## P-Value And Statistical Significance: What It Is & Why It Matters

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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.

## Hypothesis testing

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).

## What a p-value tells you

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.

## Example: Test Statistic and p-Value

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.

## P-value interpretation

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 your significance level (typically ≤ 0.05) is statistically significant.

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.

## Example: Statistical Significance

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.

## What does a p-value of 0.001 mean?

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.

## A p-value more than the significance level (typically p > 0.05) is not statistically significant and indicates strong evidence for the null 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.

## One-Tailed Test

## Two-Tailed Test

## How do you calculate the p-value ?

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.

## Example: Choosing a Statistical Test

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.

## How to report

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%).

## Example: Reporting the results

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)

- Do not use 0 before the decimal point for the statistical value p as it cannot equal 1. In other words, write p = .001 instead of p = 0.001.
- Please pay attention to issues of italics ( p is always italicized) and spacing (either side of the = sign).
- p = .000 (as outputted by some statistical packages such as SPSS) is impossible and should be written as p < .001.
- The opposite of significant is “nonsignificant,” not “insignificant.”

## Why is the p -value not enough?

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 .

## When do you reject the null hypothesis?

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.

## What does p-value of 0.05 mean?

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.

## Are all p-values below 0.05 considered statistically significant?

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.

## How does sample size affect the interpretation of p-values?

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.

## Can a non-significant p-value indicate that there is no effect or difference in the data?

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.

## Can P values be exactly zero?

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

## Further Information

- P-values and significance tests (Kahn Academy)
- Hypothesis testing and p-values (Kahn Academy)
- Wasserstein, R. L., Schirm, A. L., & Lazar, N. A. (2019). Moving to a world beyond “ p “< 0.05”.
- Criticism of using the “ p “< 0.05”.
- Publication manual of the American Psychological Association
- Statistics for Psychology Book Download

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.

## 9.1 Null and Alternative Hypotheses

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 , the — null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

## Example 9.1

H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.

## Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

- H 0 : μ __ 66
- H a : μ __ 66

## Example 9.3

We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

- H 0 : μ __ 45
- H a : μ __ 45

## Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

- H 0 : p __ 0.40
- H a : p __ 0.40

## Collaborative Exercise

Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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## Hypothesis Testing Calculator

Related: confidence interval calculator, type ii error.

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.

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.

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.

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.

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.

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## Keyboard Shortcuts

Hypothesis testing.

Key Topics:

- Basic approach
- Null and alternative hypothesis
- Decision making and the p -value
- Z-test & Nonparametric alternative

## Basic approach to hypothesis testing

- State a model describing the relationship between the explanatory variables and the outcome variable(s) in the population and the nature of the variability. State all of your assumptions .
- Specify the null and alternative hypotheses in terms of the parameters of the model.
- Invent a test statistic that will tend to be different under the null and alternative hypotheses.
- Using the assumptions of step 1, find the theoretical sampling distribution of the statistic under the null hypothesis of step 2. Ideally the form of the sampling distribution should be one of the “standard distributions”(e.g. normal, t , binomial..)
- Calculate a p -value , as the area under the sampling distribution more extreme than your statistic. Depends on the form of the alternative hypothesis.
- Choose your acceptable type 1 error rate (alpha) and apply the decision rule : reject the null hypothesis if the p-value is less than alpha, otherwise do not reject.
- \(\frac{\bar{X}-\mu_0}{\sigma / \sqrt{n}}\)
- general form is: (estimate - value we are testing)/(st.dev of the estimate)
- z-statistic follows N(0,1) distribution
- 2 × the area above |z|, area above z,or area below z, or
- compare the statistic to a critical value, |z| ≥ z α/2 , z ≥ z α , or z ≤ - z α
- Choose the acceptable level of Alpha = 0.05, we conclude …. ?

## Making the Decision

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:

- either our initial assumption is correct and we experienced an unusual event or,
- our initial assumption is incorrect

In statistics, if it is unlikely, we decide to “ reject ” our initial assumption.

## Example: Criminal Trial Analogy

First, state 2 hypotheses, the null hypothesis (“H 0 ”) and the alternative hypothesis (“H A ”)

- H 0 : Defendant is not guilty.
- H A : Defendant is guilty.

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.

- It can one-sided and two-sided.
- In two-sided we only care there is a difference, but not the direction of it. In one-sided we care about a particular direction of the relationship. We want to know if the value is strictly larger or smaller.

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.

- Defendant is innocent until proven guilty.

In statistics, we always assume the null hypothesis is true .

Then, make a decision based on the available evidence.

- If there is sufficient evidence (“beyond a reasonable doubt”), reject the null hypothesis . (Behave as if defendant is guilty.)
- If there is not enough evidence, do not reject the null hypothesis . (Behave as if defendant is not guilty.)

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).

## Using the p -value to make the decision

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 and p -value

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!

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.

- α is the maximum probability of making a Type I error.

Type II error (False negative): The null hypothesis is not rejected when it is false.

- β is the probability of making a Type II error

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:

- the effect of the sample size,
- the correctness of the underlying assumptions about the population,
- statistical vs. practical significance, etc…

(see the handout). To study the tradeoffs between the sample size, α, and Type II error we can use power and operating characteristic curves.

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.

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.

## One sample t -test

- Assume data are independently sampled from a normal distribution with unknown mean μ and variance σ 2 . Make an initial assumption, μ 0 .
- t-statistic: \(\frac{\bar{X}-\mu_0}{s / \sqrt{n}}\) where s is a sample st.dev.
- t-statistic follows t -distribution with df = n - 1
- Alpha = 0.05, we conclude ….

## Testing for the population proportion

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.

## t-test Calculator

Table of contents

Welcome to our t-test calculator! Here you can not only easily perform one-sample t-tests , but also two-sample t-tests , as well as paired t-tests .

Do you prefer to find the p-value from t-test, or would you rather find the t-test critical values? Well, this t-test calculator can do both! 😊

What does a t-test tell you? Take a look at the text below, where we explain what actually gets tested when various types of t-tests are performed. Also, we explain when to use t-tests (in particular, whether to use the z-test vs. t-test) and what assumptions your data should satisfy for the results of a t-test to be valid. If you've ever wanted to know how to do a t-test by hand, we provide the necessary t-test formula, as well as tell you how to determine the number of degrees of freedom in a t-test.

## When to use a t-test?

A t-test is one of the most popular statistical tests for location , i.e., it deals with the population(s) mean value(s).

There are different types of t-tests that you can perform:

- A one-sample t-test;
- A two-sample t-test; and
- A paired t-test.

In the next section , we explain when to use which. Remember that a t-test can only be used for one or two groups . If you need to compare three (or more) means, use the analysis of variance ( ANOVA ) method.

The t-test is a parametric test, meaning that your data has to fulfill some assumptions :

- The data points are independent; AND
- The data, at least approximately, follow a normal distribution .

If your sample doesn't fit these assumptions, you can resort to nonparametric alternatives. Visit our Mann–Whitney U test calculator or the Wilcoxon rank-sum test calculator to learn more. Other possibilities include the Wilcoxon signed-rank test or the sign test.

## Which t-test?

Your choice of t-test depends on whether you are studying one group or two groups:

One sample t-test

Choose the one-sample t-test to check if the mean of a population is equal to some pre-set hypothesized value .

The average volume of a drink sold in 0.33 l cans — is it really equal to 330 ml?

The average weight of people from a specific city — is it different from the national average?

## Two-sample t-test

Choose the two-sample t-test to check if the difference between the means of two populations is equal to some pre-determined value when the two samples have been chosen independently of each other.

In particular, you can use this test to check whether the two groups are different from one another .

The average difference in weight gain in two groups of people: one group was on a high-carb diet and the other on a high-fat diet.

The average difference in the results of a math test from students at two different universities.

This test is sometimes referred to as an independent samples t-test , or an unpaired samples t-test .

## Paired t-test

A paired t-test is used to investigate the change in the mean of a population before and after some experimental intervention , based on a paired sample, i.e., when each subject has been measured twice: before and after treatment.

In particular, you can use this test to check whether, on average, the treatment has had any effect on the population .

The change in student test performance before and after taking a course.

The change in blood pressure in patients before and after administering some drug.

## How to do a t-test?

So, you've decided which t-test to perform. These next steps will tell you how to calculate the p-value from t-test or its critical values, and then which decision to make about the null hypothesis.

Decide on the alternative hypothesis :

Use a two-tailed t-test if you only care whether the population's mean (or, in the case of two populations, the difference between the populations' means) agrees or disagrees with the pre-set value.

Use a one-tailed t-test if you want to test whether this mean (or difference in means) is greater/less than the pre-set value.

Compute your T-score value :

Formulas for the test statistic in t-tests include the sample size , as well as its mean and standard deviation . The exact formula depends on the t-test type — check the sections dedicated to each particular test for more details.

Determine the degrees of freedom for the t-test:

The degrees of freedom are the number of observations in a sample that are free to vary as we estimate statistical parameters. In the simplest case, the number of degrees of freedom equals your sample size minus the number of parameters you need to estimate . Again, the exact formula depends on the t-test you want to perform — check the sections below for details.

The degrees of freedom are essential, as they determine the distribution followed by your T-score (under the null hypothesis). If there are d degrees of freedom, then the distribution of the test statistics is the t-Student distribution with d degrees of freedom . This distribution has a shape similar to N(0,1) (bell-shaped and symmetric) but has heavier tails . If the number of degrees of freedom is large (>30), which generically happens for large samples, the t-Student distribution is practically indistinguishable from N(0,1).

💡 The t-Student distribution owes its name to William Sealy Gosset, who, in 1908, published his paper on the t-test under the pseudonym "Student". Gosset worked at the famous Guinness Brewery in Dublin, Ireland, and devised the t-test as an economical way to monitor the quality of beer. Cheers! 🍺🍺🍺

## p-value from t-test

Recall that the p-value is the probability (calculated under the assumption that the null hypothesis is true) that the test statistic will produce values at least as extreme as the T-score produced for your sample . As probabilities correspond to areas under the density function, p-value from t-test can be nicely illustrated with the help of the following pictures:

The following formulae say how to calculate p-value from t-test. By cdf t,d we denote the cumulative distribution function of the t-Student distribution with d degrees of freedom:

p-value from left-tailed t-test:

p-value = cdf t,d (t score )

p-value from right-tailed t-test:

p-value = 1 − cdf t,d (t score )

p-value from two-tailed t-test:

p-value = 2 × cdf t,d (−|t score |)

or, equivalently: p-value = 2 − 2 × cdf t,d (|t score |)

However, the cdf of the t-distribution is given by a somewhat complicated formula. To find the p-value by hand, you would need to resort to statistical tables, where approximate cdf values are collected, or to specialized statistical software. Fortunately, our t-test calculator determines the p-value from t-test for you in the blink of an eye!

## t-test critical values

Recall, that in the critical values approach to hypothesis testing, you need to set a significance level, α, before computing the critical values , which in turn give rise to critical regions (a.k.a. rejection regions).

Formulas for critical values employ the quantile function of t-distribution, i.e., the inverse of the cdf :

Critical value for left-tailed t-test: cdf t,d -1 (α)

critical region:

(-∞, cdf t,d -1 (α)]

Critical value for right-tailed t-test: cdf t,d -1 (1-α)

[cdf t,d -1 (1-α), ∞)

Critical values for two-tailed t-test: ±cdf t,d -1 (1-α/2)

(-∞, -cdf t,d -1 (1-α/2)] ∪ [cdf t,d -1 (1-α/2), ∞)

To decide the fate of the null hypothesis, just check if your T-score lies within the critical region:

If your T-score belongs to the critical region , reject the null hypothesis and accept the alternative hypothesis.

If your T-score is outside the critical region , then you don't have enough evidence to reject the null hypothesis.

## How to use our t-test calculator

Choose the type of t-test you wish to perform:

A one-sample t-test (to test the mean of a single group against a hypothesized mean);

A two-sample t-test (to compare the means for two groups); or

A paired t-test (to check how the mean from the same group changes after some intervention).

Two-tailed;

Left-tailed; or

Right-tailed.

This t-test calculator allows you to use either the p-value approach or the critical regions approach to hypothesis testing!

Enter your T-score and the number of degrees of freedom . If you don't know them, provide some data about your sample(s): sample size, mean, and standard deviation, and our t-test calculator will compute the T-score and degrees of freedom for you .

Once all the parameters are present, the p-value, or critical region, will immediately appear underneath the t-test calculator, along with an interpretation!

## One-sample t-test

The null hypothesis is that the population mean is equal to some value μ 0 \mu_0 μ 0 .

The alternative hypothesis is that the population mean is:

- different from μ 0 \mu_0 μ 0 ;
- smaller than μ 0 \mu_0 μ 0 ; or
- greater than μ 0 \mu_0 μ 0 .

One-sample t-test formula :

- μ 0 \mu_0 μ 0 — Mean postulated in the null hypothesis;
- n n n — Sample size;
- x ˉ \bar{x} x ˉ — Sample mean; and
- s s s — Sample standard deviation.

Number of degrees of freedom in t-test (one-sample) = n − 1 n-1 n − 1 .

The null hypothesis is that the actual difference between these groups' means, μ 1 \mu_1 μ 1 , and μ 2 \mu_2 μ 2 , is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the difference μ 1 − μ 2 \mu_1 - \mu_2 μ 1 − μ 2 is:

- Different from Δ \Delta Δ ;
- Smaller than Δ \Delta Δ ; or
- Greater than Δ \Delta Δ .

In particular, if this pre-determined difference is zero ( Δ = 0 \Delta = 0 Δ = 0 ):

The null hypothesis is that the population means are equal.

The alternate hypothesis is that the population means are:

- μ 1 \mu_1 μ 1 and μ 2 \mu_2 μ 2 are different from one another;
- μ 1 \mu_1 μ 1 is smaller than μ 2 \mu_2 μ 2 ; and
- μ 1 \mu_1 μ 1 is greater than μ 2 \mu_2 μ 2 .

Formally, to perform a t-test, we should additionally assume that the variances of the two populations are equal (this assumption is called the homogeneity of variance ).

There is a version of a t-test that can be applied without the assumption of homogeneity of variance: it is called a Welch's t-test . For your convenience, we describe both versions.

## Two-sample t-test if variances are equal

Use this test if you know that the two populations' variances are the same (or very similar).

Two-sample t-test formula (with equal variances) :

where s p s_p s p is the so-called pooled standard deviation , which we compute as:

- Δ \Delta Δ — Mean difference postulated in the null hypothesis;
- n 1 n_1 n 1 — First sample size;
- x ˉ 1 \bar{x}_1 x ˉ 1 — Mean for the first sample;
- s 1 s_1 s 1 — Standard deviation in the first sample;
- n 2 n_2 n 2 — Second sample size;
- x ˉ 2 \bar{x}_2 x ˉ 2 — Mean for the second sample; and
- s 2 s_2 s 2 — Standard deviation in the second sample.

Number of degrees of freedom in t-test (two samples, equal variances) = n 1 + n 2 − 2 n_1 + n_2 - 2 n 1 + n 2 − 2 .

## Two-sample t-test if variances are unequal (Welch's t-test)

Use this test if the variances of your populations are different.

Two-sample Welch's t-test formula if variances are unequal:

- s 1 s_1 s 1 — Standard deviation in the first sample;
- s 2 s_2 s 2 — Standard deviation in the second sample.

The number of degrees of freedom in a Welch's t-test (two-sample t-test with unequal variances) is very difficult to count. We can approximate it with the help of the following Satterthwaite formula :

Alternatively, you can take the smaller of n 1 − 1 n_1 - 1 n 1 − 1 and n 2 − 1 n_2 - 1 n 2 − 1 as a conservative estimate for the number of degrees of freedom.

🔎 The Satterthwaite formula for the degrees of freedom can be rewritten as a scaled weighted harmonic mean of the degrees of freedom of the respective samples: n 1 − 1 n_1 - 1 n 1 − 1 and n 2 − 1 n_2 - 1 n 2 − 1 , and the weights are proportional to the standard deviations of the corresponding samples.

As we commonly perform a paired t-test when we have data about the same subjects measured twice (before and after some treatment), let us adopt the convention of referring to the samples as the pre-group and post-group.

The null hypothesis is that the true difference between the means of pre- and post-populations is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the actual difference between these means is:

Typically, this pre-determined difference is zero. We can then reformulate the hypotheses as follows:

The null hypothesis is that the pre- and post-means are the same, i.e., the treatment has no impact on the population .

The alternative hypothesis:

- The pre- and post-means are different from one another (treatment has some effect);
- The pre-mean is smaller than the post-mean (treatment increases the result); or
- The pre-mean is greater than the post-mean (treatment decreases the result).

Paired t-test formula

In fact, a paired t-test is technically the same as a one-sample t-test! Let us see why it is so. Let x 1 , . . . , x n x_1, ... , x_n x 1 , ... , x n be the pre observations and y 1 , . . . , y n y_1, ... , y_n y 1 , ... , y n the respective post observations. That is, x i , y i x_i, y_i x i , y i are the before and after measurements of the i -th subject.

For each subject, compute the difference, d i : = x i − y i d_i := x_i - y_i d i := x i − y i . All that happens next is just a one-sample t-test performed on the sample of differences d 1 , . . . , d n d_1, ... , d_n d 1 , ... , d n . Take a look at the formula for the T-score :

Δ \Delta Δ — Mean difference postulated in the null hypothesis;

n n n — Size of the sample of differences, i.e., the number of pairs;

x ˉ \bar{x} x ˉ — Mean of the sample of differences; and

s s s — Standard deviation of the sample of differences.

Number of degrees of freedom in t-test (paired): n − 1 n - 1 n − 1

## t-test vs Z-test

We use a Z-test when we want to test the population mean of a normally distributed dataset, which has a known population variance . If the number of degrees of freedom is large, then the t-Student distribution is very close to N(0,1).

Hence, if there are many data points (at least 30), you may swap a t-test for a Z-test, and the results will be almost identical. However, for small samples with unknown variance, remember to use the t-test because, in such cases, the t-Student distribution differs significantly from the N(0,1)!

🙋 Have you concluded you need to perform the z-test? Head straight to our z-test calculator !

## What is a t-test?

A t-test is a widely used statistical test that analyzes the means of one or two groups of data. For instance, a t-test is performed on medical data to determine whether a new drug really helps.

## What are different types of t-tests?

Different types of t-tests are:

- One-sample t-test;
- Two-sample t-test; and
- Paired t-test.

## How to find the t value in a one sample t-test?

To find the t-value:

- Subtract the null hypothesis mean from the sample mean value.
- Divide the difference by the standard deviation of the sample.
- Multiply the resultant with the square root of the sample size.

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ol{padding-top:0px;}.css-4okk7a ul:not(:first-child),.css-4okk7a ol:not(:first-child){padding-top:4px;} Test setup

Choose test type

t-test for the population mean, μ, based on one independent sample . Null hypothesis H 0 : μ = μ 0

Alternative hypothesis H 1

## Test details

Significance level α

The probability that we reject a true H 0 (type I error).

Degrees of freedom

Calculated as sample size minus one.

## Test results

Statistics Made Easy

## How to Interpret a P-Value Less Than 0.05 (With Examples)

A hypothesis test is used to test whether or not some hypothesis about a population parameter is true.

Whenever we perform a hypothesis test, we always define a null and alternative hypothesis:

- Null Hypothesis (H 0 ): The sample data occurs purely from chance.
- Alternative Hypothesis (H A ): The sample data is influenced by some non-random cause.

If the p-value of the hypothesis test is less than some significance level (e.g. α = .05), then we can reject the null hypothesis and conclude that we have sufficient evidence to say that the alternative hypothesis is true.

If the p-value is not less than .05, then we fail to reject the null hypothesis and conclude that we do not have sufficient evidence to say that the alternative hypothesis is true.

The following examples explain how to interpret a p-value less than .05 and how to interpret a p-value greater than .05 in practice.

## Example: Interpret a P-Value Less Than 0.05

Suppose a factory claims that they produce tires that each weigh 200 pounds.

An auditor comes in and tests the null hypothesis that the mean weight of a tire is 200 pounds against the alternative hypothesis that the mean weight of a tire is not 200 pounds, using a 0.05 level of significance.

The null hypothesis (H 0 ): μ = 200

The alternative hypothesis: (H A ): μ ≠ 200

Upon conducting a hypothesis test for a mean, the auditor gets a p-value of 0.0154 .

Since the p-value of 0.0154 is less than the significance level of 0.05 , the auditor rejects the null hypothesis and concludes that there is sufficient evidence to say that the true average weight of a tire is not 200 pounds.

## Example: Interpret a P-Value Greater Than 0.05

Suppose a biologist believes that a certain fertilizer will cause plants to grow more during a three-month period than they normally do, which is currently 20 inches. To test this, she applies the fertilizer to each of the plants in her laboratory for three months.

She then performs a hypothesis test using the following hypotheses:

The null hypothesis (H 0 ): μ = 20 inches (the fertilizer will have no effect on the mean plant growth)

The alternative hypothesis: (H A ): μ > 20 inches (the fertilizer will cause mean plant growth to increase)

Upon conducting a hypothesis test for a mean, the biologist gets a p-value of 0.2338 .

Since the p-value of 0.2338 is greater than the significance level of 0.05 , the biologist fails to reject the null hypothesis and concludes that there is not sufficient evidence to say that the fertilizer leads to increased plant growth.

## Additional Resources

An Explanation of P-Values and Statistical Significance Statistical vs. Practical Significance P-Value vs. Alpha: What’s the Difference?

## Featured Posts

Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike. My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

## One Reply to “How to Interpret a P-Value Less Than 0.05 (With Examples)”

in case 2 where p-value is greater than 0.5, it means we should reject the null hypothesis, which means we reject this statement : “the fertilizer will have no effect on the mean plant growth”

which means : the fertilizer will have effect on the mean plant growth

why do you conclude that ” that there is not sufficient evidence to say that the fertilizer leads to increased plant growth”?

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## Statistics and probability

Course: statistics and probability > unit 12, hypothesis testing and p-values.

- One-tailed and two-tailed tests
- Z-statistics vs. T-statistics
- Small sample hypothesis test
- Large sample proportion hypothesis testing

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Peek through the statistical keyhole to discover the power of the P-value. It’s the probability that study results are due to chance if the null hypothesis holds true—essential for determining statistical significance. In this article, we will learn 2 quick methods on how to calculate p value in Microsoft Excel .

Key Takeaways:

- P-Value: The Statistical Key – It unveils the likelihood of chance influencing study outcomes if the null hypothesis were accurate.
- Data Preparation is Crucial – Before diving into P-value calculations, ensure your data is pristine, organized, and free from errors.
- Simple Methods for P-Value Calculation – Excel offers user-friendly functions like T.TEST and Data Analysis ToolPak for swift P-value determination.
- Interpreting P-Values – A P-value below 0.05 suggests significant findings, while above indicates results may be due to chance.

Table of Contents

## Unlocking the Mystery of the P-Value in Excel

What is a p-value.

Imagine you’re peering through a keyhole at the secret workings of statistical analysis . Right there, in the center of it all, is the P-value—an intriguing figure that holds the power to unlock the truth behind your data. In statistics, the P-value represents the probability that the results of your study could occur by chance if the null hypothesis, which is usually a statement of ‘no effect’ or ‘no difference’, were true.

It’s essentially the key that statisticians use to determine whether or not they’ve stumbled upon something that could be statistically significant.

## The Role of P-Value in Hypothesis Testing

When you’re delving into hypothesis testing, the P-value acts like a trusty sidekick, helping you to navigate the tricky terrain of statistical significance. Think of it as a guide that tells you whether the observational differences or relationships uncovered in your data are likely to hold true for the wider population, or if they might just be the result of random chance.

A smaller P-value, one that dips below the agreed-upon threshold (often 0.05), whispers a hint that your findings may challenge the status quo, suggesting that the null hypothesis might not hold up. On the flip side, a larger P-value, one that looms above this benchmark, suggests that the null hypothesis cannot be dismissed—the results could simply be a fluke of chance.

## Setting the Stage for P-Value Calculation

Preparing your data for analysis.

Before you get to unravel the secrets of the P-value, it’s crucial that you set the stage right by preparing your data. This means you should start off by arranging your experimental results into two clean columns, each representing different conditions or groups for comparison. Make sure that your data is neatly formatted without any stray values that could throw off your calculations. Remove duplicates , correct any entry errors, and ensure that all data points are consistent in their units of measurement.

This meticulous preparation clears the path for a smoother analysis and more reliable insights, setting a rock-solid foundation for your P-value calculations.

## The Roadmap to Calculating P-Value

Method 1: using the insert function button.

If you’re looking to calculate the P-value in Excel without getting too technical, then using the Insert Function button is like taking the scenic route on a data analysis road trip. Here’s what you need to do:

STEP 1: Park your cursor in the cell where you want the P-value to appear.

STEP 2: Click the “fx” button next to the formula bar, bringing up the Insert Function dialog box.

STEP 3: Now, type “T.TEST” into the search bar. When it pops up in the list, select it to open the door to calculate the P-value.

STEP 4: Next, you’ll be greeted by the Function Arguments dialog box. This is where you’ll input your data ranges for both groups you’re comparing into “Array1” and “Array2,” select whether you’re conducting a one-tailed or two-tailed test, and choose the type of t-test you’re running.

STEP 5: After hitting “OK,” Excel does the legwork and delivers the P-value right where you need it.

By using this friendly function, Excel handles the complex calculations in the background while you comfortably navigate through the intuitive interface.

## Method 2: Delving into Analysis ToolPAk

To calculate the P-value using this method, follow the steps below:

STEP 1: Before we start, go to the Data Tab in Excel Ribbon and look for Data Analysis. If not found, follow the steps below to install it. Otherwise, you can directly go to STEP 5.

STEP 2: Go to File > Options.

STEP 3: In the dialog box, go to Add-ins in the left pane. Select Excel Add-ins from the dropdown and click on Go.

STEP 4: Select Analysis ToolPak and Click OK.

STEP 5: Go to Data Tab > Data Analysis.

STEP 6: In the Data Analysis dialog box, select ‘t-Test: Paired Two Sample for Means’ and click ‘OK’.

STEP 7: In the ‘t-Test: Paired Two Sample for Means’ dialogue box, select –

- Select the first data set as Variable Range 1.
- Select the second data set as Variable Range 2.
- Select Output Range.

The p-value will be displayed in the t-test result.

## Specific Paths for Different Tests

Deciphering one-tailed p-value calculations.

Engage your statistical gears, because when you’re calculating a one-tailed P-value, you’re taking a more directed approach. You’re essentially investigating if one dataset is significantly greater than or less than the other, not just different. Excel simplifies this process through its functions, allowing you to focus on whether your data leans significantly in one direction.

As you unpack your one-tailed results, remember that a P-value greater than your alpha level (commonly set at 0.05) signals that you shouldn’t reject the null hypothesis. Your findings fall within the realm of normal range, with no significant slant. However, if your P-value cruises below that threshold, it’s waving a flag that the alternative hypothesis deserves some spotlight—that there’s a significant tilt in your data.

## Navigating Through Two-Tailed P-Value Computations

Navigating through two-tailed P-value computations is like setting sail in open waters—you need to be prepared for anything, as you’re testing for any significant difference without specifying a direction. In Excel, achieving this balanced approach is a matter of adjusting your sails—simply replace the tails argument with 2 when using the T.TEST function or the T Distribution functions.

In this wider sea of possibilities, a small P-value, again compared against the alpha level of 0.05 or your chosen criterion, hints at the presence of a significant difference. But unlike a one-tailed test, direction doesn’t factor in. The key is the magnitude of the difference—whether one dataset is either much greater or much less—it’s the difference itself that’s significant here.

So batten down the hatches and make sure your data is shipshape. When done correctly, two-tailed tests provide a robust analysis of your datasets, ensuring you capture any significant differences, wherever they may lie.

## Interpreting Your Results

What does your p-value mean.

Your P-value is like a secret message about your data waiting to be decoded. A small P-value, typically less than 0.05, is a whisper of a revelation—it hints strongly that your results might not be due to just luck or random chance, instead pointing to a real effect or difference. In contrast, a P-value that’s larger tells you to hold your horses—there’s not enough evidence to declare that what you’re observing is anything more than a statistical mirage.

Think of your P-value as a gauge on your dashboard, with 0.05 being a critical threshold. If the needle is to the left, switch on your “Eureka!” alert. To the right, and it’s time to keep the champagne on ice, as further investigation or new data might be necessary to draw firm conclusions.

## Making Informed Decisions Based on P-Values

Making decisions based on P-values is akin to using a compass on a data-driven expedition. It doesn’t point you to an X-marks-the-spot treasure chest directly, but it helps you avoid wandering aimlessly or falling into the quicksand of faulty conclusions.

With a P-value in hand, you can make decisions with greater confidence. If the P-value sails below the 0.05 threshold, it’s a strong wind in your sails towards rejecting the null hypothesis and considering your alternative hypothesis credible. On the flip side, if the P-value is above 0.05, it’s wise to consider that your hypothesis might need revisiting, or that more data could be needed to clarify the waters.

However, it’s essential to weigh your P-value against the context of your study, including the effect size and confidence intervals. Sailing toward the truth involves considering all instruments at your disposal—not just the P-value compass.

## FAQs on Calculating P-Value in Excel

What is p-value in excel.

The p-value in Excel is a probability score that tells you the strength of your evidence against the null hypothesis. It’s a way to measure if the data at hand could very well be a fluke or if there’s a statistically significant difference or association. In simpler terms, the p-value helps you understand whether your findings are due to actual effects or mere random chance.

## How do you find the p-value in Excel?

To find the p-value in Excel, you’d typically use built-in functions like T.TEST for t-tests or the Data Analysis tool after installing the Analysis ToolPak. These Excel features help you calculate the p-value by inputting your data range, selecting tail type, and choosing the test type, guiding you swiftly towards the statistical significances in your data.

## Is it necessary to have a strong statistics background to calculate p-values in Excel?

Not at all! While a solid grip on statistics can be beneficial, Excel’s user-friendly interface and functions greatly simplify the process of calculating p-values. You can perform these calculations effectively even with a basic understanding of statistical principles, thanks to Excel’s guided tools and wide array of resources available for support.

## How can one-tailed and two-tailed tests affect the p-value in Excel?

In Excel, a one-tailed test evaluates if a dataset is significantly greater or less than another, influencing the p-value by focusing on a single direction of interest . Meanwhile, a two-tailed test checks for any significant difference regardless of direction, effectively doubling the critical area and can result in a larger p-value. Choosing between the two hinges on your hypothesis specifics, directly impacting the p-value and your conclusions.

## Can Excel handle all types of hypothesis tests for p-value calculations?

Excel boasts a robust suite of statistical functions, able to tackle a wide range of hypothesis tests for p-value calculations. However, it may not be the best fit for extremely specialized or complex analyses, where dedicated statistical software would offer more nuanced control and options. For most standard tests, though, Excel has you covered.

## John Michaloudis

John Michaloudis is a former accountant and finance analyst at General Electric, a Microsoft MVP since 2020, an Amazon #1 bestselling author of 4 Microsoft Excel books and teacher of Microsoft Excel & Office over at his flagship Academy Online Course .

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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. Hypothesis testing

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.

Null hypothesis: µ ≥ 70 inches. Alternative hypothesis: µ < 70 inches. A two-tailed hypothesis involves making an "equal to" or "not equal to" statement. For example, suppose we assume the mean height of a male in the U.S. is equal to 70 inches. The null and alternative hypotheses in this case would be: Null hypothesis: µ = 70 inches.

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 ...

It is the alternative hypothesis that determines what "extreme" actually means, so the p-value depends on the alternative hypothesis that you state: left-tailed, right-tailed, or two-tailed. In the formulas below, S stands for a test statistic, x for the value it produced for a given sample, and Pr(event | H 0 ) is the probability of an event ...

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.

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 actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. H 0, the —null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

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 test the hypothesis in the p-value approach, compare the p-value to the level of ...

Using the p-value to make the decision. 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 p value gets smaller as the test statistic calculated from your data gets further away from the range of test statistics predicted by the null hypothesis. The p value is a proportion: if your p value is 0.05, that means that 5% of the time you would see a test statistic at least as extreme as the one you found if the null hypothesis was true.

The null and alternative hypotheses offer competing answers to your research question. When the research question asks "Does the independent variable affect the dependent variable?": The null hypothesis ( H0) answers "No, there's no effect in the population.". The alternative hypothesis ( Ha) answers "Yes, there is an effect in the ...

In this case, our t-value of 2.289 produces a p value between 0.02 and 0.05 for a two-tailed test. Our results are statistically significant, and they are consistent with the calculator's more precise results. Displaying the P value in a Chart. In the example above, you saw how to calculate a p-value starting with the sample statistics.

These next steps will tell you how to calculate the p-value from t-test or its critical values, and then which decision to make about the null hypothesis. Decide on the alternative hypothesis : Use a two-tailed t-test if you only care whether the population's mean (or, in the case of two populations, the difference between the populations ...

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 them to a percentage. For example, a p value of 0.0254 is 2.54%.

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.

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 null hypothesis (H0): μ = 200. The alternative hypothesis: (HA): μ ≠ 200. Upon conducting a hypothesis test for a mean, the auditor gets a p-value of 0.0154. Since the p-value of 0.0154 is less than the significance level of 0.05, the auditor rejects the null hypothesis and concludes that there is sufficient evidence to say that the ...

In this video there was no critical value set for this experiment. In the last seconds of the video, Sal briefly mentions a p-value of 5% (0.05), which would have a critical of value of z = (+/-) 1.96. Since the experiment produced a z-score of 3, which is more extreme than 1.96, we reject the null hypothesis.

No. The p -value only tells you how likely the data you have observed is to have occurred under the null hypothesis. If the p -value is below your threshold of significance (typically p < 0.05), then you can reject the null hypothesis, but this does not necessarily mean that your alternative hypothesis is true.

The calculated p-value is used in comparison with a predefined significance level (alpha) to make decisions about the null hypothesis. If the p-value is less than or equal to alpha, typically 0.05, the results are considered statistically significant, leading to the rejection of the null hypothesis in favor of the alternative hypothesis. If the ...

Researchers have criticized the overreliance on null-hypothesis significance testing (NHST) and common misconceptions about p values for more than half a century (e.g., Bakan, 1966; Nunnally, 1960; Rozeboom, 1960).The correct definition of a p value is the probability of observing the sample data, or more extreme data, assuming the null hypothesis is true.

There is a trade-off between power and p-value, but in reality, the p-value is fixed at p = 0.05. Therefore, it is described as unrelated in Table 2. However, power is not determined only by p-value. A more accurate null hypothesis significance test also has a higher power, because a higher power (1 − β) means a smaller β.

This video covers the following:1. Hypothesis Testing2. Null and Alternate Hypothesis3. Level of Significance4. P value 💡 Complete Statistics eB...

However, researchers always look for p-value lower than the significance level. p — value ≤ Significance Level, means that the sampled data provide enough evidence to reject the null hypothesis. In other words, the alternative hypothesis, which we want to prove true, wins the battle. The observed effect in the data is statistically significant.

A small p-value indicates only that the data observed would be unlikely under the null hypothesis. It does not validate the alternative hypothesis or imply causation.

STEP 1: Park your cursor in the cell where you want the P-value to appear. STEP 2: Click the "fx" button next to the formula bar, bringing up the Insert Function dialog box. STEP 3: Now, type "T.TEST" into the search bar. When it pops up in the list, select it to open the door to calculate the P-value.

The common threshold for significance in hypothesis testing is a p-value of 0.05. However, this cutoff is arbitrary and can be misleading. It implies a binary decision-making process where results ...

In statistics, p-values are a crucial part of hypothesis testing, a method used to determine whether there is enough evidence to reject a null hypothesis. The p-value quantifies the probability of ...