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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.

Step by step solution

01

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.

02

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.

03

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.

04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

In hypothesis testing, the null hypothesis, often denoted as \(H_0\), is a statement of no effect or no difference. It represents the assumption that any observed difference is due to sampling or experimental error.
For example, in the context of comparing the percentage of correct answers between two groups—those with a university degree and those without—the null hypothesis would claim that both groups perform equally well.
Specifically, \(H_0: p_1 = p_2\), where \(p_1\) and \(p_2\) stand for the proportions of correct responses in each group. By formulating a null hypothesis, we have a clear statement to test against real-world data.

Alternative Hypothesis

The alternative hypothesis, denoted as \(H_a\), contradicts the null hypothesis. It suggests that there is a measurable effect or difference.
For the exercise at hand, the alternative hypothesis posits that there is a significant difference in the proportions of correct answers between the two groups.
Mathematically, \(H_a: p_1 eq p_2\).
The alternative hypothesis is what researchers often hope to prove, as it indicates a new finding or insight that challenges the status quo.

Randomization Distribution

To test hypotheses, we need to understand how data could behave under the null hypothesis.
Randomization distribution helps create this understanding by simulating different scenarios where the null hypothesis holds.
In the exercise, the difference in proportions for correct responses is observed. This difference is simulated multiple times to see if such a result is rare or common, assuming there's no real difference.
Using statistical software, one can generate many samples and recompute differences, crafting a distribution that represents how often certain outcomes occur if \(H_0\) is true.

P-value

The p-value quantifies how well the observed data align with the null hypothesis. It represents the probability of obtaining results at least as extreme as the observed ones, assuming the null is true.
If the computed p-value is low (commonly less than 0.05), it indicates that such an outcome would be rare under the null hypothesis.
In our exercise, the p-value comes from the randomization distribution and signifies whether the difference in proportions is statistically significant or due to random chance.
This small number reveals whether we should question or retain our initial assumptions.

Statistical Significance

Statistical significance helps determine whether the results of our hypothesis test are meaningful.
When the p-value is below a predefined threshold (often 0.05), the results are considered statistically significant. This means the chance of observing such data if \(H_0\) were true is very low.
In the exercise, if the p-value points to statistical significance, this indicates a true difference in correct response rates between groups.
It's a valuable tool for researchers to move from observation to solid conclusion.