Question

The report "Audience Insights: Communicating to Teens (Aged 12-17)" (www.cdc.gov, 2009) described teens' attitudes about traditional media, such as TV, movies, and newspapers. In a representative sample of American teenage girls, \(41 \%\) said newspapers were boring. In a representative sample of American teenage boys, \(44 \%\) said newspapers were boring. Sample sizes were not given in the report. a. Suppose that the percentages reported had been based on a sample of 58 girls and 41 boys. Is there convincing evidence that the proportion of those who think that newspapers are boring is different for teenage girls and boys? Carry out a hypothesis test using \(\alpha=.05\) b. Suppose that the percentages reported had been based on a sample of 2000 girls and 2500 boys. Is there convincing evidence that the proportion of those who think that newspapers are boring is different for teenage girls and boys? Carry out a hypothesis test using \(\alpha=.05\). c. Explain why the hypothesis tests in Parts (a) and (b) resulted in different conclusions.

Short answer

The results of two hypothesis tests might be different due to the different sample sizes. Larger samples lead to more accurate estimation of the population, and could detect even a small difference between the two proportions, while smaller samples might fail to capture such a difference. Without the actual data, more specific conclusion can't be provided.

Step-by-step solution

State the Hypotheses

In both cases we're looking to see if there's a difference in proportion between two populations - teenage girls and boys. Therefore, the null hypothesis \(H_0: p_1 - p_2 = 0\), and the alternative hypothesis \(H_A: p_1 - p_2 \neq 0\). Where \(p_1\) is the proportion of girls who find newspapers boring and \(p_2\) is the proportion of boys who find newspapers boring.

Conduct the Hypothesis Tests

We carry out the first hypothesis test, using the sample sizes of 58 girls and 41 boys. From the information given, we know that \(x_1 = 0.41 * 58 = 24\) girls and \(x_2 = 0.44 * 41 = 18\) boys find newspapers boring. Then we calculate the sample proportions and their difference - \(\hat{p}_1 = x_1 / n_1\), \(\hat{p}_2 = x_2 / n_2\), and \(\hat{p} = x / n\) - and the standard error \(SE = \sqrt{\hat{p}*(1-\hat{p})*(1/n_1 + 1/n_2)}\). Then, we calculate the z-score \(z = (\hat{p}_1 - \hat{p}_2) / SE\). After that, we compute the p-value using a standard normal probability table. We conduct the similar steps for the second hypothesis test, using the sample sizes of 2000 girls and 2500 boys.

Decision and Conclusion

For each hypothesis test, if the computed p-value is less than \(\alpha = .05\), we reject the null hypothesis and conclude that there is a significant difference between teenage girls and boys in terms of finding newspapers boring. Otherwise, we fail to reject the null hypothesis. Then we compare the results from the two tests and explain why they differ, considering the impact of the sample sizes.

Key concepts

Proportion Difference

In statistics, when comparing two groups, we are often interested in the difference between their proportions. This can tell us whether there's a notable deviation in outcomes, attitudes, or behaviors between the two groups.

In the context of the provided exercise, we calculate the proportion of teenage girls and boys who find newspapers boring and then determine the difference between these two proportions. If this difference is significantly different from zero, it may suggest that gender plays a role in the perception of newspapers. To find this proportion difference, we use the formula \( \hat{p}_1 - \hat{p}_2 \), where \( \hat{p}_1 \) represents the sample proportion of girls, and \( \hat{p}_2 \) represents the sample proportion of boys. Accurately determining the proportion difference is critical for effective hypothesis testing.

Null Hypothesis

The null hypothesis, symbolized as \( H_0 \), is a fundamental concept in hypothesis testing. It is a statement that there is no effect or no difference, and it serves as the starting assumption for statistical tests.

In both scenarios of our exercise, the null hypothesis posits that there is no difference in the proportion of girls and boys who believe newspapers are boring, formally stated as \( H_0: p_1 - p_2 = 0 \). We assume this to be true unless we find sufficient evidence to the contrary. The null hypothesis is what we attempt to reject through our hypothesis test, and it is a crucial step in determining the statistical significance of the observed data.

Alternative Hypothesis

The alternative hypothesis, denoted as \( H_A \), represents the opposite of the null hypothesis and indicates the presence of an effect or a difference. In simple terms, it's what we suspect might be true instead of the null hypothesis.

For the exercise, the alternative hypothesis states that there is a difference in the proportion of teenage girls and boys who find newspapers boring, indicated as \( H_A: p_1 - p_2 eq 0 \). It's the hypothesis that we are trying to support by using evidence from our sample data. The alternative hypothesis is what leads us to examine the p-value and assess whether our findings are statistically significant.

P-Value

The p-value is a vital term in hypothesis testing, representing the probability of obtaining a result at least as extreme as the one observed, given that the null hypothesis is true. In essence, it helps us determine the significance of our findings.

After calculating the test statistic—often a z-score in the case of proportion differences—we look up this value in a standard normal distribution to find the corresponding p-value. In our exercise, if this p-value is less than our alpha level of \( \alpha = 0.05 \), we have strong evidence against the null hypothesis and can consider our findings significant. The lower the p-value, the higher the significance of our results in supporting the alternative hypothesis.

Statistical Significance

Statistical significance is the likelihood that the relationship or difference detected in a sample also exists in the wider population and was not due to chance. It provides assurance that our findings are meaningful and not just a fluke.

By using the threshold of \( \alpha = 0.05 \) in the exercise, we define a 5% risk as an acceptable level for making a Type I error—that is, wrongly rejecting the null hypothesis. If the p-value is below this alpha level, our results are deemed statistically significant. A vital consideration—which emerges in the exercise and its differing outcomes—is how sample size affects statistical significance. Larger samples tend to yield more reliable and detectable results, therefore enhancing statistical significance.