Question

Past experience has indicated that the true response rate is \(40 \%\) when individuals are approached with a request to fill out and return a particular questionnaire in a stamped and addressed envelope. An investigator believes that if the person distributing the questionnaire is 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 investigate this theory, a distributor is fitted with an eye patch. Of the 200 questionnaires distributed by this individual, 109 were returned. Does this strongly suggest that the response rate in this situation exceeds the rate in the past? State and test the appropriate hypotheses at significance level . 05 .

Short answer

Perform the calculation procedures as described in the steps. The null hypothesis will be rejected if the calculated p-value is less than 0.05, suggesting that the questionnaire response rate was significantly higher with the stigmatized investigator.

Step-by-step solution

Set the hypotheses

The null hypothesis, denoted \(H_0\), is that the response rate is the same as the previous response rate which is \(40 \%\). In statistical terms, if we represent the current response rate as \(p\), \(H_0: p=0.40\). The alternative hypothesis, denoted \(H_1\) or \(H_a\), is that the response rate is higher than \(40 \%\), or in statistical terms, \(H_1: p>0.40\).

Calculate Test Statistic

The test statistic for a hypothesis test of a proportion is a z-score (z). Therefore, compute the standard error (SE) and the z-score. The standard error is given by the formula: \(\sqrt{\frac{{p(1-p)}}{n}}\) where \(p\) is the sample proportion and \(n\) is the sample size. From the exercise, \(p = \frac{109}{200} = 0.545\), \(n = 200\), and hence the standard error (SE) is \(\sqrt{\frac{{0.545(1-0.545)}}{200}}\). Subsequently, the z-score can be calculated after identifying the population proportion \(\mu_p\) which is 0.4 (given in the problem). The z-score is calculated using the formula: \(z = \frac{p-\mu_p}{SE}\).

Find P-value

Using normalization tables or a calculator, look up the p-value corresponding to the calculated z-value. This p-value is the probability that we would observe such an extreme test statistic assuming the null hypothesis is true.

Compare P-value with significance level

If the calculated p-value is less than the significance level (0.05, in this case), reject the null hypothesis in favor of the alternative.

State the conclusion

Based on the analysis, if the null hypothesis is rejected, the conclusion is that there is significant evidence to suggest that the questionnaire response rate was higher with the stigmatized investigator. If the null hypothesis failed to be rejected, this indicates that the increase in response rate could be due to random chance.