1. **Problem:** Determine if there is a significant difference between the performance of males and females in Math using hypothesis testing. 2. **Step 1: State the hypotheses.** - Null hypothesis $H_0$: There is no significant difference between male and female scores, i.e., $\mu_{male} = \mu_{female}$. - Alternative hypothesis $H_a$: There is a significant difference, i.e., $\mu_{male} \neq \mu_{female}$. 3. **Step 2: Collect data and calculate sample means and standard deviations.** - Male scores: 34, 45, 38, 34, 43 - Female scores: 32, 48, 40, 30, 41 Calculate means: $$\bar{x}_{male} = \frac{34+45+38+34+43}{5} = \frac{194}{5} = 38.8$$ $$\bar{x}_{female} = \frac{32+48+40+30+41}{5} = \frac{191}{5} = 38.2$$ Calculate sample standard deviations $s$: $$s_{male} = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} = \sqrt{\frac{(34-38.8)^2 + (45-38.8)^2 + (38-38.8)^2 + (34-38.8)^2 + (43-38.8)^2}{4}}$$ $$= \sqrt{\frac{23.04 + 38.44 + 0.64 + 23.04 + 17.64}{4}} = \sqrt{25.7} = 5.07$$ Similarly for females: $$s_{female} = \sqrt{\frac{(32-38.2)^2 + (48-38.2)^2 + (40-38.2)^2 + (30-38.2)^2 + (41-38.2)^2}{4}}$$ $$= \sqrt{\frac{38.44 + 96.04 + 3.24 + 67.24 + 7.84}{4}} = \sqrt{53.7} = 7.33$$ 4. **Step 3: Use two-sample t-test formula for difference of means:** $$t = \frac{\bar{x}_{male} - \bar{x}_{female}}{\sqrt{\frac{s_{male}^2}{n_{male}} + \frac{s_{female}^2}{n_{female}}}} = \frac{38.8 - 38.2}{\sqrt{\frac{5.07^2}{5} + \frac{7.33^2}{5}}} = \frac{0.6}{\sqrt{\frac{25.7}{5} + \frac{53.7}{5}}} = \frac{0.6}{\sqrt{5.14 + 10.74}} = \frac{0.6}{\sqrt{15.88}} = \frac{0.6}{3.99} = 0.15$$ 5. **Step 4: Determine degrees of freedom (approximate):** $$df = \min(n_{male} - 1, n_{female} - 1) = 4$$ 6. **Step 5: Find critical t-value for two-tailed test at $\alpha=0.05$ and $df=4$:** $$t_{critical} \approx 2.776$$ 7. **Step 6: Compare calculated t with critical t:** $$|t| = 0.15 < 2.776$$ 8. **Step 7: Conclusion:** Since $|t|$ is less than $t_{critical}$, we fail to reject the null hypothesis. There is no significant difference between male and female math scores. **Final answer:** There is no significant difference between the performance of males and females in Math at the 5% significance level.