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The response rate is a vital concept in statistical studies, particularly when measuring how many participants in a sample return feedback or data, such as questionnaires or surveys. It is given as a percentage, signifying the proportion of returned questionnaires compared to those distributed. In our original exercise, the previous known response rate is 40%. This percentage indicates that out of every 100 people approached, 40 are expected to return the completed questionnaire.

Response rates are crucial for ensuring the reliability of data in research. A higher response rate typically suggests more robust findings because it reduces the potential for non-response bias. Non-response bias can occur when the views of respondents differ significantly from those who did not respond, potentially skewing results.

In this exercise, the investigator is testing if altering conditions, such as the distributor wearing an eye patch, impacts the response rate significantly. Specifically, they were examining if the response rate would increase beyond the established 40% norm.

The one-sample proportion z-test is a statistical method used to determine if a sample proportion is significantly different from a known population proportion. In this context, it is employed to see if the response rate of 54.5% (from the sample of distributed questionnaires) is significantly greater than the known rate of 40%.

The test involves several steps: formulating the null and alternative hypotheses, calculating the test statistic, and then interpreting the findings.

• In step one, the null hypothesis (\( H_0 \)) states that the response rate is equal to the population proportion (40%), while the alternative hypothesis (\( H_a \)) proposes that it is greater.
• The test statistic (\( z \)) is calculated using the formula: \( z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}} \) , where \( \hat{p} \) is the sample proportion (0.545), \( p \) is the population proportion (0.4), and \( n \) is the sample size (200).

The outcome of this calculation was a z-value of approximately 4.46, which is then compared against a standard normal distribution to infer statistical significance.

The significance level, often denoted as \( \alpha \), is a threshold set by the researcher to determine when to reject the null hypothesis. It reflects the degree of risk one is willing to take for a Type I error, which occurs when the null hypothesis is true, but we incorrectly reject it.

In hypothesis testing, if the p-value calculated from the test statistic is smaller than the significance level, it suggests that the observed effect is unlikely due to sampling variability alone. This prompts the rejection of the null hypothesis in favor of the alternative.

In our specific exercise, the significance level is set at 0.05, meaning there is a 5% risk of making a Type I error. Given the calculated p-value of approximately 0.000004, which is well below 0.05, the results are deemed statistically significant. Thus, the findings support the hypothesis that the response rate exceeds the previously known rate of 40% when the distributor is stigmatized. This level of significance provides strong evidence that the observed increase is not just by chance.