Question

A CBS News Poll \(^{15}\) asked the question "Compared to your parents' generation, do you think in general your opportunities to succeed in life are better than theirs, about the same as theirs, or worse than theirs?" on three separate dates over a l0-year period. All surveys involved \(n=1048\) individuals. a. Is there a significant difference among the responses to this question over time? Use \(\alpha=.05\). b. If significant differences are found in part a, describe the nature of these differences.

Short answer

Based on the results of a CBS News Poll survey conducted on three separate dates over a 10-year period, create a short answer determining if there is a significant difference among the responses to the question of respondents comparing their opportunities to succeed in life with their parents' generation, using a significance level of 𝛼=0.05.

Step-by-step solution

Set up the hypothesis

To determine if there is a significant difference among the responses over time, we will use a repeated measures ANOVA. Our null hypothesis (H0) is that the means of the responses are equal across all time points, and the alternative hypothesis (Ha) is that the means are not equal: H0: The means of the responses are equal across all time points. Ha: The means of the responses are not equal at least at one time point.

Perform the repeated measures ANOVA

To perform the repeated measures ANOVA, we will need the data from the three surveys. Unfortunately, this exercise doesn't provide the raw data or the summary statistics, which makes it impossible to perform the repeated measures ANOVA. However, we can still provide guidelines on how to do this with the necessary data. To perform the repeated measures ANOVA, you can either use statistical software, like SPSS or R, or an online calculator. Input the data for the three groups (time points), and make sure to obtain the F-value, degrees of freedom (df) within groups and between groups, and the p-value.

Compare the p-value to the significance level

Once you have the p-value from the repeated measures ANOVA, compare it to the significance level, \(\alpha = 0.05\). If the p-value is less than \(\alpha\), reject the null hypothesis and conclude that there is a significant difference among the responses across time points. Otherwise, fail to reject the null hypothesis and conclude that there is no significant difference among the responses over time.

Perform post hoc tests (if necessary)

If the null hypothesis is rejected and significant differences are found, perform post hoc tests to determine which pairs of means (time points) are significantly different. This step is important to give a detailed description of how the responses differ over time. Conduct pairwise comparisons using a suitable post hoc test, such as the Bonferroni correction or Tukey's HSD test.

Describe the results and draw conclusions

Based on the results of the repeated measures ANOVA and, if applicable, post hoc tests, describe the nature of the significant differences found. Report the F-value, degrees of freedom, and p-value, and indicate which time points have statistically significant differences in their means. Make sure to interpret these results in the context of the survey question about the opportunities to succeed in life compared to the parents' generation. Note: Since we don't have the data or the summary statistics for this exercise, we cannot provide specific results or conclusions. However, following these steps with the necessary data will allow you to solve the problem and reach a conclusion.

Key concepts

Hypothesis Testing

Hypothesis testing is a crucial statistical method used to determine if there is enough evidence in a sample of data to infer a condition for a population. The process generally involves

• Formulating a null hypothesis (H0), which posits no effect or no difference,
• And an alternative hypothesis (Ha) that indicates the presence of an effect or a difference.

In the context of the CBS News Poll, the null hypothesis was that the means of the responses regarding life opportunities are equal across all time points, implying no change in perception over the survey periods. Conversely, the alternative hypothesis suggests that there is a difference in means at least at one of the time points. Hypothesis testing helps us either reject the null hypothesis with sufficient evidence or fail to reject it if the evidence is insufficient.

Significance Level

The significance level in hypothesis testing is a threshold set by researchers before an analysis, denoted by the Greek letter alpha (\( \alpha \)). It represents the probability of rejecting the null hypothesis when it is true, also known as the Type I error rate. For most studies, a common significance level is 0.05.

In our CBS News Poll example, the significance level was defined as \( \alpha = 0.05 \).

• This means that there is a 5% risk of concluding that there is a difference in perceptions when there is none in the actual population.
• If the calculated p-value from the repeated measures ANOVA is less than this threshold, the results are deemed statistically significant, justifying rejection of the null hypothesis.

This process helps control inconsistency in decision-making due to random sampling variability.

P-Value

The p-value is crucial in the context of hypothesis testing as it helps assess the strength of the evidence against the null hypothesis. It is calculated after conducting an analysis like ANOVA and indicates the probability of observing test results at least as extreme as the results actually obtained, assuming that the null hypothesis is true.

In a repeated measures ANOVA, performing such analysis yields a p-value:

• If the p-value is lower than the predetermined significance level (e.g., 0.05), it signals strong evidence against the null hypothesis.
• It suggests that we should reject the null hypothesis, indicating that there is a significant difference in the response means over time.
• If the p-value is higher than the significance level, we cannot reject the null hypothesis, meaning there is no statistical evidence of a significant difference.

Thus, interpreting the p-value helps decide if the observed differences or relationships in the data are statistically significant.

Post Hoc Tests

Post hoc tests are conducted following an ANOVA when the null hypothesis is rejected due to a significant p-value, indicating that at least one group mean is statistically different from the others. These tests aim to determine precisely which groups are significantly different.

For the CBS News Poll problem, several post hoc tests could be employed, like:

• Bonferroni correction, which adjusts the significance level to account for multiple pairwise comparisons, thereby reducing the chances of Type I errors, or
• Tukey's Honest Significant Difference (HSD) test, which is specifically designed to control false discovery rates when comparing all possible pairs of time points.

Post hoc analysis is vital for understanding the pattern of differences among survey responses over the years, providing insightful conclusions on how perceptions have evolved regarding success opportunities in comparison to past generations.