1. **Problem Statement:** We want to test if the weight-reducing plan is effective. We have weights before and after the plan for 11 people. We will use a paired t-test at a 0.05 significance level. 2. **Hypotheses:** - Null hypothesis $H_0$: The mean difference in weight before and after is zero, i.e., $\mu_d = 0$ (no effect). - Alternative hypothesis $H_a$: The mean difference is greater than zero, i.e., $\mu_d > 0$ (weight reduced). 3. **Formula for paired t-test:** $$ t = \frac{\bar{d} - \mu_0}{s_d / \sqrt{n}} $$ where: - $\bar{d}$ = mean of differences (before - after) - $s_d$ = standard deviation of differences - $n$ = number of pairs - $\mu_0 = 0$ under $H_0$ 4. **Calculate differences (Before - After):** Person: Difference 1: $87.4 - 85.4 = 2.0$ 2: $92.9 - 88.3 = 4.6$ 3: $83.6 - 84.7 = -1.1$ 4: $81.5 - 81.2 = 0.3$ 5: $89.7 - 83.3 = 6.4$ 6: $100.5 - 94.6 = 5.9$ 7: $98.6 - 90.1 = 8.5$ 8: $88.8 - 87.2 = 1.6$ 9: $112.4 - 104.6 = 7.8$ 10: $87.6 - 88.4 = -0.8$ 11: $92.8 - 91.7 = 1.1$ 5. **Calculate mean difference $\bar{d}$:** $$ \bar{d} = \frac{2.0 + 4.6 - 1.1 + 0.3 + 6.4 + 5.9 + 8.5 + 1.6 + 7.8 - 0.8 + 1.1}{11} = \frac{36.3}{11} \approx 3.3 $$ 6. **Calculate standard deviation $s_d$:** First, find squared deviations from mean: $(2.0 - 3.3)^2 = 1.69$ $(4.6 - 3.3)^2 = 1.69$ $(-1.1 - 3.3)^2 = 19.36$ $(0.3 - 3.3)^2 = 9.0$ $(6.4 - 3.3)^2 = 9.61$ $(5.9 - 3.3)^2 = 6.76$ $(8.5 - 3.3)^2 = 27.04$ $(1.6 - 3.3)^2 = 2.89$ $(7.8 - 3.3)^2 = 20.25$ $(-0.8 - 3.3)^2 = 16.81$ $(1.1 - 3.3)^2 = 4.84$ Sum of squared deviations = 119.94$ $$ s_d = \sqrt{\frac{119.94}{11 - 1}} = \sqrt{11.994} \approx 3.46 $$ 7. **Calculate t-statistic:** $$ t = \frac{3.3 - 0}{3.46 / \sqrt{11}} = \frac{3.3}{3.46 / 3.317} = \frac{3.3}{1.043} \approx 3.16 $$ 8. **Degrees of freedom:** $df = n - 1 = 10$ 9. **Critical t-value for one-tailed test at $\alpha=0.05$ and $df=10$:** approximately 1.812 10. **Decision:** Since $t = 3.16 > 1.812$, we reject the null hypothesis. 11. **Conclusion:** There is sufficient evidence at the 0.05 significance level to conclude that the diet plan is effective in reducing weight.