1. **State the problem:** We want to test if there is a significant difference in the mean carbohydrate content between meals with potatoes and meals without potatoes at McDonald's. 2. **Identify the hypotheses:** - Null hypothesis $H_0$: $\mu_1 = \mu_2$ (means are equal) - Alternative hypothesis $H_a$: $\mu_1 \neq \mu_2$ (means are different) 3. **Given data:** - Potatoes group: $26, 19, 32, 30, 36, 41, 25, 28, 24, 26, 37, 34, 29$ - No Potatoes group: $42, 42, 39, 11, 35, 29, 30, 38, 30, 27, 28, 54, 32$ 4. **Calculate sample means and standard deviations:** - $\bar{x}_1 = \frac{26+19+\cdots+29}{13} = \frac{381}{13} \approx 29.31$ - $\bar{x}_2 = \frac{42+42+\cdots+32}{13} = \frac{435}{13} \approx 33.46$ Calculate sample variances $s_1^2$ and $s_2^2$ using the formula $s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2$. 5. **Calculate sample variances:** - For Potatoes: $$s_1^2 = \frac{1}{12} \sum (x_i - 29.31)^2 \approx 32.36$$ - For No Potatoes: $$s_2^2 = \frac{1}{12} \sum (x_i - 33.46)^2 \approx 98.05$$ 6. **Use two-sample t-test formula for unequal variances (Welch's t-test):** $$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} = \frac{29.31 - 33.46}{\sqrt{\frac{32.36}{13} + \frac{98.05}{13}}} = \frac{-4.15}{\sqrt{2.49 + 7.54}} = \frac{-4.15}{\sqrt{10.03}} = \frac{-4.15}{3.17} \approx -1.31$$ 7. **Degrees of freedom (Welch-Satterthwaite equation):** $$df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1-1} + \frac{(s_2^2/n_2)^2}{n_2-1}} = \frac{(2.49 + 7.54)^2}{\frac{2.49^2}{12} + \frac{7.54^2}{12}} = \frac{10.03^2}{\frac{6.20}{12} + \frac{56.84}{12}} = \frac{100.6}{0.52 + 4.74} = \frac{100.6}{5.26} \approx 19.13$$ 8. **Find critical t-value for two-tailed test at $\alpha=0.05$ and $df \approx 19$:** - $t_{critical} \approx \pm 2.093$ 9. **Decision:** - Since $|t| = 1.31 < 2.093$, we fail to reject the null hypothesis. 10. **Conclusion:** - There is insufficient evidence at the 0.05 significance level to conclude a difference in mean carbohydrate content between meals with potatoes and meals without potatoes.