24 ### Self-Esteem and Treatment for Bulimia: An AnalysisAn article indicates that self-esteem was an important predictor of those who will benefit from treatment for bulimia. The table provides the mean and standard deviation of self-esteem scores prior to treatment, at post-treatment, and during a follow-up.#### Table: Self-Esteem Scores| | Pretreatment | Posttreatment | Follow-up ||------------------------------|--------------|---------------|-----------|| Sample Mean (\(\bar{x}\)) | 20.2 | 26.7 | 27.8 || Standard Deviation (s) | 4.2 | 6.5 | 9.3 || Sample Size (n) | 22 | 22 | 21 |#### Instructions**(a) Hypothesis Testing**1. **Objective:** Determine whether there is sufficient evidence to conclude that the true pretreatment mean is less than 25. Use the significance level \(\alpha = 0.05\).2. **State the Hypotheses:** - \( H_0: \mu = 25 \) versus \( H_a: \mu < 25 \) - Variations in hypotheses are provided: - \( H_0: \mu = 25 \) versus \( H_a: \mu < 25 \) - \( H_0: \mu = 25 \) versus \( H_a: \mu > 25 \)3. **State the Test Statistic:** - \( t = \) (Round your answer to three decimal places.)4. **State the Rejection Region:** - If the test is one-tailed, enter NONE for the unused region and round answers to three decimal places. - \( t > \) - \( t < \)5. **State the Conclusion:** - Options include: - \( H_0 \) is not rejected or rejected based on whether there is evidence that the pretreatment mean is less than 25.**(b) Confidence Interval**- Construct a 95% confidence interval for the true posttreatment mean.- Round your answers to two decimal places.**(c) Conclusions Without Formal Statistical Testing**- Consider differences among the three sampled population means: - Options to determine visual conclusions: - The pretreatment mean

1. Denote μ as the true pretreatment mean.

An artidle indicates that self-esteem was an important predictor of those who will benefit from treatment for bulimia. The table gives the mean and standard deviation of self-esteem scores prior to treatment, at posttreatment, and during a follow-up. Pretreatment Posttreatment Follow-up Sample Mean x Standard Deviation s Sample Size n 20.2 26.7 27.8 4.2 6.5 9.3 22 22 21 A USE SALT (a) Use a test of hypothesis to determine whether there is sufficient evidence to conclude that the true pretreatment mean is less than 25. (Use a = 0.05.) State the null and alternative hypotheses. O Ho: H= 25 versus H,i = 25 O Ho: = 25 versus H: u < 25 %3D O H:H< 25 versus H: u> 25 O Ho: H< 25 versus H: u- 25 O H,: = 25 versus H: u= 25 State the test statistic. (Round your answer to three decimal places.) t- State the rejection region. (If the test is one-tailed, enter NONE for the unused region. Round your answers to three decimal places.) State the conclusion. O H, is not rejected. There is sufficient evidence to condude that the pretreatment mean is less than 25. O H, is rejected. There is sufficient evidence to conclude that the pretreatment mean is less than 25. O H, is rejected. There is insufficient evidence to condude that the pretreatment mean is less than 25. O O H, is not rejected. There is insufficient evidence to conclude that the pretreatment mean is less than 25. (b) Construct a 95% confidence interval for the true posttreatment mean. (Round your answers to two decimal places.) to (e) without a formal statistical test, what might you conclude about the differences among the three sampled population means represented by the results in the table? O The pretreatment mean looks the same as the other two means. O The pretreatment mean looks considerably larger than the posttreatment mean only. O The pretreatment mean looks considerably larger than the other two means. O The pretreatment mean looks considerably smaller than the posttreatment mean only. O The pretreatment mean looks considerably smaller than the other two means.