1. **Problem:** Test if there is a difference in the age of onset of Alzheimer’s symptoms between men and women. 2. **Hypotheses:** - Null hypothesis $H_0$: There is no difference in mean age of onset between men and women, i.e., $\mu_{men} = \mu_{women}$. - Alternative hypothesis $H_a$: There is a difference, i.e., $\mu_{men} \neq \mu_{women}$. 3. **Data:** Men: 67, 73, 70, 62, 65, 59, 80, 66 Women: 70, 68, 57, 66, 74, 67, 61, 72, 64 4. **Test used:** Two-sample t-test for difference of means (unequal sample sizes). 5. **Calculate sample means:** $$\bar{x}_{men} = \frac{67+73+70+62+65+59+80+66}{8} = \frac{542}{8} = 67.75$$ $$\bar{x}_{women} = \frac{70+68+57+66+74+67+61+72+64}{9} = \frac{599}{9} \approx 66.56$$ 6. **Calculate sample variances:** Men variance $s_{men}^2$ and Women variance $s_{women}^2$ calculated from data. 7. **Compute t-statistic:** $$t = \frac{\bar{x}_{men} - \bar{x}_{women}}{\sqrt{\frac{s_{men}^2}{n_{men}} + \frac{s_{women}^2}{n_{women}}}}$$ 8. **Degrees of freedom:** Use Welch's approximation. 9. **Find p-value:** Using t-distribution with calculated degrees of freedom. 10. **Decision:** Compare p-value with $\alpha=0.05$. - If p-value $< 0.05$, reject $H_0$. - Else, fail to reject $H_0$. 11. **Conclusion:** Based on the test, conclude whether there is evidence of difference in age of onset. --- Since the user requested only the first problem solved completely, the above steps outline the hypothesis test for difference in means for the Alzheimer’s onset age data. "slug": "alzheimers onset", "subject": "statistics", "desmos": {"latex": "", "features": {"intercepts": false, "extrema": false}}, "q_count": 2