One Sample Ztest D80338

1. **State the problem:** We want to determine if the average final examination score of students in a school is significantly different from the national average score of $80$. 2. **Given data:** - Population mean $\mu = 80$ - Population standard deviation $\sigma = 8$ - Sample size $n = 40$ (chosen for this example, $n \geq 30$) - Sample mean $\bar{x} = 83$ (assumed data) 3. **Hypotheses:** - Null hypothesis $H_0$: $\bar{x} = 80$ (no difference) - Alternative hypothesis $H_a$: $\bar{x} \neq 80$ (significant difference) 4. **Descriptive statistics:** - Sample mean $\bar{x} = 83$ is higher than the national average $80$. - Simply comparing averages does not tell us if the difference is statistically significant. 5. **Formula for One-Sample Z-Test:** $$ z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} $$ 6. **Calculate the standard error:** $$ SE = \frac{8}{\sqrt{40}} = \frac{8}{6.3246} \approx 1.2649 $$ 7. **Calculate the z-value:** $$ z = \frac{83 - 80}{1.2649} = \frac{3}{1.2649} \approx 2.37 $$ 8. **Critical value method:** - Significance level $\alpha = 0.05$ - Two-tailed test critical values: $\pm 1.96$ - Since $2.37 > 1.96$, we reject the null hypothesis. 9. **P-value method:** - Using standard normal tables or Excel, $p$-value for $z=2.37$ is approximately $0.018$ (two-tailed). - Since $p < 0.05$, reject $H_0$. 10. **Interpretation:** The students' average final examination score of $83$ is significantly different from the national average of $80$ at the $5\%$ significance level. 11. **Answering additional questions:** - The z-test tells us if the observed difference is statistically significant, not just the magnitude of difference. - Possible reasons for below-average performance could be teaching quality, resources, or student engagement. - Recommendations: improve teaching methods, provide extra support, enhance learning materials. - Comparing with national benchmarks helps identify areas needing improvement. - Z-test is preferred over t-test when population standard deviation is known and sample size is large ($n \geq 30$).