this research was the “number of alternations” (i.e., the number of times H followed T or T followed H), which is two-tailed once again. This would result in a test statistic of 9, which is extremely high and has a p-value of 1/2 8 = 1/256, or roughly 0.0039. This would be regarded as extremely significant, much beyond the 0.05 level. These findings suggest that the data set is exceedingly improbable to have happened by random in terms of one test statistic, yet they do not imply that the coin is biased towards heads or tails.
The data have a high p-value according to the first test statistic, indicating that the number of heads observed is not impossible. The data have a low p-value according to the second test statistic, indicating that the pattern of flips observed is extremely unlikely. There is no “alternative hypothesis,” (therefore only the null hypothesis can be rejected), and such evidence could have a variety of explanations – the data could be falsified, or the coin could have been flipped by a magician who purposefully swapped outcomes.
This example shows that the p-value is entirely dependent on the test statistic used and that p-values can only be used to reject a null hypothesis, not to explore an alternate hypothesis.
We Know that P-value is a statistical measure, that helps to determine whether the hypothesis is correct or not. P-value is a number that lies between 0 and 1. The level of significance(α) is a predefined threshold that should be set by the researcher. It is generally fixed as 0.05. The formula for the calculation for P-value is
Step 1: Find out the test static Z is
P0 = assumed population proportion in the null hypothesis
N = sample size
Step 2: Look at the Z-table to find the corresponding level of P from the z value obtained.
An example to find the P-value is given here.
Question: A statistician wants to test the hypothesis H 0 : μ = 120 using the alternative hypothesis Hα: μ > 120 and assuming that α = 0.05. For that, he took the sample values as
n =40, σ = 32.17 and x̄ = 105.37. Determine the conclusion for this hypothesis?
We know that,
Now substitute the given values
Now, using the test static formula, we get
t = (105.37 – 120) / 5.0865
Therefore, t = -2.8762
Using the Z-Score table , we can find the value of P(t>-2.8762)
From the table, we get
P (t<-2.8762) = P(t>2.8762) = 0.003
If P(t>-2.8762) =1- 0.003 =0.997
P- value =0.997 > 0.05
Therefore, from the conclusion, if p>0.05, the null hypothesis is accepted or fails to reject.
Hence, the conclusion is “fails to reject H 0. ”
What is meant by p-value.
The p-value is defined as the probability of obtaining the result at least as extreme as the observed result of a statistical hypothesis test, assuming that the null hypothesis is true.
The smaller the p-value, the greater the statistical significance of the observed difference, which results in the rejection of the null hypothesis in favour of alternative hypotheses.
If the p-value is greater than 0.05, then the result is not statistically significant.
P-value means probability value, which tells you the probability of achieving the result under a certain hypothesis. Since it is a probability, its value ranges between 0 and 1, and it cannot exceed 1.
If the p-value is less than 0.05, then the result is statistically significant, and hence we can reject the null hypothesis in favour of the alternative hypothesis.
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Statistics By Jim
Making statistics intuitive
By Jim Frost 45 Comments
Hypothesis testing is a vital process in inferential statistics where the goal is to use sample data to draw conclusions about an entire population . In the testing process, you use significance levels and p-values to determine whether the test results are statistically significant.
You hear about results being statistically significant all of the time. But, what do significance levels, P values, and statistical significance actually represent? Why do we even need to use hypothesis tests in statistics?
In this post, I answer all of these questions. I use graphs and concepts to explain how hypothesis tests function in order to provide a more intuitive explanation. This helps you move on to understanding your statistical results.
To start, I’ll demonstrate why we need to use hypothesis tests using an example.
A researcher is studying fuel expenditures for families and wants to determine if the monthly cost has changed since last year when the average was $260 per month. The researcher draws a random sample of 25 families and enters their monthly costs for this year into statistical software. You can download the CSV data file: FuelsCosts . Below are the descriptive statistics for this year.
We’ll build on this example to answer the research question and show how hypothesis tests work.
The researcher collected a random sample and found that this year’s sample mean (330.6) is greater than last year’s mean (260). Why perform a hypothesis test at all? We can see that this year’s mean is higher by $70! Isn’t that different?
Regrettably, the situation isn’t as clear as you might think because we’re analyzing a sample instead of the full population. There are huge benefits when working with samples because it is usually impossible to collect data from an entire population. However, the tradeoff for working with a manageable sample is that we need to account for sample error.
The sampling error is the gap between the sample statistic and the population parameter. For our example, the sample statistic is the sample mean, which is 330.6. The population parameter is μ, or mu, which is the average of the entire population. Unfortunately, the value of the population parameter is not only unknown but usually unknowable. Learn more about Sampling Error .
We obtained a sample mean of 330.6. However, it’s conceivable that, due to sampling error, the mean of the population might be only 260. If the researcher drew another random sample, the next sample mean might be closer to 260. It’s impossible to assess this possibility by looking at only the sample mean. Hypothesis testing is a form of inferential statistics that allows us to draw conclusions about an entire population based on a representative sample. We need to use a hypothesis test to determine the likelihood of obtaining our sample mean if the population mean is 260.
Background information : The Difference between Descriptive and Inferential Statistics and Populations, Parameters, and Samples in Inferential Statistics
It is very unlikely for any sample mean to equal the population mean because of sample error. In our case, the sample mean of 330.6 is almost definitely not equal to the population mean for fuel expenditures.
If we could obtain a substantial number of random samples and calculate the sample mean for each sample, we’d observe a broad spectrum of sample means. We’d even be able to graph the distribution of sample means from this process.
This type of distribution is called a sampling distribution. You obtain a sampling distribution by drawing many random samples of the same size from the same population. Why the heck would we do this?
Because sampling distributions allow you to determine the likelihood of obtaining your sample statistic and they’re crucial for performing hypothesis tests.
Luckily, we don’t need to go to the trouble of collecting numerous random samples! We can estimate the sampling distribution using the t-distribution, our sample size, and the variability in our sample.
We want to find out if the average fuel expenditure this year (330.6) is different from last year (260). To answer this question, we’ll graph the sampling distribution based on the assumption that the mean fuel cost for the entire population has not changed and is still 260. In statistics, we call this lack of effect, or no change, the null hypothesis . We use the null hypothesis value as the basis of comparison for our observed sample value.
Sampling distributions and t-distributions are types of probability distributions.
Related posts : Sampling Distributions and Understanding Probability Distributions
The graph below shows which sample means are more likely and less likely if the population mean is 260. We can place our sample mean in this distribution. This larger context helps us see how unlikely our sample mean is if the null hypothesis is true (μ = 260).
The graph displays the estimated distribution of sample means. The most likely values are near 260 because the plot assumes that this is the true population mean. However, given random sampling error, it would not be surprising to observe sample means ranging from 167 to 352. If the population mean is still 260, our observed sample mean (330.6) isn’t the most likely value, but it’s not completely implausible either.
The sampling distribution shows us that we are relatively unlikely to obtain a sample of 330.6 if the population mean is 260. Is our sample mean so unlikely that we can reject the notion that the population mean is 260?
In statistics, we call this rejecting the null hypothesis. If we reject the null for our example, the difference between the sample mean (330.6) and 260 is statistically significant. In other words, the sample data favor the hypothesis that the population average does not equal 260.
However, look at the sampling distribution chart again. Notice that there is no special location on the curve where you can definitively draw this conclusion. There is only a consistent decrease in the likelihood of observing sample means that are farther from the null hypothesis value. Where do we decide a sample mean is far away enough?
To answer this question, we’ll need more tools—hypothesis tests! The hypothesis testing procedure quantifies the unusualness of our sample with a probability and then compares it to an evidentiary standard. This process allows you to make an objective decision about the strength of the evidence.
We’re going to add the tools we need to make this decision to the graph—significance levels and p-values!
These tools allow us to test these two hypotheses:
Related post : Hypothesis Testing Overview
A significance level, also known as alpha or α, is an evidentiary standard that a researcher sets before the study. It defines how strongly the sample evidence must contradict the null hypothesis before you can reject the null hypothesis for the entire population. The strength of the evidence is defined by the probability of rejecting a null hypothesis that is true. In other words, it is the probability that you say there is an effect when there is no effect.
For instance, a significance level of 0.05 signifies a 5% risk of deciding that an effect exists when it does not exist.
Lower significance levels require stronger sample evidence to be able to reject the null hypothesis. For example, to be statistically significant at the 0.01 significance level requires more substantial evidence than the 0.05 significance level. However, there is a tradeoff in hypothesis tests. Lower significance levels also reduce the power of a hypothesis test to detect a difference that does exist.
The technical nature of these types of questions can make your head spin. A picture can bring these ideas to life!
To learn a more conceptual approach to significance levels, see my post about Understanding Significance Levels .
On the probability distribution plot, the significance level defines how far the sample value must be from the null value before we can reject the null. The percentage of the area under the curve that is shaded equals the probability that the sample value will fall in those regions if the null hypothesis is correct.
To represent a significance level of 0.05, I’ll shade 5% of the distribution furthest from the null value.
The two shaded regions in the graph are equidistant from the central value of the null hypothesis. Each region has a probability of 0.025, which sums to our desired total of 0.05. These shaded areas are called the critical region for a two-tailed hypothesis test.
The critical region defines sample values that are improbable enough to warrant rejecting the null hypothesis. If the null hypothesis is correct and the population mean is 260, random samples (n=25) from this population have means that fall in the critical region 5% of the time.
Our sample mean is statistically significant at the 0.05 level because it falls in the critical region.
Related posts : One-Tailed and Two-Tailed Tests Explained , What Are Critical Values? , and T-distribution Table of Critical Values
Let’s redo this hypothesis test using the other common significance level of 0.01 to see how it compares.
This time the sum of the two shaded regions equals our new significance level of 0.01. The mean of our sample does not fall within with the critical region. Consequently, we fail to reject the null hypothesis. We have the same exact sample data, the same difference between the sample mean and the null hypothesis value, but a different test result.
What happened? By specifying a lower significance level, we set a higher bar for the sample evidence. As the graph shows, lower significance levels move the critical regions further away from the null value. Consequently, lower significance levels require more extreme sample means to be statistically significant.
You must set the significance level before conducting a study. You don’t want the temptation of choosing a level after the study that yields significant results. The only reason I compared the two significance levels was to illustrate the effects and explain the differing results.
The graphical version of the 1-sample t-test we created allows us to determine statistical significance without assessing the P value. Typically, you need to compare the P value to the significance level to make this determination.
Related post : Step-by-Step Instructions for How to Do t-Tests in Excel
P values are the probability that a sample will have an effect at least as extreme as the effect observed in your sample if the null hypothesis is correct.
This tortuous, technical definition for P values can make your head spin. Let’s graph it!
First, we need to calculate the effect that is present in our sample. The effect is the distance between the sample value and null value: 330.6 – 260 = 70.6. Next, I’ll shade the regions on both sides of the distribution that are at least as far away as 70.6 from the null (260 +/- 70.6). This process graphs the probability of observing a sample mean at least as extreme as our sample mean.
The total probability of the two shaded regions is 0.03112. If the null hypothesis value (260) is true and you drew many random samples, you’d expect sample means to fall in the shaded regions about 3.1% of the time. In other words, you will observe sample effects at least as large as 70.6 about 3.1% of the time if the null is true. That’s the P value!
Learn more about How to Find the P Value .
If your P value is less than or equal to your alpha level, reject the null hypothesis.
The P value results are consistent with our graphical representation. The P value of 0.03112 is significant at the alpha level of 0.05 but not 0.01. Again, in practice, you pick one significance level before the experiment and stick with it!
Using the significance level of 0.05, the sample effect is statistically significant. Our data support the alternative hypothesis, which states that the population mean doesn’t equal 260. We can conclude that mean fuel expenditures have increased since last year.
P values are very frequently misinterpreted as the probability of rejecting a null hypothesis that is actually true. This interpretation is wrong! To understand why, please read my post: How to Interpret P-values Correctly .
Hypothesis tests determine whether your sample data provide sufficient evidence to reject the null hypothesis for the entire population. To perform this test, the procedure compares your sample statistic to the null value and determines whether it is sufficiently rare. “Sufficiently rare” is defined in a hypothesis test by:
There is no special significance level that correctly determines which studies have real population effects 100% of the time. The traditional significance levels of 0.05 and 0.01 are attempts to manage the tradeoff between having a low probability of rejecting a true null hypothesis and having adequate power to detect an effect if one actually exists.
The significance level is the rate at which you incorrectly reject null hypotheses that are actually true ( type I error ). For example, for all studies that use a significance level of 0.05 and the null hypothesis is correct, you can expect 5% of them to have sample statistics that fall in the critical region. When this error occurs, you aren’t aware that the null hypothesis is correct, but you’ll reject it because the p-value is less than 0.05.
This error does not indicate that the researcher made a mistake. As the graphs show, you can observe extreme sample statistics due to sample error alone. It’s the luck of the draw!
Related posts : Statistical Significance: Definition & Meaning and Types of Errors in Hypothesis Testing
Hypothesis tests are crucial when you want to use sample data to make conclusions about a population because these tests account for sample error. Using significance levels and P values to determine when to reject the null hypothesis improves the probability that you will draw the correct conclusion.
Keep in mind that statistical significance doesn’t necessarily mean that the effect is important in a practical, real-world sense. For more information, read my post about Practical vs. Statistical Significance .
If you like this post, read the companion post: How Hypothesis Tests Work: Confidence Intervals and Confidence Levels .
You can also read my other posts that describe how other tests work:
To see an alternative approach to traditional hypothesis testing that does not use probability distributions and test statistics, learn about bootstrapping in statistics !
December 11, 2022 at 10:56 am
For very easy concept about level of significance & p-value 1.Teacher has given a one assignment to student & asked how many error you have doing this assignment? Student reply, he can has error ≤ 5% (it is level of significance). After completion of assignment, teacher checked his error which is ≤ 5% (may be 4% or 3% or 2% even less, it is p-value) it means his results are significant. Otherwise he has error > 5% (may be 6% or 7% or 8% even more, it is p-value) it means his results are non-significant. 2. Teacher has given a one assignment to student & asked how many error you have doing this assignment? Student reply, he can has error ≤ 1% (it is level of significance). After completion of assignment, teacher checked his error which is ≤ 1% (may be 0.9% or 0.8% or 0.7% even less, it is p-value) it means his results are significant. Otherwise he has error > 1% (may be 1.1% or 1.5% or 2% even more, it is p-value) it means his results are non-significant. p-value is significant or not mainly dependent upon the level of significance.
December 11, 2022 at 7:50 pm
I think that approach helps explain how to determine statistical significance–is the p-value less than or equal to the significance level. However, it doesn’t really explain what statistical significance means. I find that comparing the p-value to the significance level is the easy part. Knowing what it means and how to choose your significance level is the harder part!
December 3, 2022 at 5:54 pm
What would you say to someone who believes that a p-value higher than the level of significance (alpha) means the null hypothesis has been proven? Should you support that statement or deny it?
December 3, 2022 at 10:18 pm
Hi Emmanuel,
When the p-value is greater than the significance level, you fail to reject the null hypothesis . That is different than proving it. To learn why and what it means, click the link to read a post that I’ve written that will answer your question!
April 19, 2021 at 12:27 am
Thank you so much Sir
April 18, 2021 at 2:37 pm
Hi sir, your blogs are much more helpful for clearing the concepts of statistics, as a researcher I find them much more useful. I have some quarries:
1. In many research papers I have seen authors using the statement ” means or values are statically at par at p = 0.05″ when they do some pair wise comparison between the treatments (a kind of post hoc) using some value of CD (critical difference) or we can say LSD which is calculated using alpha not using p. So with this article I think this should be alpha =0.05 or 5%, not p = 0.05 earlier I thought p and alpha are same. p it self is compared with alpha 0.05. Correct me if I am wrong.
2. When we can draw a conclusion using critical value based on critical values (CV) which is based on alpha values in different tests (e.g. in F test CV is at F (0.05, t-1, error df) when alpha is 0.05 which is table value of F and is compared with F calculated for drawing the conclusion); then why we go for p values, and draw a conclusion based on p values, even many online software do not give p value, they just mention CD (LSD)
3. can you please help me in interpreting interaction in two factor analysis (Factor A X Factor b) in Anova.
Thank You so much!
(Commenting again as I have not seen my comment in comment list; don’t know why)
April 18, 2021 at 10:57 pm
Hi Himanshu,
I manually approve comments so there will be some time lag involved before they show up.
Regarding your first question, yes, you’re correct. Test results are significant at particular significance levels or alpha. They should not use p to define the significance level. You’re also correct in that you compare p to alpha.
Critical values are a different (but related) approach for determining significance. It was more common before computer analysis took off because it reduced the calculations. Using this approach in its simplest form, you only know whether a result is significant or not at the given alpha. You just determine whether the test statistic falls within a critical region to determine statistical significance or not significant. However, it is ok to supplement this type of result with the actual p-value. Knowing the precise p-value provides additional information that significant/not significant does not provide. The critical value and p-value approaches will always agree too. For more information about why the exact p-value is useful, read my post about Five Tips for Interpreting P-values .
Finally, I’ve written about two-way ANOVA in my post, How to do Two-Way ANOVA in Excel . Additionally, I write about it in my Hypothesis Testing ebook .
January 28, 2021 at 3:12 pm
Thank you for your answer, Jim, I really appreciate it. I’m taking a Coursera stats course and online learning without being able to ask questions of a real teacher is not my forte!
You’re right, I don’t think I’m ready for that calculation! However, I think I’m struggling with something far more basic, perhaps even the interpretation of the t-table? I’m just not sure how you came up with the p-value as .03112, with the 24 degrees of freedom. When I pull up a t-table and look at the 24-degrees of freedom row, I’m not sure how any of those numbers correspond with your answer? Either the single tail of 0.01556 or the combined of 0.03112. What am I not getting? (which, frankly, could be a lot!!) Again, thank you SO much for your time.
January 28, 2021 at 11:19 pm
Ah ok, I see! First, let me point you to several posts I’ve written about t-values and the t-distribution. I don’t cover those in this post because I wanted to present a simplified version that just uses the data in its regular units. The basic idea is that the hypothesis tests actually convert all your raw data down into one value for a test statistic, such as the t-value. And then it uses that test statistic to determine whether your results are statistically significant. To be significant, the t-value must exceed a critical value, which is what you lookup in the table. Although, nowadays you’d typically let your software just tell you.
So, read the following two posts, which covers several aspects of t-values and distributions. And then if you have more questions after that, you can post them. But, you’ll have a lot more information about them and probably some of your questions will be answered! T-values T-distributions
January 27, 2021 at 3:10 pm
Jim, just found your website and really appreciate your thoughtful, thorough way of explaining things. I feel very dumb, but I’m struggling with p-values and was hoping you could help me.
Here’s the section that’s getting me confused:
“First, we need to calculate the effect that is present in our sample. The effect is the distance between the sample value and null value: 330.6 – 260 = 70.6. Next, I’ll shade the regions on both sides of the distribution that are at least as far away as 70.6 from the null (260 +/- 70.6). This process graphs the probability of observing a sample mean at least as extreme as our sample mean.
** I’m good up to this point. Draw the picture, do the subtraction, shade the regions. BUT, I’m not sure how to figure out the area of the shaded region — even with a T-table. When I look at the T-table on 24 df, I’m not sure what to do with those numbers, as none of them seem to correspond in any way to what I’m looking at in the problem. In the end, I have no idea how you calculated each shaded area being 0.01556.
I feel like there’s a (very simple) step that everyone else knows how to do, but for some reason I’m missing it.
Again, dumb question, but I’d love your help clarifying that.
thank you, Sara
January 27, 2021 at 9:51 pm
That’s not a dumb question at all. I actually don’t show or explain the calculations for figuring out the area. The reason for that is the same reason why students never calculate the critical t-values for their test, instead you look them up in tables or use statistical software. The common reason for all that is because calculating these values is extremely complicated! It’s best to let software do that for you or, when looking critical values, use the tables!
The principal though is that percentage of the area under the curve equals the probability that values will fall within that range.
And then, for this example, you’d need to figure out the area under the curve for particular ranges!
January 15, 2021 at 10:57 am
HI Jim, I have a question related to Hypothesis test.. in Medical imaging, there are different way to measure signal intensity (from a tumor lesion for example). I tested for the same 100 patients 4 different ways to measure tumor captation to a injected dose. So for the 100 patients, i got 4 linear regression (relation between injected dose and measured quantity at tumor sites) = so an output of 4 equations Condition A output = -0,034308 + 0,0006602*input Condition B output = 0,0117631 + 0,0005425*input Condition C output = 0,0087871 + 0,0005563*input Condition D output = 0,001911 + 0,0006255*input
My question : i want to compare the 4 methods to find the best one (compared to others) : do Hypothesis test good to me… and if Yes, i do not find test to perform it. Can you suggest me a software. I uselly used JMP for my stats… but open to other softwares…
THank for your time G
November 16, 2020 at 5:42 am
Thank you very much for writing about this topic!
Your explanation made more sense to me about: Why we reject Null Hypothesis when p value < significance level
Kind greetings, Jalal
September 25, 2020 at 1:04 pm
Hi Jim, Your explanations are so helpful! Thank you. I wondered about your first graph. I see that the mean of the graph is 260 from the null hypothesis, and it looks like the standard deviation of the graph is about 31. Where did you get 31 from? Thank you
September 25, 2020 at 4:08 pm
Hi Michelle,
That is a great question. Very observant. And it gets to how these tests work. The hypothesis test that I’m illustrating here is the one-sample t-test. And this graph illustrates the sampling distribution for the t-test. T-tests use the t-distribution to determine the sampling distribution. For the t-distribution, you need to specify the degrees of freedom, which entirely defines the distribution (i.e., it’s the only parameter). For 1-sample t-tests, the degrees of freedom equal the number of observations minus 1. This dataset has 25 observations. Hence, the 24 DF you see in the graph.
Unlike the normal distribution, there is no standard deviation parameter. Instead, the degrees of freedom determines the spread of the curve. Typically, with t-tests, you’ll see results discussed in terms of t-values, both for your sample and for defining the critical regions. However, for this introductory example, I’ve converted the t-values into the raw data units (t-value * SE mean).
So, the standard deviation you’re seeing in the graph is a result of the spread of the underlying t-distribution that has 24 degrees of freedom and then applying the conversion from t-values to raw values.
September 10, 2020 at 8:19 am
Your blog is incredible.
I am having difficulty understanding why the phrase ‘as extreme as’ is required in the definition of p-value (“P values are the probability that a sample will have an effect at least as extreme as the effect observed in your sample if the null hypothesis is correct.”)
Why can’t P-Values simply be defined as “The probability of sample observation if the null hypothesis is correct?”
In your other blog titled ‘Interpreting P values’ you have explained p-values as “P-values indicate the believability of the devil’s advocate case that the null hypothesis is correct given the sample data”. I understand (or accept) this explanation. How does one move from this definition to one that contains the phrase ‘as extreme as’?
September 11, 2020 at 5:05 pm
Thanks so much for your kind words! I’m glad that my website has been helpful!
The key to understanding the “at least as extreme” wording lies in the probability plots for p-values. Using probability plots for continuous data, you can calculate probabilities, but only for ranges of values. I discuss this in my post about understanding probability distributions . In a nutshell, we need a range of values for these probabilities because the probabilities are derived from the area under a distribution curve. A single value just produces a line on these graphs rather than an area. Those ranges are the shaded regions in the probability plots. For p-values, the range corresponds to the “at least as extreme” wording. That’s where it comes from. We need a range to calculate a probability. We can’t use the single value of the observed effect because it doesn’t produce an area under the curve.
I hope that helps! I think this is a particularly confusing part of understanding p-values that most people don’t understand.
August 7, 2020 at 5:45 pm
Hi Jim, thanks for the post.
Could you please clarify the following excerpt from ‘Graphing Significance Levels as Critical Regions’:
“The percentage of the area under the curve that is shaded equals the probability that the sample value will fall in those regions if the null hypothesis is correct.”
I’m not sure if I understood this correctly. If the sample value fall in one of the shaded regions, doesn’t mean that the null hypothesis can be rejected, hence that is not correct?
August 7, 2020 at 10:23 pm
Think of it this way. There are two basic reasons for why a sample value could fall in a critical region:
You don’t know which one is true. Remember, just because you reject the null hypothesis it doesn’t mean the null is false. However, by using hypothesis tests to determine statistical significance, you control the chances of #1 occurring. The rate at which #1 occurs equals your significance level. On the hand, you don’t know the probability of the sample value falling in a critical region if the alternative hypothesis is correct (#2). It depends on the precise distribution for the alternative hypothesis and you usually don’t know that, which is why you’re testing the hypotheses in the first place!
I hope I answered the question you were asking. If not, feel free to ask follow up questions. Also, this ties into how to interpret p-values . It’s not exactly straightforward. Click the link to learn more.
June 4, 2020 at 6:17 am
Hi Jim, thank you very much for your answer. You helped me a lot!
June 3, 2020 at 5:23 pm
Hi, Thanks for this post. I’ve been learning a lot with you. My question is regarding to lack of fit. The p-value of my lack of fit is really low, making my lack of fit significant, meaning my model does not fit well. Is my case a “false negative”? given that my pure error is really low, making the computation of the lack of fit low. So it means my model is good. Below I show some information, that I hope helps to clarify my question.
SumSq DF MeanSq F pValue ________ __ ________ ______ __________
Total 1246.5 18 69.25 Model 1241.7 6 206.94 514.43 9.3841e-14 . Linear 1196.6 3 398.87 991.53 1.2318e-14 . Nonlinear 45.046 3 15.015 37.326 2.3092e-06 Residual 4.8274 12 0.40228 . Lack of fit 4.7388 7 0.67698 38.238 0.0004787 . Pure error 0.088521 5 0.017704
June 3, 2020 at 7:53 pm
As you say, a low p-value for a lack of fit test indicates that the model doesn’t fit your data adequately. This is a positive result for the test, which means it can’t be a “false negative.” At best, it could be a false positive, meaning that your data actually fit model well despite the low p-value.
I’d recommend graphing the residuals and looking for patterns . There is probably a relationship between variables that you’re not modeling correctly, such as curvature or interaction effects. There’s no way to diagnose the specific nature of the lack-of-fit problem by using the statistical output. You’ll need the graphs.
If there are no patterns in the residual plots, then your lack-of-fit results might be a false positive.
I hope this helps!
May 30, 2020 at 6:23 am
First of all, I have to say there are not many resources that explain a complicated topic in an easier manner.
My question is, how do we arrive at “if p value is less than alpha, we reject the null hypothesis.”
Is this covered in a separate article I could read?
Thanks Shekhar
May 25, 2020 at 12:21 pm
Hi Jim, terrific website, blog, and after this I’m ordering your book. One of my biggest challenges is nomenclature, definitions, context, and formulating the hypotheses. Here’s one I want to double-be-sure I understand: From above you write: ” These tools allow us to test these two hypotheses:
Null hypothesis: The population mean equals the null hypothesis mean (260). Alternative hypothesis: The population mean does not equal the null hypothesis mean (260). ” I keep thinking that 260 is the population mean mu, the underlying population (that we never really know exactly) and that the Null Hypothesis is comparing mu to x-bar (the sample mean of the 25 families randomly sampled w mean = sample mean = x-bar = 330.6).
So is the following incorrect, and if so, why? Null hypothesis: The population mean mu=260 equals the null hypothesis mean x-bar (330.6). Alternative hypothesis: The population mean mu=269 does not equal the null hypothesis mean x-bar (330.6).
And my thinking is that usually the formulation of null and alternative hypotheses is “test value” = “mu current of underlying population”, whereas I read the formulation on the webpage above to be the reverse.
Any comments appreciated. Many Thanks,
May 26, 2020 at 8:56 pm
The null hypothesis states that population value equals the null value. Now, I know that’s not particularly helpful! But, the null value varies based on test and context. So, in this example, we’re setting the null value aa $260, which was the mean from the previous year. So, our null hypothesis states:
Null: the population mean (mu) = 260. Alternative: the population mean ≠ 260.
These hypothesis statements are about the population parameter. For this type of one-sample analysis, the target or reference value you specify is the null hypothesis value. Additionally, you don’t include the sample estimate in these statements, which is the X-bar portion you tacked on at the end. It’s strictly about the value of the population parameter you’re testing. You don’t know the value of the underlying distribution. However, given the mutually exclusive nature of the null and alternative hypothesis, you know one or the other is correct. The null states that mu equals 260 while the alternative states that it doesn’t equal 260. The data help you decide, which brings us to . . .
However, the procedure does compare our sample data to the null hypothesis value, which is how it determines how strong our evidence is against the null hypothesis.
I hope I answered your question. If not, please let me know!
May 8, 2020 at 6:00 pm
Really using the interpretation “In other words, you will observe sample effects at least as large as 70.6 about 3.1% of the time if the null is true”, our head seems to tie a knot. However, doing the reverse interpretation, it is much more intuitive and easier. That is, we will observe the sample effect of at least 70.6 in about 96.9% of the time, if the null is false (that is, our hypothesis is true).
May 8, 2020 at 7:25 pm
Your phrasing really isn’t any simpler. And it has the additional misfortune of being incorrect.
What you’re essentially doing is creating a one-sided confidence interval by using the p-value from a two-sided test. That’s incorrect in two ways.
So, what you need is a two-sided 95% CI (1-alpha). You could then state the results are statistically significant and you have 95% confidence that the population effect is between X and Y. If you want a lower bound as you propose, then you’ll need to use a one-sided hypothesis test with a 95% Lower Bound. That’ll give you a different value for the lower bound than the one you use.
I like confidence intervals. As I write elsewhere, I think they’re easier to understand and provide more information than a binary test result. But, you need to use them correctly!
One other point. When you are talking about p-values, it’s always under the assumption that the null hypothesis is correct. You *never* state anything about the p-value in relation to the null being false (i.e. alternative is true). But, if you want to use the type of phrasing you suggest, use it in the context of CIs and incorporate the points I cover above.
February 10, 2020 at 11:13 am
Muchas gracias profesor por compartir sus conocimientos. Un saliud especial desde Colombia.
August 6, 2019 at 11:46 pm
i found this really helpful . also can you help me out ?
I’m a little confused Can you tell me if level of significance and pvalue are comparable or not and if they are what does it mean if pvalue < LS . Do we reject the null hypothesis or do we accept the null hypothesis ?
August 7, 2019 at 12:49 am
Hi Divyanshu,
Yes, you compare the p-value to the significance level. When the p-value is less than the significance level (alpha), your results are statistically significant and you reject the null hypothesis.
I’d suggest re-reading the “Using P values and Significance Levels Together” section near the end of this post more closely. That describes the process. The next section describes what it all means.
July 1, 2019 at 4:19 am
sure.. I will use only in my class rooms that too offline with due credits to your orginal page. I will encourage my students to visit your blog . I have purchased your eBook on Regressions….immensely useful.
July 1, 2019 at 9:52 am
Hi Narasimha, that sounds perfect. Thanks for buying my ebook as well. I’m thrilled to hear that you’ve found it to be helpful!
June 28, 2019 at 6:22 am
I have benefited a lot by your writings….Can I share the same with my students in the classroom?
June 30, 2019 at 8:44 pm
Hi Narasimha,
Yes, you can certainly share with your students. Please attribute my original page. And please don’t copy whole sections of my posts onto another webpage as that can be bad with Google! Thanks!
February 11, 2019 at 7:46 pm
Hello, great site and my apologies if the answer to the following question exists already.
I’ve always wondered why we put the sampling distribution about the null hypothesis rather than simply leave it about the observed mean. I can see mathematically we are measuring the same distance from the null and basically can draw the same conclusions.
For example we take a sample (say 50 people) we gather an observation (mean wage) estimate the standard error in that observation and so can build a sampling distribution about the observed mean. That sampling distribution contains a confidence interval, where say, i am 95% confident the true mean lies (i.e. in repeated sampling the true mean would reside within this interval 95% of the time).
When i use this for a hyp-test, am i right in saying that we place the sampling dist over the reference level simply because it’s mathematically equivalent and it just seems easier to gauge how far the observation is from 0 via t-stats or its likelihood via p-values?
It seems more natural to me to look at it the other way around. leave the sampling distribution on the observed value, and then look where the null sits…if it’s too far left or right then it is unlikely the true population parameter is what we believed it to be, because if the null were true it would only occur ~ 5% of the time in repeated samples…so perhaps we need to change our opinion.
Can i interpret a hyp-test that way? Or do i have a misconception?
February 12, 2019 at 8:25 pm
The short answer is that, yes, you can draw the interval around the sample mean instead. And, that is, in fact, how you construct confidence intervals. The distance around the null hypothesis for hypothesis tests and the distance around the sample for confidence intervals are the same distance, which is why the results will always agree as long as you use corresponding alpha levels and confidence levels (e.g., alpha 0.05 with a 95% confidence level). I write about how this works in a post about confidence intervals .
I prefer confidence intervals for a number of reasons. They’ll indicate whether you have significant results if they exclude the null value and they indicate the precision of the effect size estimate. Corresponding with what you’re saying, it’s easier to gauge how far a confidence interval is from the null value (often zero) whereas a p-value doesn’t provide that information. See Practical versus Statistical Significance .
So, you don’t have any misconception at all! Just refer to it as a confidence interval rather than a hypothesis test, but, of course, they are very closely related.
January 9, 2019 at 10:37 pm
Hi Jim, Nice Article.. I have a question… I read the Central limit theorem article before this article…
Coming to this article, During almost every hypothesis test, we draw a normal distribution curve assuming there is a sampling distribution (and then we go for test statistic, p value etc…). Do we draw a normal distribution curve for hypo tests because of the central limit theorem…
Thanks in advance, Surya
January 10, 2019 at 1:57 am
These distributions are actually the t-distribution which are different from the normal distribution. T-distributions only have one parameter–the degrees of freedom. As the DF of increases, the t-distribution tightens up. Around 25 degrees of freedom, the t-distribution approximates the normal distribution. Depending on the type of t-test, this corresponds to a sample size of 26 or 27. Similarly, the sampling distribution of the means also approximate the normal distribution at around these sample sizes. With a large enough sample size, both the t-distribution and the sample distribution converge to a normal distribution regardless (largely) of the underlying population distribution. So, yes, the central limit theorem plays a strong role in this.
It’s more accurate to say that central limit theorem causes the sampling distribution of the means to converge on the same distribution that the t-test uses, which allows you to assume that the test produces valid results. But, technically, the t-test is based on the t-distribution.
Problems can occur if the underlying distribution is non-normal and you have a small sample size. In that case, the sampling distribution of the means won’t approximate the t-distribution that the t-test uses. However, the test results will assume that it does and produce results based on that–which is why it causes problems!
November 19, 2018 at 9:15 am
Dear Jim! Thank you very much for your explanation. I need your help to understand my data. I have two samples (about 300 observations) with biased distributions. I did the ttest and obtained the p-value, which is quite small. Can I draw the conclusion that the effect size is small even when the distribution of my data is not normal? Thank you
November 19, 2018 at 9:34 am
Hi Tetyana,
First, when you say that your p-value is small and that you want to “draw the conclusion that the effect size is small,” I assume that you mean statistically significant. When the p-value is low, the null hypothesis must go! In other words, you reject the null and conclude that there is a statistically significant effect–not a small effect.
Now, back to the question at hand! Yes, When you have a sufficiently large sample-size, t-tests are robust to departures from normality. For a 2-sample t-test, you should have at least 15 samples per group, which you exceed by quite a bit. So, yes, you can reliably conclude that your results are statistically significant!
You can thank the central limit theorem! 🙂
September 10, 2018 at 12:18 am
Hello Jim, I am very sorry; I have very elementary of knowledge of stats. So, would you please explain how you got a p- value of 0.03112 in the above calculation/t-test? By looking at a chart? Would you also explain how you got the information that “you will observe sample effects at least as large as 70.6 about 3.1% of the time if the null is true”?
July 6, 2018 at 7:02 am
A quick question regarding your use of two-tailed critical regions in the article above: why? I mean, what is a real-world scenario that would warrant a two-tailed test of any kind (z, t, etc.)? And if there are none, why keep using the two-tailed scenario as an example, instead of the one-tailed which is both more intuitive and applicable to most if not all practical situations. Just curious, as one person attempting to educate people on stats to another (my take on the one vs. two-tailed tests can be seen here: http://blog.analytics-toolkit.com/2017/one-tailed-two-tailed-tests-significance-ab-testing/ )
Thanks, Georgi
July 6, 2018 at 12:05 pm
There’s the appropriate time and place for both one-tailed and two-tailed tests. I plan to write a post on this issue specifically, so I’ll keep my comments here brief.
So much of statistics is context sensitive. People often want concrete rules for how to do things in statistics but that’s often hard to provide because the answer depends on the context, goals, etc. The question of whether to use a one-tailed or two-tailed test falls firmly in this category of it depends.
I did read the article you wrote. I’ll say that I can see how in the context of A/B testing specifically there might be a propensity to use one-tailed tests. You only care about improvements. There’s probably not too much downside in only caring about one direction. In fact, in a post where I compare different tests and different options , I suggest using a one-tailed test for a similar type of casing involving defects. So, I’m onboard with the idea of using one-tailed tests when they’re appropriate. However, I do think that two-tailed tests should be considered the default choice and that you need good reasons to move to a one-tailed test. Again, your A/B testing area might supply those reasons on a regular basis, but I can’t make that a blanket statement for all research areas.
I think your article mischaracterizes some of the pros and cons of both types of tests. Just a couple of for instances. In a two-tailed test, you don’t have to take the same action regardless of which direction the results are significant (example below). And, yes, you can determine the direction of the effect in a two-tailed test. You simply look at the estimated effect. Is it positive or negative?
On the other hand, I do agree that one-tailed tests don’t increase the overall Type I error. However, there is a big caveat for that. In a two-tailed test, the Type I error rate is evenly split in both tails. For a one-tailed test, the overall Type I error rate does not change, but the Type I errors are redistributed so they all occur in the direction that you are interested in rather than being split between the positive and negative directions. In other words, you’ll have twice as many Type I errors in the specific direction that you’re interested in. That’s not good.
My big concerns with one-tailed tests are that it makes it easier to obtain the results that you want to obtain. And, all of the Type I errors (false positives) are in that direction too. It’s just not a good combination.
To answer your question about when you might want to use two-tailed tests, there are plenty of reasons. For one, you might want to avoid the situation I describe above. Additionally, in a lot of scientific research, the researchers truly are interested in detecting effects in either direction for the sake of science. Even in cases with a practical application, you might want to learn about effects in either direction.
For example, I was involved in a research study that looked at the effects of an exercise intervention on bone density. The idea was that it might be a good way to prevent osteoporosis. I used a two-tailed test. Obviously, we’re hoping that there was positive effect. However, we’d be very interested in knowing whether there was a negative effect too. And, this illustrates how you can have different actions based on both directions. If there was a positive effect, you can recommend that as a good approach and try to promote its use. If there’s a negative effect, you’d issue a warning to not do that intervention. You have the potential for learning both what is good and what is bad. The extra false-positives would’ve cause problems because we’d think that there’d be health benefits for participants when those benefits don’t actually exist. Also, if we had performed only a one-tailed test and didn’t obtain significant results, we’d learn that it wasn’t a positive effect, but we would not know whether it was actually detrimental or not.
Here’s when I’d say it’s OK to use a one-tailed test. Consider a one-tailed test when you’re in situation where you truly only need to know whether an effect exists in one direction, and the extra Type I errors in that direction are an acceptable risk (false positives don’t cause problems), and there’s no benefit in determining whether an effect exists in the other direction. Those conditions really restrict when one-tailed tests are the best choice. Again, those restrictions might not be relevant for your specific field, but as for the usage of statistics as a whole, they’re absolutely crucial to consider.
On the other hand, according to this article, two-tailed tests might be important in A/B testing !
March 30, 2018 at 5:29 am
Dear Sir, please confirm if there is an inadvertent mistake in interpretation as, “We can conclude that mean fuel expenditures have increased since last year.” Our null hypothesis is =260. If found significant, it implies two possibilities – both increase and decrease. Please let us know if we are mistaken here. Many Thanks!
March 30, 2018 at 9:59 am
Hi Khalid, the null hypothesis as it is defined for this test represents the mean monthly expenditure for the previous year (260). The mean expenditure for the current year is 330.6 whereas it was 260 for the previous year. Consequently, the mean has increased from 260 to 330.7 over the course of a year. The p-value indicates that this increase is statistically significant. This finding does not suggest both an increase and a decrease–just an increase. Keep in mind that a significant result prompts us to reject the null hypothesis. So, we reject the null that the mean equals 260.
Let’s explore the other possible findings to be sure that this makes sense. Suppose the sample mean had been closer to 260 and the p-value was greater than the significance level, those results would indicate that the results were not statistically significant. The conclusion that we’d draw is that we have insufficient evidence to conclude that mean fuel expenditures have changed since the previous year.
If the sample mean was less than the null hypothesis (260) and if the p-value is statistically significant, we’d concluded that mean fuel expenditures have decreased and that this decrease is statistically significant.
When you interpret the results, you have to be sure to understand what the null hypothesis represents. In this case, it represents the mean monthly expenditure for the previous year and we’re comparing this year’s mean to it–hence our sample suggests an increase.
Saul Mcleod, PhD
Editor-in-Chief for Simply Psychology
BSc (Hons) Psychology, MRes, PhD, University of Manchester
Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.
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On This Page:
The p-value in statistics quantifies the evidence against a null hypothesis. A low p-value suggests data is inconsistent with the null, potentially favoring an alternative hypothesis. Common significance thresholds are 0.05 or 0.01.
When you perform a statistical test, a p-value helps you determine the significance of your results in relation to the null hypothesis.
The null hypothesis (H0) states no relationship exists between the two variables being studied (one variable does not affect the other). It states the results are due to chance and are not significant in supporting the idea being investigated. Thus, the null hypothesis assumes that whatever you try to prove did not happen.
The alternative hypothesis (Ha or H1) is the one you would believe if the null hypothesis is concluded to be untrue.
The alternative hypothesis states that the independent variable affected the dependent variable, and the results are significant in supporting the theory being investigated (i.e., the results are not due to random chance).
A p-value, or probability value, is a number describing how likely it is that your data would have occurred by random chance (i.e., that the null hypothesis is true).
The level of statistical significance is often expressed as a p-value between 0 and 1.
The smaller the p -value, the less likely the results occurred by random chance, and the stronger the evidence that you should reject the null hypothesis.
Remember, a p-value doesn’t tell you if the null hypothesis is true or false. It just tells you how likely you’d see the data you observed (or more extreme data) if the null hypothesis was true. It’s a piece of evidence, not a definitive proof.
Suppose you’re conducting a study to determine whether a new drug has an effect on pain relief compared to a placebo. If the new drug has no impact, your test statistic will be close to the one predicted by the null hypothesis (no difference between the drug and placebo groups), and the resulting p-value will be close to 1. It may not be precisely 1 because real-world variations may exist. Conversely, if the new drug indeed reduces pain significantly, your test statistic will diverge further from what’s expected under the null hypothesis, and the p-value will decrease. The p-value will never reach zero because there’s always a slim possibility, though highly improbable, that the observed results occurred by random chance.
The significance level (alpha) is a set probability threshold (often 0.05), while the p-value is the probability you calculate based on your study or analysis.
A p-value less than or equal to a predetermined significance level (often 0.05 or 0.01) indicates a statistically significant result, meaning the observed data provide strong evidence against the null hypothesis.
This suggests the effect under study likely represents a real relationship rather than just random chance.
For instance, if you set α = 0.05, you would reject the null hypothesis if your p -value ≤ 0.05.
It indicates strong evidence against the null hypothesis, as there is less than a 5% probability the null is correct (and the results are random).
Therefore, we reject the null hypothesis and accept the alternative hypothesis.
Upon analyzing the pain relief effects of the new drug compared to the placebo, the computed p-value is less than 0.01, which falls well below the predetermined alpha value of 0.05. Consequently, you conclude that there is a statistically significant difference in pain relief between the new drug and the placebo.
A p-value of 0.001 is highly statistically significant beyond the commonly used 0.05 threshold. It indicates strong evidence of a real effect or difference, rather than just random variation.
Specifically, a p-value of 0.001 means there is only a 0.1% chance of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is correct.
Such a small p-value provides strong evidence against the null hypothesis, leading to rejecting the null in favor of the alternative hypothesis.
This means we retain the null hypothesis and reject the alternative hypothesis. You should note that you cannot accept the null hypothesis; we can only reject it or fail to reject it.
Note : when the p-value is above your threshold of significance, it does not mean that there is a 95% probability that the alternative hypothesis is true.
Most statistical software packages like R, SPSS, and others automatically calculate your p-value. This is the easiest and most common way.
Online resources and tables are available to estimate the p-value based on your test statistic and degrees of freedom.
These tables help you understand how often you would expect to see your test statistic under the null hypothesis.
Understanding the Statistical Test:
Different statistical tests are designed to answer specific research questions or hypotheses. Each test has its own underlying assumptions and characteristics.
For example, you might use a t-test to compare means, a chi-squared test for categorical data, or a correlation test to measure the strength of a relationship between variables.
Be aware that the number of independent variables you include in your analysis can influence the magnitude of the test statistic needed to produce the same p-value.
This factor is particularly important to consider when comparing results across different analyses.
If you’re comparing the effectiveness of just two different drugs in pain relief, a two-sample t-test is a suitable choice for comparing these two groups. However, when you’re examining the impact of three or more drugs, it’s more appropriate to employ an Analysis of Variance ( ANOVA) . Utilizing multiple pairwise comparisons in such cases can lead to artificially low p-values and an overestimation of the significance of differences between the drug groups.
A statistically significant result cannot prove that a research hypothesis is correct (which implies 100% certainty).
Instead, we may state our results “provide support for” or “give evidence for” our research hypothesis (as there is still a slight probability that the results occurred by chance and the null hypothesis was correct – e.g., less than 5%).
In our comparison of the pain relief effects of the new drug and the placebo, we observed that participants in the drug group experienced a significant reduction in pain ( M = 3.5; SD = 0.8) compared to those in the placebo group ( M = 5.2; SD = 0.7), resulting in an average difference of 1.7 points on the pain scale (t(98) = -9.36; p < 0.001).
The 6th edition of the APA style manual (American Psychological Association, 2010) states the following on the topic of reporting p-values:
“When reporting p values, report exact p values (e.g., p = .031) to two or three decimal places. However, report p values less than .001 as p < .001.
The tradition of reporting p values in the form p < .10, p < .05, p < .01, and so forth, was appropriate in a time when only limited tables of critical values were available.” (p. 114)
A lower p-value is sometimes interpreted as meaning there is a stronger relationship between two variables.
However, statistical significance means that it is unlikely that the null hypothesis is true (less than 5%).
To understand the strength of the difference between the two groups (control vs. experimental) a researcher needs to calculate the effect size .
In statistical hypothesis testing, you reject the null hypothesis when the p-value is less than or equal to the significance level (α) you set before conducting your test. The significance level is the probability of rejecting the null hypothesis when it is true. Commonly used significance levels are 0.01, 0.05, and 0.10.
Remember, rejecting the null hypothesis doesn’t prove the alternative hypothesis; it just suggests that the alternative hypothesis may be plausible given the observed data.
The p -value is conditional upon the null hypothesis being true but is unrelated to the truth or falsity of the alternative hypothesis.
If your p-value is less than or equal to 0.05 (the significance level), you would conclude that your result is statistically significant. This means the evidence is strong enough to reject the null hypothesis in favor of the alternative hypothesis.
No, not all p-values below 0.05 are considered statistically significant. The threshold of 0.05 is commonly used, but it’s just a convention. Statistical significance depends on factors like the study design, sample size, and the magnitude of the observed effect.
A p-value below 0.05 means there is evidence against the null hypothesis, suggesting a real effect. However, it’s essential to consider the context and other factors when interpreting results.
Researchers also look at effect size and confidence intervals to determine the practical significance and reliability of findings.
Sample size can impact the interpretation of p-values. A larger sample size provides more reliable and precise estimates of the population, leading to narrower confidence intervals.
With a larger sample, even small differences between groups or effects can become statistically significant, yielding lower p-values. In contrast, smaller sample sizes may not have enough statistical power to detect smaller effects, resulting in higher p-values.
Therefore, a larger sample size increases the chances of finding statistically significant results when there is a genuine effect, making the findings more trustworthy and robust.
No, a non-significant p-value does not necessarily indicate that there is no effect or difference in the data. It means that the observed data do not provide strong enough evidence to reject the null hypothesis.
There could still be a real effect or difference, but it might be smaller or more variable than the study was able to detect.
Other factors like sample size, study design, and measurement precision can influence the p-value. It’s important to consider the entire body of evidence and not rely solely on p-values when interpreting research findings.
While a p-value can be extremely small, it cannot technically be absolute zero. When a p-value is reported as p = 0.000, the actual p-value is too small for the software to display. This is often interpreted as strong evidence against the null hypothesis. For p values less than 0.001, report as p < .001
Bland, J. M., & Altman, D. G. (1994). One and two sided tests of significance: Authors’ reply. BMJ: British Medical Journal , 309 (6958), 874.
Goodman, S. N., & Royall, R. (1988). Evidence and scientific research. American Journal of Public Health , 78 (12), 1568-1574.
Goodman, S. (2008, July). A dirty dozen: twelve p-value misconceptions . In Seminars in hematology (Vol. 45, No. 3, pp. 135-140). WB Saunders.
Lang, J. M., Rothman, K. J., & Cann, C. I. (1998). That confounded P-value. Epidemiology (Cambridge, Mass.) , 9 (1), 7-8.
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10.15.2021 • 9 min read
Subject Matter Expert
In this article, we'll take a deep dive on p-values, beginning with a description and definition of this key component of statistical hypothesis testing, before moving on to look at how to calculate it for different types of variables.
What is a p-value, calculating p-values for discrete random variables, calculating p-values for continuous random variables.
A p-value (short for probability value) is a probability used in hypothesis testing. It represents the probability of observing sample data that is at least as extreme as the observed sample data, assuming that the null hypothesis is true .
In a hypothesis test, you have two competing hypotheses: a null (or starting) hypothesis, H 0 H_0 H 0 and an alternative hypothesis, H a H_a H a . The goal of a hypothesis test is to use statistical evidence from a sample or multiple samples to determine which of the hypotheses is more likely to be true. The p-value can be used in the final stage of the test to make this determination.
Interpreting a p-value
Because it is a probability, the p-value can be expressed as a decimal or a percentage ranging from 0 to 1 or 0% to 100%. The closer the p-value is to zero, the stronger the evidence is in support of the alternative hypothesis, H a H_a H a .
Reject or Fail to Reject the Null Hypothesis?
When the p-value is below a certain threshold, the null hypothesis is rejected in favor of the alternative hypothesis. This threshold is known as the significance level (or alpha level) of the test.
The most commonly used significance level is 0.05 or 5%, but the choice of the significance level is up to the researcher. You could just as easily use a significance level of 0.1 or 0.01, for example. Remember, however, that the lower the p-value, the stronger the evidence is in support of the alternative hypothesis. For this reason, choosing a lower significance level means that you can have more confidence in your decision to reject a null hypothesis.
When the p-value is greater than the significance level, the evidence favors the null hypothesis, and the researcher or statistician must fail to reject the null hypothesis.
As mentioned earlier, the p-value is the probability of observing sample data that’s at least as extreme as the observed sample data, assuming that the null hypothesis is true.
If your data consists of a discrete random variable, you can map out the entire set of possible outcomes and their respective probabilities in order to calculate the p-value.
The p-value will then be the sum of three things:
the probability of the observed outcome
the probability of all outcomes that are just as likely as the observed outcome
and the probability of any outcome that is less likely than the observed outcome
Here is an example.
A stranger invites you to play a game of dice, and claims her dice are fair. The rules of the game are as follows: You roll a single die. If you roll an even number, you count that as a win (or success) and earn $1. If you roll an odd number, you count that as a loss (or failure) and lose $0.80. You can play the game for as many rounds as you like.
Let’s say you play four rounds of the game, and you lose all four rounds. This leaves you $3.20 poorer than before you started playing.
Given your losses, you may be interested in conducting a hypothesis test. The null hypothesis will be that the dice used in the game are indeed fair and that there is an equal chance of rolling an even or odd number with each roll. Your alternative hypothesis is that the dice are weighted towards landing on odd numbers.
To calculate the p-value, we map all of the possible outcomes of playing four rounds of the game. In each round, there are only two possible outcomes (odd or even), and after four rounds, there are a total of 2 4 2^4 2 4 , or 16, outcomes. If we assume the null hypothesis is true—that the dice are fair)—each of these outcomes is equally likely, with a probability of 1/16.
Since we are only concerned about the total number of wins and losses, and not concerned at all with their order, the outcomes and probabilities we care about are the following:
the probability of getting 4 wins and 0 losses = 1/16
the probability of getting 3 wins and 1 loss = 4/16
the probability of getting 2 wins and 2 losses = 6/16
the probability of getting 1 win and 3 losses = 4/16
the probability of getting 0 wins and 4 losses = 1/16
To calculate the p-value, we sum up the following:
the probability of the observed outcome (0 wins and 4 losses)
the probability of any outcome that is just as likely as the observed outcome (4 wins and 0 losses)
the probability of any outcome that is less likely than the observed outcome (in this example, there are no outcomes that are less likely than the observed outcome, so this value is zero)
p-Value = 1/16 + 1/16 = 1/8 or 0.125
The p-value we found is 0.125. Surprisingly, this is still well above a 0.05 significance level. It is even above a 0.10 (or 10%) significance level. Regardless of which of these thresholds you choose, you must fail to reject the null hypothesis. In other words, despite four losses in a row, the evidence still favors the hypothesis that the dice are fair! It may be a different story if you experience 10 or even 5 losses in a row. Calculate the p-value to find out!
When the hypothesis test involves a continuous random variable, we use a test statistic and the area under the probability density function to determine the p-value. The intuition behind the p-value is the same as in the discrete case. Assuming that the null hypothesis is true, we are calculating the probability of observing sample data that is at least as extreme as the sample data we have observed.
Let’s take a look at another example.
Say you have an orange grove, and you’re convinced that your oranges now grow larger than when you first started growing citrus. You happen to know that the standard deviation of the weights of your oranges, σ \sigma σ , is equal to 0.8 oz. This is the perfect opportunity to conduct a hypothesis test.
Your null hypothesis, in this case, is that the mean weight of your oranges has remained unchanged over the years and is equal to 5 oz (the null hypothesis typically represents the hypothesis that you are trying to move away from). Your alternative hypothesis is that the average weight of your oranges is now greater than 5 oz.
Because you can’t weigh every orange in your grove, you pick a large random sample of oranges (with a sample size of 100), weigh those, and observe that the average weight in your sample, x ‾ \overline x x , is equal to 5.2 oz.
Does this result support the null hypothesis or the alternative hypothesis? It’s not immediately clear. By pure chance, you could have had a handful of extra-large oranges in your sample, and this could have pushed your sample mean above a population mean of 5 oz. Alternatively, the sample mean could indicate that the population mean is, in fact, greater than 5 oz.
Here is where we begin the hypothesis test. We’ll conduct the test at a 0.05 significance level.
We start by asking the following question: Assuming that the null hypothesis is true, how likely or unlikely is it to observe a sample mean x ‾ \overline x x = 5.2 oz?
From the central limit theorem, we know that if our sample is randomly drawn and large enough, we can assume that the sampling distribution of the sample means is normally distributed with a mean equal to the true population mean, μ \mu μ , and a standard error equal to σ n \frac\sigma{\sqrt n} n σ . This means that if the null hypothesis is true, the sampling distribution for the sample mean of our orange weights will be normally distributed, with a mean equal to 5 and a standard error equal to 0.08.
From here, we can convert our sample mean of 5.2 into what is known as a test statistic. To do this we use the exact same process we use when calculating standardized units such as z-scores or t-scores. Since we know the sampling distribution is approximately normal, and since we know the population standard deviation σ \sigma σ and the standard error σ n \frac\sigma{\sqrt n} n σ of the sampling distribution, we can calculate a Z-test statistic in the same way that we would calculate a z-score (if we did not know σ \sigma σ , we would use the sample standard deviation, s, to calculate a t-test statistic in the same way that we calculate t-scores).
The test statistic is telling us that if our null hypothesis is true, then our observed sample mean, x ‾ \overline x x , is 2.5 standard deviations above the mean of the sampling distribution. To put the p-value to work we can do one of two things.
1. We can calculate the p-value associated with the test statistic. This can be done by finding the area under the standard normal distribution that lies to the right of 2.5. This gives us a p-value of 0.0062. The p-value is telling us that if the null hypothesis is true, we would only observe a sample mean of 5.2 or greater 0.0062 (or 0.62%) of the time. Because this probability is so low, it’s likely that the null hypothesis is false.
Since the p-value of 0.0062 is less than the significance level of 0.05, we can reject the null hypothesis at the 0.05 significance level. We can even reject it at the 0.01 significance level! You’re likely to be right about your oranges: the average weights have likely increased over time.
2. If you are familiar with standard normal distributions you may have realized that the significance level of our test (alpha = 0.05) is associated with the 95th percentile of the standard normal distribution. You may also know that the 95th percentile of a standard normal distribution is associated with a Z-score of 1.64. Since the test statistic 2.5 lies to the right of the Z-score, we can assume that the p-value will be less than 0.05. This is another way to complete the hypothesis test without having to do additional calculations.
Two-sided, upper-tailed, and lower-tailed hypothesis tests
In the orange grove example above, we conducted an upper-tailed hypothesis test, because the alternative hypothesis H a H_a H a was of the form μ > μ 0 \mu>\mu_0 μ > μ 0 . It’s important to know, however, how the calculation of p-values differs when you have a two-tailed or a lower-tailed hypothesis test.
For a two-tailed test (when the alternative hypothesis, H a H_a H a , stipulates that a population parameter is ≠ to some number), the p-value is equal to twice the probability associated with the test statistic. If we had conducted a two-tailed test in the orange grove example ( H a H_a H a : μ ≠ 5 \mu\neq5 μ = 5 ), the p-value would be equal to the probability that x ‾ \overline x x was greater than 2.5 plus the probability that x ‾ \overline x x is less than -2.5. Because the standard normal is symmetric about the mean, this is equal to (0.0062 * 2 = 0.0124).
For a lower-tailed test (when the alternative hypothesis, H a H_a H a , stipulates that a population parameter is ≤ to some number) the process is similar to the upper-tailed test, but the p-value will be the probability of getting a sample statistic that lies to the left of the test-statistic, rather than to the right of it.
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Revision note.
What is a hypothesis test.
A hypothesis test is carried out at the 5% level of significance to test if a normal coin is fair or not.
How do we decide whether to reject or accept the null hypothesis.
For the following situations, state at the 1% and 5% significance levels whether the null hypothesis should be rejected or not.
How is a hypothesis test carried out.
A teacher carried out a hypothesis test at the 10% significance level to test if her students perform better in exams after using a new revision technique. The p – value for her test statistic is 0.09142. Write a conclusion for her hypothesis test.
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Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.
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To find the p value for your sample, do the following: Identify the correct test statistic. Calculate the test statistic using the relevant properties of your sample. Specify the characteristics of the test statistic's sampling distribution. Place your test statistic in the sampling distribution to find the p value.
The P -value is, therefore, the area under a tn - 1 = t14 curve to the left of -2.5 and to the right of 2.5. It can be shown using statistical software that the P -value is 0.0127 + 0.0127, or 0.0254. The graph depicts this visually. Note that the P -value for a two-tailed test is always two times the P -value for either of the one-tailed tests.
Formally, the p-value is the probability that the test statistic will produce values at least as extreme as the value it produced for your sample.It is crucial to remember that this probability is calculated under the assumption that the null hypothesis H 0 is true!. More intuitively, p-value answers the question: Assuming that I live in a world where the null hypothesis holds, how probable is ...
The p value is a number, calculated from a statistical test, that describes how likely you are to have found a particular set of observations if the null hypothesis were true. P values are used in hypothesis testing to help decide whether to reject the null hypothesis. The smaller the p value, the more likely you are to reject the null ...
A p value is used in hypothesis testing to help you support or reject the null hypothesis. The p value is the evidence against a null hypothesis. The smaller the p-value, the stronger the evidence that you should reject the null hypothesis. P values are expressed as decimals although it may be easier to understand what they are if you convert ...
A P-value calculator is used to determine the statistical significance of an observed result in hypothesis testing. It takes as input the observed test statistic, the null hypothesis, and the relevant parameters of the statistical test (such as degrees of freedom), and computes the p-value. The p-value represents the probability of obtaining ...
Then, if the null hypothesis is wrong, then the data will tend to group at a point that is not the value in the null hypothesis (1.2), and then our p-value will wind up being very small. If the null hypothesis is correct, or close to being correct, then the p-value will be larger, because the data values will group around the value we hypothesized.
The p-value is a crucial concept in statistical hypothesis testing, providing a quantitative measure of the strength of evidence against the null hypothesis. It guides decision-making by comparing the p-value to a chosen significance level, typically 0.05.
Here is the technical definition of P values: P values are the probability of observing a sample statistic that is at least as extreme as your sample statistic when you assume that the null hypothesis is true. Let's go back to our hypothetical medication study. Suppose the hypothesis test generates a P value of 0.03.
P-Value: The p-value is the level of marginal significance within a statistical hypothesis test representing the probability of the occurrence of a given event. The p-value is used as an ...
About. Transcript. We compare a P-value to a significance level to make a conclusion in a significance test. Given the null hypothesis is true, a p-value is the probability of getting a result as or more extreme than the sample result by random chance alone. If a p-value is lower than our significance level, we reject the null hypothesis.
The test statistic is, therefore: Z = p ^ − p 0 p 0 ( 1 − p 0) n = 0.853 − 0.90 0.90 ( 0.10) 150 = − 1.92. And, the rejection region is: Z P lesson 9.3 α = 0.05 -1.645 0 0.90. Since the test statistic Z = −1.92 < −1.645, we reject the null hypothesis. There is sufficient evidence at the α = 0.05 level to conclude that the rate has ...
The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. ... In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a ...
Given the importance of the p-value, it is essential to ensure its interpretation is correct. Here are five essential tips for ensuring the p-value from a hypothesis test is understood correctly. 1. Know What the P-value Represents. First, it is essential to understand what a p-value is. In hypothesis testing, the p-value is defined as the ...
The p-value is a probability computed assuming the null hypothesis is true, that the test statistic would take a value as extreme or more extreme than that actually observed. Since it's a probability, it is a number between 0 and 1. The closer the number is to 0 means the event is "unlikely." So if p-value is "small," (typically, less ...
The P-value is known as the level of marginal significance within the hypothesis testing that represents the probability of occurrence of the given event. The P-value is used as an alternative to the rejection point to provide the least significance at which the null hypothesis would be rejected. If the P-value is small, then there is stronger ...
Using P values and Significance Levels Together. If your P value is less than or equal to your alpha level, reject the null hypothesis. The P value results are consistent with our graphical representation. The P value of 0.03112 is significant at the alpha level of 0.05 but not 0.01.
A p-value, or probability value, is a number describing how likely it is that your data would have occurred by random chance (i.e., that the null hypothesis is true). The level of statistical significance is often expressed as a p-value between 0 and 1. The smaller the p -value, the less likely the results occurred by random chance, and the ...
Onward! We use p -values to make conclusions in significance testing. More specifically, we compare the p -value to a significance level α to make conclusions about our hypotheses. If the p -value is lower than the significance level we chose, then we reject the null hypothesis H 0 in favor of the alternative hypothesis H a .
The p-value can be used in the final stage of the test to make this determination. Interpreting a p-value. Because it is a probability, the p-value can be expressed as a decimal or a percentage ranging from 0 to 1 or 0% to 100%. The closer the p-value is to zero, the stronger the evidence is in support of the alternative hypothesis, H a H_a H a .
There are a number of ways that a hypothesis test can be carried out for different models, however the following steps should form the base for your test: Step 1. Define the test statistic and population parameter ; Step 2. Write the null and alternative hypotheses clearly; Step 3. Calculate the critical value(s) or the p - value for the test ...
Unit test. Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values to make conclusions about hypotheses.
To find the p-value, we can use a standard normal distribution table or a calculator to determine the cumulative probability to the left of -1.86. ... In hypothesis testing, the null hypothesis (H0) is the hypothesis being tested, while the alternative hypothesis (Ha) is the one the test attempts to support. The goal of hypothesis testing is to ...