1. **Problem Statement:** We want to test the null hypothesis that there is no difference in anxiety levels between junior and senior high school students caused by the learning modules in Mathematics. Given data: - Means: $\bar{x} = 50$, $\bar{y} = 44$ - Standard deviations: $S_x = 5$, $S_y = 10$ - Sample sizes: $n_x = 12$, $n_y = 16$ - Significance level: $\alpha = 0.01$ 2. **Hypotheses:** - Null hypothesis $H_0$: $\mu_x = \mu_y$ (no difference in means) - Alternative hypothesis $H_a$: $\mu_x \neq \mu_y$ 3. **Test statistic formula for two independent samples with unequal variances (Welch's t-test):** $$ t = \frac{\bar{x} - \bar{y}}{\sqrt{\frac{S_x^2}{n_x} + \frac{S_y^2}{n_y}}} $$ 4. **Calculate the test statistic:** - Calculate variances: $S_x^2 = 5^2 = 25$, $S_y^2 = 10^2 = 100$ - Calculate standard error: $$ SE = \sqrt{\frac{25}{12} + \frac{100}{16}} = \sqrt{2.0833 + 6.25} = \sqrt{8.3333} \approx 2.887 $$ - Calculate $t$: $$ t = \frac{50 - 44}{2.887} = \frac{6}{2.887} \approx 2.078 $$ 5. **Calculate degrees of freedom (df) using the given formula:** $$ df = \frac{\left(\frac{S_x^2}{n_x} + \frac{S_y^2}{n_y}\right)^2}{\frac{S_x^4}{n_x^2 (n_x - 1)} + \frac{S_y^4}{n_y^2 (n_y - 1)}} $$ - Calculate numerator: $$ \left(\frac{25}{12} + \frac{100}{16}\right)^2 = (2.0833 + 6.25)^2 = 8.3333^2 = 69.4444 $$ - Calculate denominator: $$ \frac{25^2}{12^2 \times 11} + \frac{100^2}{16^2 \times 15} = \frac{625}{144 \times 11} + \frac{10000}{256 \times 15} = \frac{625}{1584} + \frac{10000}{3840} \approx 0.394 + 2.604 = 2.998 $$ - Calculate $df$: $$ df = \frac{69.4444}{2.998} \approx 23.17 $$ 6. **Decision rule:** At $\alpha = 0.01$ and $df \approx 23$, the two-tailed critical $t$ value is approximately $\pm 2.807$. 7. **Conclusion:** Since the calculated $t = 2.078$ is less than the critical value $2.807$, we fail to reject the null hypothesis. **Final answer:** There is not enough evidence at the 0.01 significance level to conclude a difference in anxiety levels between junior and senior high school students caused by the learning modules.