To determine if there is sufficient statistical evidence to support the teacher's claim, we can conduct a two-sample $t$-test (or $z$-test since the sample sizes are large). This will test whether the mean score of group A is significantly higher than that of group B. Here's the step-by-step solution:

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1. Formulate the hypotheses

• Null hypothesis ($H_0$): There is no difference in mean scores between the two groups, i.e., $\mu_A = \mu_B$ or $\mu_A - \mu_B = 0$.
• Alternative hypothesis ($H_a$): The mean score for group A is higher than that for group B, i.e., $\mu_A > \mu_B$.

This is a one-tailed test since we are testing for a specific direction (group A > group B).

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2. Determine the test statistic

The test statistic is calculated using the formula: $z = \frac{\bar{x}_A - \bar{x}_B}{\sqrt{\frac{s_A^2}{n_A} + \frac{s_B^2}{n_B}}}$ where:

• $\bar{x}_A$ and $\bar{x}_B$ are the sample means,
• $s_A$ and $s_B$ are the standard deviations,
• $n_A$ and $n_B$ are the sample sizes.

Substituting the given values:

• $\bar{x}_A = 7.5$, $\bar{x}_B = 6.0$,
• $s_A = s_B = 0.7$,
• $n_A = n_B = 200$.

= \frac{1.5}{\sqrt{\frac{0.49}{200} + \frac{0.49}{200}}} = \frac{1.5}{\sqrt{0.0049 + 0.0049}} = \frac{1.5}{\sqrt{0.0098}} = \frac{1.5}{0.099} \approx 15.15$$ --- ### **3. Find the critical value** At a significance level ($$\alpha$$) of 0.05 for a one-tailed test, the critical $$z$$-value is: $$z_{\text{critical}} = 1.645$$ --- ### **4. Compare the test statistic with the critical value** The calculated $$z$$-value is $$15.15$$, which is much greater than $$z_{\text{critical}} = 1.645$$. --- ### **5. Conclusion** Since $$z > z_{\text{critical}}$$, we reject the null hypothesis. There is strong statistical evidence to support the teacher's claim that group A performs significantly better than group B. --- Would you like further details or additional examples? Here are 5 related questions: 1. How does a $$t$$-test differ from a $$z$$-test, and when should each be used? 2. What is the meaning of the $$p$$-value in hypothesis testing, and how can it be interpreted? 3. How do sample size and variability affect the power of a statistical test? 4. Can you explain why this test assumes normal distribution and how to handle non-normal data? 5. How can effect size be calculated and interpreted in this context? **Tip:** When performing hypothesis tests, always double-check the assumptions (e.g., normality, equal variances) to ensure the validity of the results.