Answer to Question #237316 in Statistics and Probability for Aurithra

Question

The average score of a sixth-grander

in a certain school district on a math aptitude exam is 89 with a standard deviation of 6.4.

A random sample of 28 students in one school was taken. The mean score of these

students was 84. Does this indicate that the students of this school are significantly

slower in their mathematical ability? Use 5% Level of significance.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\geq89$

$H_1:\mu<89$

This corresponds to a left-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 left-tailed test is $z_c=-1.6449.$

The rejection region for this left-tailed test is $R=\{z:z<-1.6449\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}=\dfrac{84-89}{6.4/\sqrt{28}}=-4.1340$$

Since it is observed that  $z=-4.1340<-1.6449=z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is $p=P(Z<-4.1340)=0.000018,$ and since $p=0.000018<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is less than 89, at the $\alpha=0.05$ significance level.

Therefore, there is enough evidence to claim that the students of this school are significantly slower in their mathematical ability at the $\alpha=0.05$ significance level.