1. **State the problem:** Determine whether there is a significant difference between the performance of males and females in Math using hypothesis testing. 2. **State the hypotheses:** - Null hypothesis ($H_0$): There is no significant difference in Math performance between males and females. - Alternative hypothesis ($H_a$): There is a significant difference in Math performance between males and females. 3. **Set the significance level:** $\alpha = 0.05$ (two-tailed test). 4. **Choose the test:** Independent-samples t-test because we compare means of two independent groups. 5. **Given data:** - Male scores: $34, 45, 38, 34, 43$ - Female scores: $32, 48, 40, 30, 41$ 6. **Calculate sample means:** $$\bar{x}_m = \frac{34 + 45 + 38 + 34 + 43}{5} = \frac{194}{5} = 38.8$$ $$\bar{x}_f = \frac{32 + 48 + 40 + 30 + 41}{5} = \frac{191}{5} = 38.2$$ 7. **Calculate sample variances:** $$s_m^2 = \frac{\sum (x_i - \bar{x}_m)^2}{n_m - 1} = \frac{(34-38.8)^2 + (45-38.8)^2 + (38-38.8)^2 + (34-38.8)^2 + (43-38.8)^2}{4}$$ $$= \frac{23.04 + 38.44 + 0.64 + 23.04 + 17.64}{4} = \frac{102.8}{4} = 25.7$$ $$s_f^2 = \frac{(32-38.2)^2 + (48-38.2)^2 + (40-38.2)^2 + (30-38.2)^2 + (41-38.2)^2}{4}$$ $$= \frac{38.44 + 96.04 + 3.24 + 67.24 + 7.84}{4} = \frac{212.8}{4} = 53.2$$ 8. **Calculate pooled variance:** $$s_p^2 = \frac{(n_m - 1)s_m^2 + (n_f - 1)s_f^2}{n_m + n_f - 2} = \frac{4 \times 25.7 + 4 \times 53.2}{8} = \frac{102.8 + 212.8}{8} = \frac{315.6}{8} = 39.45$$ 9. **Calculate standard error:** $$SE = \sqrt{s_p^2 \left(\frac{1}{n_m} + \frac{1}{n_f}\right)} = \sqrt{39.45 \left(\frac{1}{5} + \frac{1}{5}\right)} = \sqrt{39.45 \times 0.4} = \sqrt{15.78} = 3.97$$ 10. **Calculate t-statistic:** $$t = \frac{\bar{x}_m - \bar{x}_f}{SE} = \frac{38.8 - 38.2}{3.97} = \frac{0.6}{3.97} = 0.151$$ 11. **Degrees of freedom:** $$df = n_m + n_f - 2 = 5 + 5 - 2 = 8$$ 12. **Critical t-value:** For $\alpha=0.05$ two-tailed and $df=8$, $t_{critical} \approx 2.306$. 13. **Decision:** Since $|t| = 0.151 < 2.306$, we fail to reject the null hypothesis. 14. **Conclusion:** There is no significant difference in Math performance between males and females at the 5% significance level.