ONE-SAMPLE TEST OF A HYPOTHESIS

A.  Overview of One-Sample Hypothesis Testing
B.  Step-By-Step Instructions for Performing a One-Sample Hypothesis Test in Excel
C.  Interpreting the Results of the Test

A.  Overview of One-Sample Hypothesis Testing

In statistical terms, a hypothesis is a statement about a population parameter and hypothesis testing is simply a test of the statement about the population parameter.  For example, suppose that a new weight loss program claims that participants lose at least 5 pounds by participating in the program.  In this example, the population is all of the participants in the weight loss program and the population parameter of interest is the population's mean or average weight loss.  "The mean weight loss of participants is greater than 5 pounds" is a hypothesis that can be tested using a statistical analysis.  Step-by-step instructions for performing such a test are described in the following section. 

B.  Step-By-Step Instructions for Performing a One-Sample Hypothesis Test in Excel

Refer to the Speedy Oil Change example discussed in the Descriptive Statistics section.  In the Speedy Oil Change example, the oil change company claims that it will change customer's oil in less than 30 minutes on average.  However, several complaints have been filed from customers stating that their oil change took longer than 30 minutes and upper-level management at Speedy Oil Change headquarters has requested that you, the manager,  investigate the complaints.  You have completed the Descriptive Statistics analysis in Excel and are ready to further investigate the validity of the customer complaints.

Step 1.  The first step is to state the hypothesis to be tested, called the null hypothesis, and the alternative hypothesis.  The null and alternative hypothesis are denoted Ho and Ha, respectively.  If a claim is being tested, as in this example, the claim is stated in the alternative hypothesis. For this example, the null and alternative hypotheses are:

Step 2.  Select the level of significance to be used in the test.  The level of significance is the probability of rejecting the null hypothesis when it is true. Common significance levels are .10, .05, and .01.  Suppose you chose a .05 level of significance, meaning there is a 5% chance that you will reject the null hypothesis when it is true.

Step 3.  Select the test statistic that is appropriate for this test. 

In general, you will need to decide between using a z test statistic or a t test statistic. A z test statistic is based on the normal distribution and the t test statistic is based on the t-distribution Your choice comes down to how large your sample size is.  Using the t distribution, you locate your critical value based on the degrees of freedom (n-1 for a one-sample test) and the significance level you chose in step 2 (see t table)When n > 30, it is acceptable to use the z value. Why? Remember, as the sample size increases, all sampling distributions converge towards the normal (z).  This is the Central Limit Theorem. The generally accepted rule of thumb is that, when n > 30, any sampling distribution can be assumed to be approximately normal. That is why you can use the z values when n > 30.  For proof of this, notice that the critical ts, when df=infinity are exactly the same as the critical zís.

In the Speedy Oil Change example, the sample size is 36, so it is acceptable to use a z test statistic.

The z test statistic for this example is shown below.

is the population mean, s is the sample standard deviation, and n is the number of observations in the sample. Note that if you were performing a  t test, you would use a similar formula and proceed in the same manner:

Step 4.  Determine the rejection region.  The rejection region defines the conditions under which the null hypothesis is rejected.  The rejection region depends on the alternative hypothesis stated in Step 1 and the significance level chosen in Step 2.  In this example, the rejection region will be found in the lower tail of the standard normal distribution.  Why?  The rejection region for a one-tailed test can always be located by looking at the direction of the inequality in the alternative hypothesis.  Since the alternative hypothesis states μ < 30, the rejection region is in the lower tail of the standard normal distribution.  We use the standard normal distribution in this case because we are using the z test statistic which has a standard normal distribution.  As stated above, the rejection region also depends on the significance level chosen in Step 2.  In this example, a .05 significance level indicates that the rejection region is defined by all the z values that are less than -1.645, which is called the critical value (notice that, if you had chosen to use the t-distribution rather than the z, your critical values would have been -1.69.  As you can see, it makes no practical difference whether you use the z or the t when you have large sample sizes. The critical value is the dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.  This critical value is found by finding the z value (from a standard normal table) that cuts off the area equal to .05 in the lower tail of the standard normal distribution.

Step 5.  Perform the hypothesis test.  First, the z test statistic needs to be calculated.  The Speedy Oil Change results from the Descriptive Statistics analysis are shown below.

 

The Descriptive Statistics analysis results indicate that

Next, we need to compare the z test statistic of -3.5874 to the critical value of -1.645.  Because our test statistic is smaller than the critical value and, therefore falls into the rejection region, we reject the null hypothesis in favor of the alternative hypothesis.

C.  Interpreting the Results of the Test

As the manager of Speedy Oil Change, you were asked to investigate the customer complaints that oil changes at Speedy Oil Change take more than 30 minutes on average.  You formulated a hypothesis, defined a significance level, determined the appropriate test statistic and rejection region, and then performed the hypothesis test.  The results of the test indicate that the null hypothesis should be rejected in favor of the alternative.  The null hypothesis is that the mean oil change time is more than 30 minutes.  Because this hypothesis is rejected, you can report to upper-level management that with 95% confidence [100(1 - significance level)] the mean oil change time is less than 30 minutes. 


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Last Modified: 08/05/03
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