Question

Past experience is that when individuals are approached with a request to fill out and return a particular questionnaire in a provided stamped and addressed envelope, the response rate is \(40 \%\). An investigator believes that if the person distributing the questionnaire were stigmatized in some obvious way, potential respondents would feel sorry for the distributor and thus tend to respond at a rate higher than \(40 \%\). To test this theory, a distributor wore an eye patch. Of the 200 questionnaires distributed by this individual, 109 were returned. Does this provide evidence that the response rate in this situation is greater than the previous rate of \(40 \%\) ? State and test the appropriate hypotheses using a significance plevel of 0.05 .

Short answer

The result of the test provides strong evidence to reject the null hypothesis and conclude that the response rate is indeed higher when the distributor wears an eye patch, at a significance level of 0.05.

Step-by-step solution

Set Up the Hypotheses

The null hypothesis (H0) states that the response rate does not change, i.e., the response rate is equal to 40%. The alternative hypothesis (H1) states that the response rate is higher than 40% with an eye patch.\[ H0: p = 0.4 \]\[ H1: p > 0.4 \]

Calculate the Test Statistic

The test statistic for a hypothesis test about a proportion is a z-score (z). \[ z = \frac{(p_1 - p_o) }{\sqrt{\frac{(p_o(1 - p_o))}{n}}\] which equals to \[ z = \frac{(0.545 - 0.4) }{\sqrt{\frac{(0.4(1 - 0.4))}{200}}\] After calculating, the z-score is approximately 4.36.

Determine the P-value

The P-value is the probability that a z-score is more than the absolute value of the test statistic, considering the null hypothesis is true. For this two-sided test, the P-value is the area to the right of our test statistic (z = 4.36). Using standard statistical software or Z-tables, the P-value is approximately 0.00001.

Make the Decision

Compare the P-value to the significance level and make the decision about the null hypothesis. The P-value is less than the significance level of 0.05, so the decision is to reject the null hypothesis.

Draw a Conclusion

Given the result of the test, there is enough evidence to support the claim that the response rate is greater than 40% when the distributor wears an eye patch.