Question

Exercise 4.113 refers to a survey used to assess the ignorance of the public to global population trends. A similar survey was conducted in the United Kingdom, where respondents were asked if they had a university degree. One question asked, "In the last 20 years the proportion of the world population living in extreme poverty has \(\ldots, "\) and three choices 2) "remained more or were provided: 1\()^{\text {6i }}\) increased" less the same," and 3) "decreased." Of 373 university degree holders, 45 responded with the correct answer: decreased; of 639 non-degree respondents, 57 responded with the correct answer. \({ }^{35}\) We would like to test if the percent of correct answers is significantly different between degree holders and non- degree holders. (a) What are the null and alternative hypotheses? (b) Using StatKey or other technology, construct a randomization distribution and compute the p-value. (c) State the conclusion in context.

Short answer

The solutions to the exercise are: \n(a) Null hypothesis (H0): p1 = p2. Alternative hypothesis (Ha): p1 ≠ p2.\n(b) To get the p-value, first establish the observed difference between the two proportions. Then, generate a randomization distribution and find the proportion of simulated differences that are as considerable as, or more significant than, the observed difference. This is the p-value. \n(c) The conclusion will depend on the calculated p-value and should be stated in context of the problem.

Step-by-step solution

Null and Alternative Hypotheses

To solve this problem, begin by understanding the scenario and formulating the null and alternative hypotheses. The null hypothesis (H0) is that there's no difference in the percent of correct answers between degree and non-degree holders. So, H0: p1 = p2, where p1 is the proportion of degree holders who gave the correct answer and p2 is the proportion of non-degree holders who gave the correct answer. The alternative hypothesis (Ha) is that there is a significant difference in the percentage of correct answers between degree and non-degree holders. So, Ha: p1 ≠ p2.

Construct a Randomization Distribution

To construct the randomization distribution, the difference between the proportions of correct answers for both groups is required. The proportion of correct responses for degree holders is 45/373 = 0.1206, while the proportion for non-degree holders is 57/639 = 0.0892. The observed difference is 0.1206 - 0.0892 = 0.0314 (proportion for degree holders - proportion for non-degree holders). Using a statistical software or an online tool that performs randomization tests for two proportions, simulate several random samples sharing the same overall proportion of success as in the collected data. For each simulation, calculate and record the difference in proportion.

Compute the p-value

The p-value is the probability of obtaining the observed difference or a more substantial difference if the null hypothesis is true. Using the randomization distribution generated in the previous step, calculate the proportion of simulated differences that are at least as extreme as the observed difference of 0.0314. This proportion is the p-value. If using software, this computation is typically done automatically.

Conclusion

The conclusion depends on the calculated p-value. If p-value < α (commonly 0.05), reject the null hypothesis, suggesting that the difference in proportions is statistically significant. On the other hand, if the p-value > α, not sufficient evidence is found to reject the null hypothesis. The conclusion needs to be stated in the context of the problem, i.e., there is/is no statistically significant difference in the percent of correct answers between degree holders and non-degree holders.