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Hypothesis Testing for Proportions Study Guide (page 3)

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Updated on Oct 5, 2011

Step 4: Decide Whether or Not to Reject the Null Hypothesis

Before beginning the study, the significance level of the test is set. The significance level a is the largest acceptable probability of a type I error. If the p-value is less than α, the null hypothesis is rejected; otherwise, the null is not rejected. In statistical hypothesis testing, we control the probability of a type I error. We often do not know the probability of a type II error. If we reject the null hypothesis, we know the probability of making an error. If we do not reject the null hypothesis, we do not know the probability of having made an error. This is why we would not accept the null hypothesis; we only fail to reject it. In science, a significance level of α = 0.05 is generally the standard. That is, we reject H0 if p < α = 0.05. A p-value less than α = 0.01 is usually viewed as highly significant. However, the researcher can set the significance level that is most appropriate for his or her study. Once the decision is made to reject or not to reject the null hypothesis, it is important to state what conclusions have been drawn.

Step 5: State Conclusions in the Context of the Study

Statistical tests of hypotheses are conducted to determine whether or not sufficient evidence exists to reject the null hypothesis in favor of the alternative hypothesis. Once the decision is made to reject or not to reject the null hypothesis, it is important to state what conclusions have been drawn.

Example of Conducting Hypothesis Tests on Proportions

Example:

A sleep researcher believes that most (more than half) of all college students take naps during the afternoon or early evening. He randomly selects 60 students from a large university. He asks each selected student, "Do you regularly take naps during the afternoon or early evening?" Of the 60 students, 34 responded yes. Does sufficient statistical evidence exist to conclude that more than half of the students at this university regularly take afternoon or early evening naps?

Solution:

We will follow the five steps of hypothesis testing.

Step 1: Specifying the Hypotheses

The parameter of interest in the study is p, the proportion of students at this university who regularly take afternoon or early evening naps. The sleep researcher believes more than 50% of the students take afternoon or early evening naps regularly, so this is the alternative hypothesis. Thus, the set of hypotheses to be tested are:

H0: p = 0.50

Ha: p > 0.50

Note: Equality appears in H0. This is necessary to know the distribution of the test statistic under H0. Also, a one-sided alternative (p > 0.50) is used instead of a two-sided alternative (p ≠ 0.50). The reason for this is that the sleep scientist wants to conclude that more than half of the students take naps, not that some proportion other than half take naps (the meaning of the two-sided alternative here).

Step 2: Verify Necessary Conditions for a Test and, if Satisfied, Construct the Test Statistic

The sleep researcher took a random sample of students from those attending the university, so the first condition for inference is satisfied. To check the second condition, we have and . Thus, the second condition for inference is also satisfied. Note that

The test statistic is

.

Step 3: Find the p-Value Associated with the Test Statistic

If the null hypothesis is true, the test statistic has an approximate standard normal distribution. The p-value is the probability of determining the probability of observing a value as extreme or more extreme as zT from a random selection of the standard normal distribution. For this study, there would have been more support for the alternative if the sample proportion of nap-taking students had been greater than the observed = 0.0567. This would have led to a larger value of zT. Thus, the p-value is:

p = p(z > zT)
  = p(z > 1.03)
  =1 – p(z ≤ 1.03)
  = 1 – 0.8485
  = 0.1515

 

The 0.8485 was obtained from the standard normal table (see Figure 16.4).

Figure 16.4

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