Question

All freshmen in a particular school were found to
have variability in grades expressed as a standard
deviation of 3. Two samples among these
freshmen, made up of 20 and 50 student’s each,
were found to have means of 88 and 85
respectively. Based on their grades, is the first
group really brighter than the second group at
£=0.01?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu_1\leq\mu_2$

$H_1:\mu_1>\mu_2$

This corresponds to a right-tailed test, for which a z-test for two population means, with known population standard deviations will be used.

Based on the information provided, the significance level is $\alpha=0.01,$ and the critical value for a right-tailed test is $z_c=2.33.$

The rejection region for this right-tailed test is $R=\{z:z>2.33\}$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{X_1}-\bar{X_2}}{\sqrt{\sigma_1^2/n_1+\sigma_2^2/n_2}}$$

$$=\dfrac{88-85}{\sqrt{3^2/20+3^2/20}}\approx3.1623$$

Since it is observed that $z=3.1623>2.33=z_c,$ it is then concluded that the null hypothesis is rejected.

The first group really brighter than the second group at $\alpha=0.01.$

Using the P-value approach: The p-value is $p=P(z>3.1623)=0.00078,$

and since $p=0.00078<0.01,$ it is concluded that the null hypothesis is rejected.

The first group really brighter than the second group at $\alpha=0.01.$