1. Observational Study or Experiment?
(a) Was this an observational study or an experiment?
Section 24.3 Investigation 5.2: Teaching Morals
Exercises The Study
Lee et al. (2014) examined whether classic stories about moral behavior influence whether children lie. They studied three moral stories (Pinocchio, The Boy Who Cried Wolf, and George Washington and the Cherry Tree) and one control story (The Tortoise and the Hare). Children ages 3-7 completed a temptation-resistance task and then were asked whether they had peeked. Suppose the results for children who peeked were:
| Tortoise and Hare (control) | George Washington | Pinocchio | Boy Who Cried Wolf | Total | |
|---|---|---|---|---|---|
| Confessed | 20 | 22 | 13 | 16 | 71 |
| Did not (lied) | 44 | 22 | 31 | 30 | 127 |
| Total | 64 | 44 | 44 | 46 | 198 |
2. Identify Variables.
(b) Identify the explanatory variable and response variable. Classify each as quantitative or categorical.
3. State Hypotheses.
(c) State appropriate null and alternative hypotheses for this research question in symbols and/or words. Define any symbols used.
The following table displays both observed and expected counts:
| Tortoise and Hare (control) | George Washington | Pinocchio | Boy Who Cried Wolf | Total | |
|---|---|---|---|---|---|
| Confessed | 20 (22.95) | 22 (15.78) | 13 (15.78) | 16 (16.49) | 71 |
| Did not (lied) | 44 (41.05) | 22 (28.22) | 31 (29.22) | 30 (29.51) | 127 |
| Total | 64 | 44 | 44 | 46 | 198 |
To model this randomized experiment under the null, we shuffle confession outcomes among story groups while preserving the group sizes.
4. Simulation with Applet.
(d) Use the Analyzing Two-way Tables applet and enter the two-way table (without totals, using one-word labels). Use story as explanatory variable, run a large number of shuffles, choose chi-squared as the statistic, and compute an empirical p-value.
5. Overlay Chi-square Model.
(e) Overlay the chi-square distribution (df = 3). Does it appear to model the simulated null distribution reasonably well? Explain.
When data arise from a randomized experiment, the chi-squared model is generally appropriate if at least 80% of expected counts are at least 5 and all are at least 1.
6. Technology Output.
(f) Use technology to carry out the chi-squared test. Report the chi-squared statistic, degrees of freedom, and p-value.
7. Largest Cell Contributions.
(g) Examine chi-squared cell contributions (or residuals). Which cell(s) contribute most to the chi-squared sum? Compare observed to expected counts in those cells. What do these comparisons suggest about which stories may make children more likely to confess?
Study Conclusions.
The observed confession proportions were 0.313 (control), 0.500 (George Washington), 0.295 (Pinocchio), and 0.348 (Boy Who Cried Wolf). Although the pattern suggests higher confession in the George Washington group, a chi-squared test with these data is not statistically significant (\(\chi^2 = 5.202\text{,}\) \(p = 0.158\)). Thus these data do not provide convincing evidence that story type influences the probability of confession in this setting.
8. Practice Problem 5.2.
Suppose we compare only George Washington to the control (Tortoise and the Hare).
(a) Would a chi-squared test be valid for these data? Explain.
(b) What are the degrees of freedom?
(c) Use technology to calculate the chi-squared statistic and p-value. What do you conclude?
(d) Use technology for a two-sample \(z\)-test. How does the two-sided p-value compare with part (c)?
(e) Use Fisher’s Exact Test. How does that p-value compare?
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