1. **Problem Statement:** We want to test the claim that retaking the SAT increases the score on average by more than 30 points. 2. **Given Data:** - Sample size $n = 200$ - Sample mean difference $\bar{x} = 33$ - Sample standard deviation $s = 28.2843$ - Hypothesized mean difference $\mu_0 = 30$ 3. **Hypotheses:** - Null hypothesis $H_0: \mu \leq 30$ - Alternative hypothesis $H_a: \mu > 30$ 4. **Test Statistic Formula:** $$ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} $$ This formula calculates how many standard errors the sample mean is from the hypothesized mean. 5. **Calculate the test statistic:** $$ t = \frac{33 - 30}{28.2843 / \sqrt{200}} = \frac{3}{28.2843 / 14.1421} = \frac{3}{2} = 1.5 $$ 6. **Critical values and decision rule:** - For significance level $\alpha = 0.10$ and degrees of freedom $df = 199$, the critical value $t_{0.10,199} \approx 1.29$ - For significance level $\alpha = 0.01$ and $df = 199$, the critical value $t_{0.01,199} \approx 2.36$ 7. **Test at $\alpha = 0.10$:** - Since $t = 1.5 > 1.29$, we reject $H_0$. - **Conclusion:** There is sufficient evidence at the 0.10 level to support the claim that retaking the SAT increases the score by more than 30 points. 8. **Test at $\alpha = 0.01$:** - Since $t = 1.5 < 2.36$, we fail to reject $H_0$. - **Conclusion:** There is not sufficient evidence at the 0.01 level to support the claim. **Final answers:** - (a) Test statistic $t = 1.5$ - (b) At 0.10 significance level, critical value $1.29$, sufficient evidence: **Yes** - (c) At 0.01 significance level, critical value $2.36$, sufficient evidence: **No**