Question

The mean Verbal SAT score for the population of first students at Radford is 520. The

standard deviation of scores in this population is 95. An investigator believes that the mean

Verbal SAT of first year psychology majors is significantly different from the mean score of the

population. The mean of a sample of 36 first year psychology majors is 548. Test the

investigator's prediction using an alpha level of .05.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0: \mu=520$

$H_1:\mu\not=520$

This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

Based on the information provided, the significance level is $\alpha=0.05,$ and the critical value for a two-tailed test is $z_c=1.96.$

The rejection region for this two-tailed test is $R = \{z: |z| > 1.96\}.$

The z-statistic is computed as follows:

$$z=\dfrac{x-\mu}{\sigma/\sqrt{n}}=\dfrac{548-520}{95/\sqrt{36}}=1.7684$$

Since it is observed that $|z|=1.7684<1.96=z_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is $p=2P(z<-1.7684)=0.076994,$ and since $p=0.076994>0.05=\alpha,$ 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 $520,$ at the $\alpha = 0.05$ significance level.