Question

An education organization claims that the mean SAT scores for male athletes and male non-athletes at a college are different. A random sample of 26 male athletes at the college has a mean SAT score of 1189 and a standard deviation of 218. A random sample of 18 male non-athletes at the college has a mean SAT score of 1376 and a standard deviation of 186. At 	α=0.05, can you support the organization’s claim? Interpret the decision in the context of the original claim. Assume the populations are normally distributed and the population variances are equal.

Accepted Answer

Identify the hypotheses for the test. The null hypothesis $$H_0$$ states that the mean SAT scores for male athletes and male non-athletes are equal: $$\mu_1$ = \mu_2$. The alternative hypothesis H_a$ states that the means are different: $$\$$mu_1$$ 
eq \$$mu_2. $$Since the population variances are assumed equal and the samples are independent, use the two-sample t-test for equal variances. Calculate the pooled standard deviation $$s_p$$ using the formula:

$$s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$$

where $$n_1$$ and $$n_2$$ are the sample sizes, and $$s_1$$ and $$s_2$$ are the sample standard deviations. Calculate the test statistic $$t$$ using the formula:

$$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$$

where $$\bar{x}_1$$ and $$\bar{x}_2$$ are the sample means. Determine the degrees of freedom for the test, which is $$df = n_1$ + n_2$ - 2. $$Then, find the critical t-value(s) from the t-distribution table for a two-tailed test at significance level $$\alpha = 0.05. $$Compare the absolute value of the calculated test statistic $$|t| to $$the critical t-value. If $$|t| is $$greater than the critical value, reject the null hypothesis $$H_0$$ and conclude there is sufficient evidence to support the claim that the mean SAT scores differ. Otherwise, do not reject $$H_0. $$Interpret this decision in the context of the original claim about the difference in mean SAT scores.

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Key Concepts

Two-Sample t-Test for Means

Assumption of Equal Population Variances

Significance Level and Hypothesis Testing