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The previous two chapters introduced methods for organizing and summarizing sample data, and using sample statistics to estimate population parameters. This chapter introduces the next major topic of inferential statistics: hypothesis testing.
A hypothesis is a statement or claim about a property of a population.
When conducting scientific research, typically there is some known information, perhaps from some past work or from a long accepted idea. We want to test whether this claim is believable. This is the basic idea behind a hypothesis test:
For example, past research tells us that the average life span for a hummingbird is about four years. You have been studying the hummingbirds in the southeastern United States and find a sample mean lifespan of 4.8 years. Should you reject the known or accepted information in favor of your results? How confident are you in your estimate? At what point would you say that there is enough evidence to reject the known information and support your alternative claim? How far from the known mean of four years can the sample mean be before we reject the idea that the average lifespan of a hummingbird is four years?
Hypothesis testing is a procedure, based on sample evidence and probability, used to test claims regarding a characteristic of a population.
A hypothesis is a claim or statement about a characteristic of a population of interest to us. A hypothesis test is a way for us to use our sample statistics to test a specific claim.
The population mean weight is known to be 157 lb. We want to test the claim that the mean weight has increased.
Two years ago, the proportion of infected plants was 37%. We believe that a treatment has helped, and we want to test the claim that there has been a reduction in the proportion of infected plants.
The null hypothesis is a statement about the value of a population parameter, such as the population mean (µ) or the population proportion ( p ). It contains the condition of equality and is denoted as H 0 (H-naught).
H 0 : µ = 157 or H 0 : p = 0.37
The alternative hypothesis is the claim to be tested, the opposite of the null hypothesis. It contains the value of the parameter that we consider plausible and is denoted as H 1 .
H 1 : µ > 157 or H 1 : p ≠ 0.37
The test statistic is a value computed from the sample data that is used in making a decision about the rejection of the null hypothesis. The test statistic converts the sample mean ( x̄ ) or sample proportion ( p̂ ) to a Z- or t-score under the assumption that the null hypothesis is true . It is used to decide whether the difference between the sample statistic and the hypothesized claim is significant.
The p-value is the area under the curve to the left or right of the test statistic. It is compared to the level of significance ( α ).
The critical value is the value that defines the rejection zone (the test statistic values that would lead to rejection of the null hypothesis). It is defined by the level of significance.
The level of significance ( α ) is the probability that the test statistic will fall into the critical region when the null hypothesis is true. This level is set by the researcher.
The conclusion is the final decision of the hypothesis test. The conclusion must always be clearly stated, communicating the decision based on the components of the test. It is important to realize that we never prove or accept the null hypothesis. We are merely saying that the sample evidence is not strong enough to warrant the rejection of the null hypothesis. The conclusion is made up of two parts:
1) Reject or fail to reject the null hypothesis, and 2) there is or is not enough evidence to support the alternative claim.
Option 1) Reject the null hypothesis (H 0 ). This means that you have enough statistical evidence to support the alternative claim (H 1 ).
Option 2) Fail to reject the null hypothesis (H 0 ). This means that you do NOT have enough evidence to support the alternative claim (H 1 ).
Another way to think about hypothesis testing is to compare it to the US justice system. A defendant is innocent until proven guilty (Null hypothesis—innocent). The prosecuting attorney tries to prove that the defendant is guilty (Alternative hypothesis—guilty). There are two possible conclusions that the jury can reach. First, the defendant is guilty (Reject the null hypothesis). Second, the defendant is not guilty (Fail to reject the null hypothesis). This is NOT the same thing as saying the defendant is innocent! In the first case, the prosecutor had enough evidence to reject the null hypothesis (innocent) and support the alternative claim (guilty). In the second case, the prosecutor did NOT have enough evidence to reject the null hypothesis (innocent) and support the alternative claim of guilty.
There are three different pairs of null and alternative hypotheses:
where c is some known value.
This tests whether the population parameter is equal to, versus not equal to, some specific value.
H o : μ = 12 vs. H 1 : μ ≠ 12
The critical region is divided equally into the two tails and the critical values are ± values that define the rejection zones.
A forester studying diameter growth of red pine believes that the mean diameter growth will be different if a fertilization treatment is applied to the stand.
This is a two-sided question, as the forester doesn’t state whether population mean diameter growth will increase or decrease.
This tests whether the population parameter is equal to, versus greater than, some specific value.
H o : μ = 12 vs. H 1 : μ > 12
The critical region is in the right tail and the critical value is a positive value that defines the rejection zone.
A biologist believes that there has been an increase in the mean number of lakes infected with milfoil, an invasive species, since the last study five years ago.
This is a right-sided question, as the biologist believes that there has been an increase in population mean number of infected lakes.
This tests whether the population parameter is equal to, versus less than, some specific value.
H o : μ = 12 vs. H 1 : μ < 12
The critical region is in the left tail and the critical value is a negative value that defines the rejection zone.
A scientist’s research indicates that there has been a change in the proportion of people who support certain environmental policies. He wants to test the claim that there has been a reduction in the proportion of people who support these policies.
This is a left-sided question, as the scientist believes that there has been a reduction in the true population proportion.
When the observed results (the sample statistics) are unlikely (a low probability) under the assumption that the null hypothesis is true, we say that the result is statistically significant, and we reject the null hypothesis. This result depends on the level of significance, the sample statistic, sample size, and whether it is a one- or two-sided alternative hypothesis.
When testing, we arrive at a conclusion of rejecting the null hypothesis or failing to reject the null hypothesis. Such conclusions are sometimes correct and sometimes incorrect (even when we have followed all the correct procedures). We use incomplete sample data to reach a conclusion and there is always the possibility of reaching the wrong conclusion. There are four possible conclusions to reach from hypothesis testing. Of the four possible outcomes, two are correct and two are NOT correct.
A Type I error is when we reject the null hypothesis when it is true. The symbol α (alpha) is used to represent Type I errors. This is the same alpha we use as the level of significance. By setting alpha as low as reasonably possible, we try to control the Type I error through the level of significance.
A Type II error is when we fail to reject the null hypothesis when it is false. The symbol β (beta) is used to represent Type II errors.
In general, Type I errors are considered more serious. One step in the hypothesis test procedure involves selecting the significance level ( α ), which is the probability of rejecting the null hypothesis when it is correct. So the researcher can select the level of significance that minimizes Type I errors. However, there is a mathematical relationship between α, β , and n (sample size).
The natural inclination is to select the smallest possible value for α, thinking to minimize the possibility of causing a Type I error. Unfortunately, this forces an increase in Type II errors. By making the rejection zone too small, you may fail to reject the null hypothesis, when, in fact, it is false. Typically, we select the best sample size and level of significance, automatically setting β .
A Type II error ( β ) is the probability of failing to reject a false null hypothesis. It follows that 1- β is the probability of rejecting a false null hypothesis. This probability is identified as the power of the test, and is often used to gauge the test’s effectiveness in recognizing that a null hypothesis is false.
The probability that at a fixed level α significance test will reject H 0 , when a particular alternative value of the parameter is true is called the power of the test.
Power is also directly linked to sample size. For example, suppose the null hypothesis is that the mean fish weight is 8.7 lb. Given sample data, a level of significance of 5%, and an alternative weight of 9.2 lb., we can compute the power of the test to reject μ = 8.7 lb. If we have a small sample size, the power will be low. However, increasing the sample size will increase the power of the test. Increasing the level of significance will also increase power. A 5% test of significance will have a greater chance of rejecting the null hypothesis than a 1% test because the strength of evidence required for the rejection is less. Decreasing the standard deviation has the same effect as increasing the sample size: there is more information about μ .
We are going to examine two equivalent ways to perform a hypothesis test: the classical approach and the p-value approach. The classical approach is based on standard deviations. This method compares the test statistic (Z-score) to a critical value (Z-score) from the standard normal table. If the test statistic falls in the rejection zone, you reject the null hypothesis. The p-value approach is based on area under the normal curve. This method compares the area associated with the test statistic to alpha ( α ), the level of significance (which is also area under the normal curve). If the p-value is less than alpha, you would reject the null hypothesis.
As a past student poetically said: If the p-value is a wee value, Reject Ho
Both methods must have:
There are four steps required for a hypothesis test:
A forester studying diameter growth of red pine believes that the mean diameter growth will be different from the known mean growth of 1.35 inches/year if a fertilization treatment is applied to the stand. He conducts his experiment, collects data from a sample of 32 plots, and gets a sample mean diameter growth of 1.6 in./year. The population standard deviation for this stand is known to be 0.46 in./year. Does he have enough evidence to support his claim?
Step 1) State the null and alternative hypotheses.
Step 2) State the level of significance and the critical value.
Step 3) Compute the test statistic.
Step 4) State a conclusion.
In this problem, the test statistic falls in the red rejection zone. The test statistic of 3.07 is greater than the critical value of 1.96.We will reject the null hypothesis. We have enough evidence to support the claim that the mean diameter growth is different from (not equal to) 1.35 in./year.
A researcher believes that there has been an increase in the average farm size in his state since the last study five years ago. The previous study reported a mean size of 450 acres with a population standard deviation ( σ ) of 167 acres. He samples 45 farms and gets a sample mean of 485.8 acres. Is there enough information to support his claim?
We fail to reject the null hypothesis. We do not have enough evidence to support the claim that the mean farm size has increased from 450 acres.
A researcher believes that there has been a reduction in the mean number of hours that college students spend preparing for final exams. A national study stated that students at a 4-year college spend an average of 23 hours preparing for 5 final exams each semester with a population standard deviation of 7.3 hours. The researcher sampled 227 students and found a sample mean study time of 19.6 hours. Does this indicate that the average study time for final exams has decreased? Use a 1% level of significance to test this claim.
We reject the null hypothesis. We have sufficient evidence to support the claim that the mean final exam study time has decreased below 23 hours.
The p-value is the probability of observing our sample mean given that the null hypothesis is true. It is the area under the curve to the left or right of the test statistic. If the probability of observing such a sample mean is very small (less than the level of significance), we would reject the null hypothesis. Computations for the p-value depend on whether it is a one- or two-sided test.
Steps for a hypothesis test using p-values:
Instead of comparing Z-score test statistic to Z-score critical value, as in the classical method, we compare area of the test statistic to area of the level of significance.
The Decision Rule: If the p-value is less than alpha, we reject the null hypothesis
If it is a two-sided test (the alternative claim is ≠), the p-value is equal to two times the probability of the absolute value of the test statistic. If the test is a left-sided test (the alternative claim is “<”), then the p-value is equal to the area to the left of the test statistic. If the test is a right-sided test (the alternative claim is “>”), then the p-value is equal to the area to the right of the test statistic.
Let’s look at Example 6 again.
A forester studying diameter growth of red pine believes that the mean diameter growth will be different from the known mean growth of 1.35 in./year if a fertilization treatment is applied to the stand. He conducts his experiment, collects data from a sample of 32 plots, and gets a sample mean diameter growth of 1.6 in./year. The population standard deviation for this stand is known to be 0.46 in./year. Does he have enough evidence to support his claim?
Step 2) State the level of significance.
The p-value is two times the area of the absolute value of the test statistic (because the alternative claim is “not equal”).
Step 4) Compare the p-value to alpha and state a conclusion.
Let’s look at Example 7 again.
The p-value is the area to the right of the Z-score 1.44 (the hatched area).
We fail to reject the null hypothesis. We do not have enough evidence to support the claim that the mean farm size has increased.
Let’s look at Example 8 again.
The p-value is the area to the left of the test statistic (the little black area to the left of -7.02). The Z-score of -7.02 is not on the standard normal table. The smallest probability on the table is 0.0002. We know that the area for the Z-score -7.02 is smaller than this area (probability). Therefore, the p-value is <0.0002.
We reject the null hypothesis. We have enough evidence to support the claim that the mean final exam study time has decreased below 23 hours.
Both the classical method and p-value method for testing a hypothesis will arrive at the same conclusion. In the classical method, the critical Z-score is the number on the z-axis that defines the level of significance ( α ). The test statistic converts the sample mean to units of standard deviation (a Z-score). If the test statistic falls in the rejection zone defined by the critical value, we will reject the null hypothesis. In this approach, two Z-scores, which are numbers on the z-axis, are compared. In the p-value approach, the p-value is the area associated with the test statistic. In this method, we compare α (which is also area under the curve) to the p-value. If the p-value is less than α , we reject the null hypothesis. The p-value is the probability of observing such a sample mean when the null hypothesis is true. If the probability is too small (less than the level of significance), then we believe we have enough statistical evidence to reject the null hypothesis and support the alternative claim.
(referring to Ex. 8)
Test of mu = 23 vs. < 23 |
The assumed standard deviation = 7.3 |
99% Upper | |||||
N | Mean | SE Mean | Bound | Z | P |
227 | 19.600 | 0.485 | 20.727 | -7.02 | 0.000 |
Excel does not offer 1-sample hypothesis testing.
Frequently, the population standard deviation (σ) is not known. We can estimate the population standard deviation (σ) with the sample standard deviation (s). However, the test statistic will no longer follow the standard normal distribution. We must rely on the student’s t-distribution with n-1 degrees of freedom. Because we use the sample standard deviation (s), the test statistic will change from a Z-score to a t-score.
Steps for a hypothesis test are the same that we covered in Section 2.
Just as with the hypothesis test from the previous section, the data for this test must be from a random sample and requires either that the population from which the sample was drawn be normal or that the sample size is sufficiently large (n≥30). A t-test is robust, so small departures from normality will not adversely affect the results of the test. That being said, if the sample size is smaller than 30, it is always good to verify the assumption of normality through a normal probability plot.
We will still have the same three pairs of null and alternative hypotheses and we can still use either the classical approach or the p-value approach.
Selecting the correct critical value from the student’s t-distribution table depends on three factors: the type of test (one-sided or two-sided alternative hypothesis), the sample size, and the level of significance.
For a two-sided test (“not equal” alternative hypothesis), the critical value (t α /2 ), is determined by alpha ( α ), the level of significance, divided by two, to deal with the possibility that the result could be less than OR greater than the known value.
For a one-sided test (“a less than” or “greater than” alternative hypothesis), the critical value (t α ) , is determined by alpha ( α ), the level of significance, being all in the one side.
Find the critical value you would use to test the claim that μ ≠ 112 with a sample size of 18 and a 5% level of significance.
In this case, the critical value (t α /2 ) would be 2.110. This is a two-sided question (≠) so you would divide alpha by 2 (0.05/2 = 0.025) and go down the 0.025 column to 17 degrees of freedom.
What would the critical value be if you wanted to test that μ < 112 for the same data?
In this case, the critical value would be 1.740. This is a one-sided question (<) so alpha would be divided by 1 (0.05/1 = 0.05). You would go down the 0.05 column with 17 degrees of freedom to get the correct critical value.
In 2005, the mean pH level of rain in a county in northern New York was 5.41. A biologist believes that the rain acidity has changed. He takes a random sample of 11 rain dates in 2010 and obtains the following data. Use a 1% level of significance to test his claim.
4.70, 5.63, 5.02, 5.78, 4.99, 5.91, 5.76, 5.54, 5.25, 5.18, 5.01
The sample size is small and we don’t know anything about the distribution of the population, so we examine a normal probability plot. The distribution looks normal so we will continue with our test.
The sample mean is 5.343 with a sample standard deviation of 0.397.
We will fail to reject the null hypothesis. We do not have enough evidence to support the claim that the mean rain pH has changed.
Cadmium, a heavy metal, is toxic to animals. Mushrooms, however, are able to absorb and accumulate cadmium at high concentrations. The government has set safety limits for cadmium in dry vegetables at 0.5 ppm. Biologists believe that the mean level of cadmium in mushrooms growing near strip mines is greater than the recommended limit of 0.5 ppm, negatively impacting the animals that live in this ecosystem. A random sample of 51 mushrooms gave a sample mean of 0.59 ppm with a sample standard deviation of 0.29 ppm. Use a 5% level of significance to test the claim that the mean cadmium level is greater than the acceptable limit of 0.5 ppm.
The sample size is greater than 30 so we are assured of a normal distribution of the means.
Step 4) State a Conclusion.
The test statistic falls in the rejection zone. We will reject the null hypothesis. We have enough evidence to support the claim that the mean cadmium level is greater than the acceptable safe limit.
BUT, what happens if the significance level changes to 1%?
The critical value is now found by going down the 0.01 column with 50 degrees of freedom. The critical value is 2.403. The test statistic is now LESS THAN the critical value. The test statistic does not fall in the rejection zone. The conclusion will change. We do NOT have enough evidence to support the claim that the mean cadmium level is greater than the acceptable safe limit of 0.5 ppm.
The level of significance is the probability that you, as the researcher, set to decide if there is enough statistical evidence to support the alternative claim. It should be set before the experiment begins.
We can also use the p-value approach for a hypothesis test about the mean when the population standard deviation ( σ ) is unknown. However, when using a student’s t-table, we can only estimate the range of the p-value, not a specific value as when using the standard normal table. The student’s t-table has area (probability) across the top row in the table, with t-scores in the body of the table.
Estimating P-value from a Student’s T-table
If your test statistic is 3.789 with 3 degrees of freedom, you would go across the 3 df row. The value 3.789 falls between the values 3.482 and 4.541 in that row. Therefore, the p-value is between 0.02 and 0.01. The p-value will be greater than 0.01 but less than 0.02 (0.01<p<0.02).
If your level of significance is 5%, you would reject the null hypothesis as the p-value (0.01-0.02) is less than alpha ( α ) of 0.05.
If your level of significance is 1%, you would fail to reject the null hypothesis as the p-value (0.01-0.02) is greater than alpha ( α ) of 0.01.
Software packages typically output p-values. It is easy to use the Decision Rule to answer your research question by the p-value method.
(referring to Ex. 12)
Test of mu = 0.5 vs. > 0.5
95% Lower | ||||||
N | Mean | StDev | SE Mean | Bound | T | P |
51 | 0.5900 | 0.2900 | 0.0406 | 0.5219 | 2.22 | 0.016 |
Additional example: www.youtube.com/watch?v=WwdSjO4VUsg .
Frequently, the parameter we are testing is the population proportion.
Recall that the best point estimate of p , the population proportion, is given by
when np (1 – p )≥10. We can use both the classical approach and the p-value approach for testing.
The steps for a hypothesis test are the same that we covered in Section 2.
The test statistic follows the standard normal distribution. Notice that the standard error (the denominator) uses p instead of p̂ , which was used when constructing a confidence interval about the population proportion. In a hypothesis test, the null hypothesis is assumed to be true, so the known proportion is used.
A botanist has produced a new variety of hybrid soy plant that is better able to withstand drought than other varieties. The botanist knows the seed germination for the parent plants is 75%, but does not know the seed germination for the new hybrid. He tests the claim that it is different from the parent plants. To test this claim, 450 seeds from the hybrid plant are tested and 321 have germinated. Use a 5% level of significance to test this claim that the germination rate is different from 75%.
This is a two-sided question so alpha is divided by 2.
The test statistic does not fall in the rejection zone. We fail to reject the null hypothesis. We do not have enough evidence to support the claim that the germination rate of the hybrid plant is different from the parent plants.
Let’s answer this question using the p-value approach. Remember, for a two-sided alternative hypothesis (“not equal”), the p-value is two times the area of the test statistic. The test statistic is -1.81 and we want to find the area to the left of -1.81 from the standard normal table.
Now compare the p-value to alpha. The Decision Rule states that if the p-value is less than alpha, reject the H 0 . In this case, the p-value (0.0702) is greater than alpha (0.05) so we will fail to reject H 0 . We do not have enough evidence to support the claim that the germination rate of the hybrid plant is different from the parent plants.
You are a biologist studying the wildlife habitat in the Monongahela National Forest. Cavities in older trees provide excellent habitat for a variety of birds and small mammals. A study five years ago stated that 32% of the trees in this forest had suitable cavities for this type of wildlife. You believe that the proportion of cavity trees has increased. You sample 196 trees and find that 79 trees have cavities. Does this evidence support your claim that there has been an increase in the proportion of cavity trees?
Use a 10% level of significance to test this claim.
This is a one-sided question so alpha is divided by 1.
The test statistic is larger than the critical value (it falls in the rejection zone). We will reject the null hypothesis. We have enough evidence to support the claim that there has been an increase in the proportion of cavity trees.
Now use the p-value approach to answer the question. This is a right-sided question (“greater than”), so the p-value is equal to the area to the right of the test statistic. Go to the positive side of the standard normal table and find the area associated with the Z-score of 2.49. The area is 0.9936. Remember that this table is cumulative from the left. To find the area to the right of 2.49, we subtract from one.
p-value = (1 – 0.9936) = 0.0064
The p-value is less than the level of significance (0.10), so we reject the null hypothesis. We have enough evidence to support the claim that the proportion of cavity trees has increased.
(referring to Ex. 15)
Test of p = 0.32 vs. p > 0.32
90% Lower | ||||||
Sample | X | N | Sample p | Bound | Z-Value | p-Value |
1 | 79 | 196 | 0.403061 | 0.358160 | 2.49 | 0.006 |
Using the normal approximation. |
When people think of statistical inference, they usually think of inferences involving population means or proportions. However, the particular population parameter needed to answer an experimenter’s practical questions varies from one situation to another, and sometimes a population’s variability is more important than its mean. Thus, product quality is often defined in terms of low variability.
Sample variance S 2 can be used for inferences concerning a population variance σ 2 . For a random sample of n measurements drawn from a normal population with mean μ and variance σ 2 , the value S 2 provides a point estimate for σ 2 . In addition, the quantity ( n – 1) S 2 / σ 2 follows a Chi-square ( χ 2 ) distribution, with df = n – 1.
The properties of Chi-square ( χ 2 ) distribution are:
Alternative hypothesis:
where the χ 2 critical value in the rejection region is based on degrees of freedom df = n – 1 and a specified significance level of α .
As with previous sections, if the test statistic falls in the rejection zone set by the critical value, you will reject the null hypothesis.
A forester wants to control a dense understory of striped maple that is interfering with desirable hardwood regeneration using a mist blower to apply an herbicide treatment. She wants to make sure that treatment has a consistent application rate, in other words, low variability not exceeding 0.25 gal./acre (0.06 gal. 2 ). She collects sample data (n = 11) on this type of mist blower and gets a sample variance of 0.064 gal. 2 Using a 5% level of significance, test the claim that the variance is significantly greater than 0.06 gal. 2
H 0 : σ 2 = 0.06
H 1 : σ 2 >0.06
The critical value is 18.307. Any test statistic greater than this value will cause you to reject the null hypothesis.
The test statistic is
We fail to reject the null hypothesis. The forester does NOT have enough evidence to support the claim that the variance is greater than 0.06 gal. 2 You can also estimate the p-value using the same method as for the student t-table. Go across the row for degrees of freedom until you find the two values that your test statistic falls between. In this case going across the row 10, the two table values are 4.865 and 15.987. Now go up those two columns to the top row to estimate the p-value (0.1-0.9). The p-value is greater than 0.1 and less than 0.9. Both are greater than the level of significance (0.05) causing us to fail to reject the null hypothesis.
(referring to Ex. 16)
Test and CI for One Variance
Method | ||
Null hypothesis | Sigma-squared | = 0.06 |
Alternative hypothesis | Sigma-squared | > 0.06 |
The chi-square method is only for the normal distribution.
Test | |||
Method | Statistic | DF | P-Value |
Chi-Square | 10.67 | 10 | 0.384 |
Excel does not offer 1-sample χ 2 testing.
To test a claim about μ when σ is known.
Natural Resources Biometrics Copyright © 2014 by Diane Kiernan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.
Step 1: define the hypothesis, step 2: set the criteria, step 3: calculate the statistic, step 4: reach a conclusion, types of errors, the bottom line.
Charlene Rhinehart is a CPA , CFE, chair of an Illinois CPA Society committee, and has a degree in accounting and finance from DePaul University.
Your investment advisor proposes you a monthly income investment plan that promises a variable return each month. You will invest in it only if you are assured of an average $180 monthly income. Your advisor also tells you that for the past 300 months, the scheme had investment returns with an average value of $190 and a standard deviation of $75. Should you invest in this scheme? Hypothesis testing comes to the aid for such decision-making.
Hypothesis or significance testing is a mathematical model for testing a claim, idea or hypothesis about a parameter of interest in a given population set, using data measured in a sample set. Calculations are performed on selected samples to gather more decisive information about the characteristics of the entire population, which enables a systematic way to test claims or ideas about the entire dataset.
Here is a simple example: A school principal reports that students in their school score an average of 7 out of 10 in exams. To test this “hypothesis,” we record marks of say 30 students (sample) from the entire student population of the school (say 300) and calculate the mean of that sample. We can then compare the (calculated) sample mean to the (reported) population mean and attempt to confirm the hypothesis.
To take another example, the annual return of a particular mutual fund is 8%. Assume that mutual fund has been in existence for 20 years. We take a random sample of annual returns of the mutual fund for, say, five years (sample) and calculate its mean. We then compare the (calculated) sample mean to the (claimed) population mean to verify the hypothesis.
This article assumes readers' familiarity with concepts of a normal distribution table, formula, p-value and related basics of statistics.
Different methodologies exist for hypothesis testing, but the same four basic steps are involved:
Usually, the reported value (or the claim statistics) is stated as the hypothesis and presumed to be true. For the above examples, the hypothesis will be:
This stated description constitutes the “ Null Hypothesis (H 0 ) ” and is assumed to be true – the way a defendant in a jury trial is presumed innocent until proven guilty by the evidence presented in court. Similarly, hypothesis testing starts by stating and assuming a “ null hypothesis ,” and then the process determines whether the assumption is likely to be true or false.
The important point to note is that we are testing the null hypothesis because there is an element of doubt about its validity. Whatever information that is against the stated null hypothesis is captured in the Alternative Hypothesis (H 1 ). For the above examples, the alternative hypothesis will be:
In other words, the alternative hypothesis is a direct contradiction of the null hypothesis.
As in a trial, the jury assumes the defendant's innocence (null hypothesis). The prosecutor has to prove otherwise (alternative hypothesis). Similarly, the researcher has to prove that the null hypothesis is either true or false. If the prosecutor fails to prove the alternative hypothesis, the jury has to let the defendant go (basing the decision on the null hypothesis). Similarly, if the researcher fails to prove an alternative hypothesis (or simply does nothing), then the null hypothesis is assumed to be true.
The decision-making criteria have to be based on certain parameters of datasets.
The decision-making criteria have to be based on certain parameters of datasets and this is where the connection to normal distribution comes into the picture.
As per the standard statistics postulate about sampling distribution , for any sample size n, the sampling distribution of X is normal if the X from which the sample is drawn is normally distributed. Hence, the probabilities of all other possible sample mean that one could select are normally distributed.
For e.g., determine if the average daily return, of any stock listed on XYZ stock market , around New Year's Day is greater than 2%.
H 0 : Null Hypothesis: mean = 2%
H 1 : Alternative Hypothesis: mean > 2% (this is what we want to prove)
Take the sample (say of 50 stocks out of total 500) and compute the mean of the sample.
For a normal distribution, 95% of the values lie within two standard deviations of the population mean. Hence, this normal distribution and central limit assumption for the sample dataset allows us to establish 5% as a significance level. It makes sense as, under this assumption, there is less than a 5% probability (100-95) of getting outliers that are beyond two standard deviations from the population mean. Depending upon the nature of datasets, other significance levels can be taken at 1%, 5% or 10%. For financial calculations (including behavioral finance), 5% is the generally accepted limit. If we find any calculations that go beyond the usual two standard deviations, then we have a strong case of outliers to reject the null hypothesis.
Graphically, it is represented as follows:
In the above example, if the mean of the sample is much larger than 2% (say 3.5%), then we reject the null hypothesis. The alternative hypothesis (mean >2%) is accepted, which confirms that the average daily return of the stocks is indeed above 2%.
However, if the mean of the sample is not likely to be significantly greater than 2% (and remains at, say, around 2.2%), then we CANNOT reject the null hypothesis. The challenge comes on how to decide on such close range cases. To make a conclusion from selected samples and results, a level of significance is to be determined, which enables a conclusion to be made about the null hypothesis. The alternative hypothesis enables establishing the level of significance or the "critical value” concept for deciding on such close range cases.
According to the textbook standard definition, “A critical value is a cutoff value that defines the boundaries beyond which less than 5% of sample means can be obtained if the null hypothesis is true. Sample means obtained beyond a critical value will result in a decision to reject the null hypothesis." In the above example, if we have defined the critical value as 2.1%, and the calculated mean comes to 2.2%, then we reject the null hypothesis. A critical value establishes a clear demarcation about acceptance or rejection.
This step involves calculating the required figure(s), known as test statistics (like mean, z-score , p-value , etc.), for the selected sample. (We'll get to these in a later section.)
With the computed value(s), decide on the null hypothesis. If the probability of getting a sample mean is less than 5%, then the conclusion is to reject the null hypothesis. Otherwise, accept and retain the null hypothesis.
There can be four possible outcomes in sample-based decision-making, with regard to the correct applicability to the entire population:
|
| |
| Correct | Incorrect (TYPE 1 Error - a) |
| Incorrect (TYPE 2 Error - b) | Correct |
The “Correct” cases are the ones where the decisions taken on the samples are truly applicable to the entire population. The cases of errors arise when one decides to retain (or reject) the null hypothesis based on the sample calculations, but that decision does not really apply for the entire population. These cases constitute Type 1 ( alpha ) and Type 2 ( beta ) errors, as indicated in the table above.
Selecting the correct critical value allows eliminating the type-1 alpha errors or limiting them to an acceptable range.
Alpha denotes the error on the level of significance and is determined by the researcher. To maintain the standard 5% significance or confidence level for probability calculations, this is retained at 5%.
According to the applicable decision-making benchmarks and definitions:
A few more examples will demonstrate this and other calculations.
A monthly income investment scheme exists that promises variable monthly returns. An investor will invest in it only if they are assured of an average $180 monthly income. The investor has a sample of 300 months’ returns which has a mean of $190 and a standard deviation of $75. Should they invest in this scheme?
Let’s set up the problem. The investor will invest in the scheme if they are assured of the investor's desired $180 average return.
H 0 : Null Hypothesis: mean = 180
H 1 : Alternative Hypothesis: mean > 180
Identify a critical value X L for the sample mean, which is large enough to reject the null hypothesis – i.e. reject the null hypothesis if the sample mean >= critical value X L
P (identify a Type I alpha error) = P (reject H 0 given that H 0 is true),
This would be achieved when the sample mean exceeds the critical limits.
= P (given that H 0 is true) = alpha
Graphically, it appears as follows:
Taking alpha = 0.05 (i.e. 5% significance level), Z 0.05 = 1.645 (from the Z-table or normal distribution table)
= > X L = 180 +1.645*(75/sqrt(300)) = 187.12
Since the sample mean (190) is greater than the critical value (187.12), the null hypothesis is rejected, and the conclusion is that the average monthly return is indeed greater than $180, so the investor can consider investing in this scheme.
One can also use standardized value z.
Test Statistic, Z = (sample mean – population mean) / (std-dev / sqrt (no. of samples).
Then, the rejection region becomes the following:
Z= (190 – 180) / (75 / sqrt (300)) = 2.309
Our rejection region at 5% significance level is Z> Z 0.05 = 1.645.
Since Z= 2.309 is greater than 1.645, the null hypothesis can be rejected with a similar conclusion mentioned above.
We aim to identify P (sample mean >= 190, when mean = 180).
= P (Z >= (190- 180) / (75 / sqrt (300))
= P (Z >= 2.309) = 0.0084 = 0.84%
The following table to infer p-value calculations concludes that there is confirmed evidence of average monthly returns being higher than 180:
p-value | Inference |
less than 1% | supporting alternative hypothesis |
between 1% and 5% | supporting alternative hypothesis |
between 5% and 10% | supporting alternative hypothesis |
greater than 10% | supporting alternative hypothesis |
A new stockbroker (XYZ) claims that their brokerage fees are lower than that of your current stock broker's (ABC). Data available from an independent research firm indicates that the mean and std-dev of all ABC broker clients are $18 and $6, respectively.
A sample of 100 clients of ABC is taken and brokerage charges are calculated with the new rates of XYZ broker. If the mean of the sample is $18.75 and std-dev is the same ($6), can any inference be made about the difference in the average brokerage bill between ABC and XYZ broker?
H 0 : Null Hypothesis: mean = 18
H 1 : Alternative Hypothesis: mean <> 18 (This is what we want to prove.)
Rejection region: Z <= - Z 2.5 and Z>=Z 2.5 (assuming 5% significance level, split 2.5 each on either side).
Z = (sample mean – mean) / (std-dev / sqrt (no. of samples))
= (18.75 – 18) / (6/(sqrt(100)) = 1.25
This calculated Z value falls between the two limits defined by:
- Z 2.5 = -1.96 and Z 2.5 = 1.96.
This concludes that there is insufficient evidence to infer that there is any difference between the rates of your existing broker and the new broker.
Alternatively, The p-value = P(Z< -1.25)+P(Z >1.25)
= 2 * 0.1056 = 0.2112 = 21.12% which is greater than 0.05 or 5%, leading to the same conclusion.
Graphically, it is represented by the following:
Criticism Points for the Hypothetical Testing Method:
Hypothesis testing allows a mathematical model to validate a claim or idea with a certain confidence level. However, like the majority of statistical tools and models, it is bound by a few limitations. The use of this model for making financial decisions should be considered with a critical eye, keeping all dependencies in mind. Alternate methods like Bayesian Inference are also worth exploring for similar analysis.
Gregory J. Privitera. " Chapter 8: Introduction to Hypothesis Testing ." Statistics for Behavioral Sciences, Part III: Probability and the Foundations of Inferential Statistics. Sage Publications , pp. 4-5.
Rice University, OpenStax. " Introductory Statistics 2e: 7.1 The Central Limit Theorem for Sample Means (Averages) ."
Gregory J. Privitera. " Chapter 8: Introduction to Hypothesis Testing ." Statistics for Behavioral Sciences, Part III: Probability and the Foundations of Inferential Statistics. Sage Publications , pp. 5-6.
Gregory J. Privitera. " Chapter 8: Introduction to Hypothesis Testing ." Statistics for Behavioral Sciences, Part III: Probability and the Foundations of Inferential Statistics. Sage Publications , pp. 13.
Gregory J. Privitera. " Chapter 8: Introduction to Hypothesis Testing ." Statistics for Behavioral Sciences, Part III: Probability and the Foundations of Inferential Statistics. Sage Publications , pp. 6.
Gregory J. Privitera. " Chapter 8: Introduction to Hypothesis Testing ." Statistics for Behavioral Sciences, Part III: Probability and the Foundations of Inferential Statistics. Sage Publications , pp. 6-7.
Gregory J. Privitera. " Chapter 8: Introduction to Hypothesis Testing ." Statistics for Behavioral Sciences, Part III: Probability and the Foundations of Inferential Statistics. Sage Publications , pp. 10.
Gregory J. Privitera. " Chapter 8: Introduction to Hypothesis Testing ." Statistics for Behavioral Sciences, Part III: Probability and the Foundations of Inferential Statistics. Sage Publications , pp. 11.
Gregory J. Privitera. " Chapter 8: Introduction to Hypothesis Testing ." Statistics for Behavioral Sciences, Part III: Probability and the Foundations of Inferential Statistics. Sage Publications , pp. 7, 10-11.
Hypothesis testing allows us to make data-driven decisions by testing assertions about populations. It is the backbone behind scientific research, business analytics, financial modeling, and more.
This comprehensive guide aims to solidify your understanding with:
So let‘s get comfortable with making statements, gathering evidence, and letting the data speak!
Hypothesis testing is structured around making a claim in the form of competing hypotheses, gathering data, performing statistical tests, and making decisions about which hypothesis the evidence supports.
Here are some key terms about hypotheses and the testing process:
Null Hypothesis ($H_0$): The default statement about a population parameter. Generally asserts that there is no statistical significance between two data sets or that a sample parameter equals some claimed population parameter value. The statement being tested that is either rejected or supported.
Alternative Hypothesis ($H_1$): The statement that sample observations indicate statistically significant effect or difference from what the null hypothesis states. $H_1$ and $H_0$ are mutually exclusive, meaning if statistical tests support rejecting $H_0$, then you conclude $H_1$ has strong evidence.
Significance Level ($\alpha$): The probability of incorrectly rejecting a true null hypothesis, known as making a Type I error. Common significance levels are 90%, 95%, and 99%. The lower significance level, the more strict the criteria is for rejecting $H_0$.
Test Statistic: Summary calculations of sample data including mean, proportion, correlation coefficient, etc. Used to determine statistical significance and improbability under $H_0$.
P-value: Probability of obtaining sample results at least as extreme as the test statistic, assuming $H_0$ is true. Small p-values indicate strong statistical evidence against the null hypothesis.
Type I Error: Incorrectly rejecting a true null hypothesis
Type II Error : Failing to reject a false null hypothesis
These terms set the stage for the overall process:
1. Make Hypotheses
Define the null ($H_0$) and alternative hypothesis ($H_1$).
2. Set Significance Level
Typical significance levels are 90%, 95%, and 99%. Higher significance means more strict burden of proof for rejecting $H_0$.
3. Collect Data
Gather sample and population data related to the hypotheses under examination.
4. Determine Test Statistic
Calculate relevant test statistics like p-value, z-score, t-statistic, etc along with degrees of freedom.
5. Compare to Significance Level
If the test statistic falls in the critical region based on the significance, reject $H_0$, otherwise fail to reject $H_0$.
6. Draw Conclusions
Make determinations about hypotheses given the statistical evidence and context of the situation.
Now that you know the process and objectives, let’s apply this to some concrete examples.
We‘ll demonstrate hypothesis testing using Numpy, Scipy, Pandas and simulated data sets. Specifically, we‘ll conduct and interpret:
These represent some of the most widely used methods for determining statistical significance between groups.
We‘ll plot the data distributions to check normality assumptions where applicable. And determine if evidence exists to reject the null hypotheses across several scenarios.
Two sample t-tests determine whether the mean of a numerical variable differs significantly across two independent groups. It assumes observations follow approximate normal distributions within each group, but not that variances are equal.
Let‘s test for differences in reported salaries at hypothetical Company X vs Company Y:
$H_0$ : Average reported salaries are equal at Company X and Company Y
$H_1$ : Average reported salaries differ between Company X and Company Y
First we‘ll simulate salary samples for each company based on random normal distributions, set a 95% confidence level, run the t-test using NumPy, then interpret.
The t-statistic of 9.35 shows the difference between group means is nearly 9.5 standard errors. The very small p-value rejects the idea the salaries are equal across a randomly sampled population of employees.
Since the test returned a p-value lower than the significance level, we reject $H_0$, meaning evidence supports $H_1$ that average reported salaries differ between these hypothetical companies.
While an independent groups t-test analyzes mean differences between distinct groups, a paired t-test looks for significant effects pre vs post some treatment within the same set of subjects. This helps isolate causal impacts by removing effects from confounding individual differences.
Let‘s analyze Amazon purchase data to determine if spending increases during the holiday months of November and December.
$H_0$ : Average monthly spending is equal pre-holiday and during the holiday season
$H_1$ : Average monthly spending increases during the holiday season
We‘ll import transaction data using Pandas, add seasonal categories, then run and interpret the paired t-test.
Since the p-value is below the 0.05 significance level, we reject $H_0$. The output shows statistically significant evidence at 95% confidence that average spending increases during November-December relative to January-October.
Visualizing the monthly trend helps confirm the spike during the holiday months.
A single sample z-test allows testing whether a sample mean differs significantly from a population mean. It requires knowing the population standard deviation.
Let‘s test if recently surveyed shoppers differ significantly in their reported ages from the overall customer base:
$H_0$ : Sample mean age equals population mean age of 39
$H_1$ : Sample mean age does not equal population mean of 39
Here the absolute z-score over 2 and p-value under 0.05 indicates statistically significant evidence that recently surveyed shopper ages differ from the overall population parameter.
Chi-squared tests help determine independence between categorical variables. The test statistic measures deviations between observed and expected outcome frequencies across groups to determine magnitude of relationship.
Let‘s test if credit card application approvals are independent across income groups using simulated data:
$H_0$ : Credit card approvals are independent of income level
$H_1$ : Credit approvals and income level are related
Since the p-value is greater than the 0.05 significance level, we fail to reject $H_0$. There is not sufficient statistical evidence to conclude that credit card approval rates differ by income categories.
Analysis of variance (ANOVA) hypothesis tests determine if mean differences exist across more than two groups. ANOVA expands upon t-tests for multiple group comparisons.
Let‘s test if average debt obligations vary depending on highest education level attained.
$H_0$ : Average debt obligations are equal across education levels
$H_1$ : Average debt obligations differ based on education level
We‘ll simulate ordered education and debt data for visualization via box plots and then run ANOVA.
The ANOVA output shows an F-statistic of 91.59 that along with a tiny p-value leads to rejecting $H_0$. We conclude there are statistically significant differences in average debt obligations based on highest degree attained.
The box plots visualize these distributions and means vary across four education attainment groups.
Hypothesis testing forms the backbone of data-driven decision making across science, research, business, public policy and more by allowing practitioners to draw statistically-validated conclusions.
Here is a sample of hypotheses commonly tested:
Pharmaceuticals
Politics & Social Sciences
This represents just a sample of the wide ranging real-world applications. Properly formulated hypotheses, statistical testing methodology, reproducible analysis, and unbiased interpretation helps ensure valid reliable findings.
However, hypothesis testing does still come with some limitations worth addressing.
While hypothesis testing empowers huge breakthroughs across disciplines, the methodology does come with some inherent restrictions:
Over-reliance on p-values
P-values help benchmark statistical significance, but should not be over-interpreted. A large p-value does not necessarily mean the null hypothesis is 100% true for the entire population. And small p-values do not directly prove causality as confounding factors always exist.
Significance also does not indicate practical real-world effect size. Statistical power calculations should inform necessary sample sizes to detect desired effects.
Errors from Multiple Tests
Running many hypothesis tests by chance produces some false positives due to randomness. Analysts should account for this by adjusting significance levels, pre-registering testing plans, replicating findings, and relying more on meta-analyses.
Poor Experimental Design
Bad data, biased samples, unspecified variables, and lack of controls can completely undermine results. Findings can only be reasonably extended to populations reflected by the test samples.
Garbage in, garbage out definitely applies to statistical analysis!
Assumption Violations
Most common statistical tests make assumptions about normality, homogeneity of variance, independent samples, underlying variable relationships. Violating these premises invalidates reliability.
Transformations, bootstrapping, or non-parametric methods can help navigate issues for sound methodology.
Lack of Reproducibility
The replication crisis impacting scientific research highlights issues around lack of reproducibility, especially involving human participants and high complexity systems. Randomized controlled experiments with strong statistical power provide much more reliable evidence.
While hypothesis testing methodology is rigorously developed, applying concepts correctly proves challenging even among academics and experts!
We‘ve covered core concepts, Python implementations, real-world use cases, and inherent limitations around hypothesis testing. What should you master next?
Parametric vs Non-parametric
Learn assumptions and application differences between parametric statistics like z-tests and t-tests that assume normal distributions versus non-parametric analogs like Wilcoxon signed-rank tests and Mann-Whitney U tests.
Effect Size and Power
Look beyond just p-values to determine practical effect magnitude using indexes like Cohen‘s D. And ensure appropriate sample sizes to detect effects using prospective power analysis.
Alternatives to NHST
Evaluate Bayesian inference models and likelihood ratios that move beyond binary reject/fail-to-reject null hypothesis outcomes toward more integrated evidence.
Tiered Testing Framework
Construct reusable classes encapsulating data processing, visualizations, assumption checking, and statistical tests for maintainable analysis code.
Big Data Integration
Connect statistical analysis to big data pipelines pulling from databases, data lakes and APIs at scale. Productionize analytics.
I hope this end-to-end look at hypothesis testing methodology, Python programming demonstrations, real-world grounding, inherent restrictions and next level considerations provides a launchpad for practically applying core statistics! Please subscribe using the form below for more data science tutorials.
Dr. Alex Mitchell is a dedicated coding instructor with a deep passion for teaching and a wealth of experience in computer science education. As a university professor, Dr. Mitchell has played a pivotal role in shaping the coding skills of countless students, helping them navigate the intricate world of programming languages and software development.
Beyond the classroom, Dr. Mitchell is an active contributor to the freeCodeCamp community, where he regularly shares his expertise through tutorials, code examples, and practical insights. His teaching repertoire includes a wide range of languages and frameworks, such as Python, JavaScript, Next.js, and React, which he presents in an accessible and engaging manner.
Dr. Mitchell’s approach to teaching blends academic rigor with real-world applications, ensuring that his students not only understand the theory but also how to apply it effectively. His commitment to education and his ability to simplify complex topics have made him a respected figure in both the university and online learning communities.
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Present the findings in your results and discussion section. Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps. Table of contents. Step 1: State your null and alternate hypothesis. Step 2: Collect data. Step 3: Perform a statistical test.
If the biologist set her significance level \(\alpha\) at 0.05 and used the critical value approach to conduct her hypothesis test, she would reject the null hypothesis if her test statistic t* were less than -1.6939 (determined using statistical software or a t-table):s-3-3. Since the biologist's test statistic, t* = -4.60, is less than -1.6939, the biologist rejects the null hypothesis.
Step 1: Determine the hypotheses. The hypotheses are claims about the population mean, µ. The null hypothesis is a hypothesis that the mean equals a specific value, µ 0. The alternative hypothesis is the competing claim that µ is less than, greater than, or not equal to the .
The researchers write their hypotheses. These statements apply to the population, so they use the mu (μ) symbol for the population mean parameter.. Null Hypothesis (H 0): The population means of the test scores for the two groups are equal (μ 1 = μ 2).; Alternative Hypothesis (H A): The population means of the test scores for the two groups are unequal (μ 1 ≠ μ 2).
Worked Example. Imagine we have a textile manufacturer investigating a new yarn, which claims it has a thread elongation of 12 kilograms with a standard deviation of 0.5 kilograms. Using a random sample of 4 specimens, the manufacturer wishes to test the claim that the mean thread elongation is less than 12 kilograms.
Hypothesis Testing Step 1: State the Hypotheses. In all three examples, our aim is to decide between two opposing points of view, Claim 1 and Claim 2. In hypothesis testing, Claim 1 is called the null hypothesis (denoted " Ho "), and Claim 2 plays the role of the alternative hypothesis (denoted " Ha ").
In a hypothesis test, sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis, typically denoted with H0.
S.3 Hypothesis Testing. In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail. The general idea of hypothesis testing involves: Making an initial assumption. Collecting evidence (data).
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 population.". The null and alternative are always claims about the population. That's because the goal of hypothesis testing is to make inferences about a population based on a sample.
Below these are summarized into six such steps to conducting a test of a hypothesis. Set up the hypotheses and check conditions: Each hypothesis test includes two hypotheses about the population. One is the null hypothesis, notated as H 0, which is a statement of a particular parameter value. This hypothesis is assumed to be true until there is ...
Example 1: Biology. Hypothesis tests are often used in biology to determine whether some new treatment, fertilizer, pesticide, chemical, etc. causes increased growth, stamina, immunity, etc. in plants or animals. For example, suppose a biologist believes that a certain fertilizer will cause plants to grow more during a one-month period than ...
Hypothesis testing is a technique that helps scientists, researchers, or for that matter, anyone test the validity of their claims or hypotheses about real-world or real-life events in order to establish new knowledge. Hypothesis testing techniques are often used in statistics and data science to analyze whether the claims about the occurrence of the events are true, whether the results ...
Step 2: State the Alternate Hypothesis. The claim is that the students have above average IQ scores, so: H 1: μ > 100. The fact that we are looking for scores "greater than" a certain point means that this is a one-tailed test. Step 3: Draw a picture to help you visualize the problem. Step 4: State the alpha level.
Hypothesis testing is a procedure, based on sample evidence and probability, used to test claims regarding a characteristic of a population. A hypothesis is a claim or statement about a characteristic of a population of interest to us. A hypothesis test is a way for us to use our sample statistics to test a specific claim.
A teacher believes that 85% of students in the class will want to go on a field trip to the local zoo. The teacher performs a hypothesis test to determine if the percentage is the same or different from 85%. The teacher samples 50 students and 39 reply that they would want to go to the zoo. For the hypothesis test, use a 1% level of significance.
Statistics: Hypothesis Testing . A hypothesis is a claim made about a population. A hypothesis test uses sample data to test the validity of the claim. This handout will define the basic elements of hypothesis testing and provide the steps to perform hypothesis tests using the P-value method and the critical value method.
Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid. A null hypothesis and an alternative ...
Let's understand this with an example. A sanitizer manufacturer claims that its product kills 95 percent of germs on average. To put this company's claim to the test, create a null and alternate hypothesis. H0 (Null Hypothesis): Average = 95%. Alternative Hypothesis (H1): The average is less than 95%.
Hypothesis testing is a procedure, based on sample evidence and probability, used to test claims regarding a characteristic of a population. A hypothesis is a claim or statement about a characteristic of a population of interest to us. A hypothesis test is a way for us to use our sample statistics to test a specific claim.
Example \(\PageIndex{1}\) basics of hypothesis testing Suppose a manufacturer of the XJ35 battery claims the mean life of the battery is 500 days with a standard deviation of 25 days. You are the buyer of this battery and you think this claim is inflated.
Hypothesis testing is the process that an analyst uses to test a statistical hypothesis. ... Example of Hypothesis Testing . ... helping to avoid false claims and conclusions. Hypothesis testing ...
Hypothesis or significance testing is a mathematical model for testing a claim, idea or hypothesis about a parameter of interest in a given population set, using data measured in a sample set.
9.6: Additional Information and Full Hypothesis Test Examples. For each of the word problems, use a solution sheet to do the hypothesis test. The solution sheet is found in ... However, the variation among prices remains steady with a standard deviation of 20¢. A study was done to test the claim that the mean cost of a daily newspaper is $1.00 ...
Hypothesis testing allows us to make data-driven decisions by testing assertions about populations. It is the backbone behind scientific research, business ... Python code examples for t-tests, z-tests, chi-squared, and other methods; ... Hypothesis testing is structured around making a claim in the form of competing hypotheses, gathering data ...