Question

A college claims that students in its football program boast an average GPA of 3.0. A sample one year of 110 players reveals an average GPA of 2.86 with a standard deviation of 0.52. We want to test this claim, but are only concerned with whether the true GPA of football players is less than 3.0 (if it’s higher, that’s a good thing).

(a) What are the null and alternative hypotheses?

(b) Calculate the t-value to test the college’s claim.

(c) Conduct the hypothesis test. At α = 0.05, does this sample contradict the claim that the college’s football players have a GPA of at least 3.0 on average? (when consulting the t-table, remember to look at one-tail values, not two-tail)

Answer 1

The provided sample mean is $\bar X = 2.86$ and the sample standard deviation is $s = 0.52$, and the sample size is $n = 110$.

(a) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ = 3

Ha: μ < 3

This corresponds to a left-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used.

(b) Test Statistics

The t-statistic is computed as follows:

$t = \frac{\bar X - \mu_0}{s/\sqrt{n}} = \frac{ 2.86 - 3}{ 0.52/\sqrt{ 110}} = -2.824$

(c)

Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a left-tailed test is tc​=−1.659.

The rejection region for this left-tailed test is R=t:t<−1.659

Decision about the null hypothesis

Since it is observed that t=−2.824<tc​=−1.659, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0028, and since p=0.0028<0.05, it is concluded that the null hypothesis is rejected.

Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is less than 3, at the 0.05 significance level.

Graphically

T-Test Results: t-stats 2.824, p-value0.0028 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 4.0 -3.5-3.0 -2.5 -2.0 15 1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

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