Answer to Question #347491 in Statistics and Probability for jouuuuuu

Question

The high school sports coordinator is asked if soccer players are doing as well academically as the other student athletes. From the previous study, the GPA for student athletes is 3.10. After the intervention to help improve GPA of student athletes, the sports coordinator randomly samples 20 soccer players and finds that the average GPA of the sample is 3.18 with a sample standard deviation of 0.54.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=3.10$

$H_1:\mu\not=3.10$

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

Based on the information provided, the significance level is $\alpha = 0.05,$ $df=n-1=19$ and the critical value for a two-tailed test is $t_c =2.093024.$

The rejection region for this two-tailed test is $R = \{t:|t|>2.093024\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{3.18-3.10}{0.54/\sqrt{20}}=0.6625$$

Since it is observed that $|t|=0.6625<2.093024=t_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value for two-tailed, $df=19$ degrees of freedom, $t=0.6625$ is $p=0.515607,$ and since $p= 0.515607>0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population mean $\mu$

is different than 3.10, at the $\alpha = 0.05$ significance level.