Question

1. The teacher would like to find out if there is significant difference in the performance of the male and female students in 35 item test in English. He wants to consider 0.5% level of significance. The result is shown below:

MALE STUDENTS

sample size = 8

sample mean = 24.27

standard deviation = 9.36

FEMALE STUDENTS

sample size = 7

sample mean = 21.07

standard deviation = 8.25

Answer 1

Testing for Equality of Variances

A F-test is used to test for the equality of variances. The following F-ratio is obtained:

$$F=\dfrac{s_1^2}{s_2^2}=\dfrac{9.36^2}{8.25^2}=1.2872$$

The critical values for $\alpha=0.5,$ $df_1=n_1-1=7$ degrees of freedom, $df_2=n_2-1=6$ degrees of freedom are $F_L = 0.5862$ and $F_U = 1.7789,$ and since $F = 1.2872$ then the null hypothesis of equal variances is not rejected.

The following null and alternative hypotheses need to be tested:

$H_0:\mu_1=\mu_2$

$H_1:\mu_1\not=\mu_2$

This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

The degrees of freedom are computed as follows, assuming that the population variances are equal:

$$df=df_1+df_2=n_1+n_2-2=13$$

Based on the information provided, the significance level is $\alpha = 0.5,$ and the degrees of freedom are $df = 13.$

The critical value for this two-tailed test, $\alpha = 0.5, df=13$ degrees of freedom is $t_c =0.6938.$

The rejection region for this two-tailed test is $R = \{t: |t|> 0.6938\}.$

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:

$$t=\dfrac{\bar{X}_1-\bar{X}_2}{\sqrt{\dfrac{(n_1-1)(s_1)^2+(n_2-1)(s_2)^2}{n_1+n_2-2}(\dfrac{1}{n_1}+\dfrac{1}{n_2})}}$$ $$=\dfrac{24.27-21.07}{\sqrt{\dfrac{(8-1)(9.36)^2+(7-1)(8.25)^2}{8+7-2}(\dfrac{1}{8}+\dfrac{1}{7})}}$$ $$=0.6975$$

Since it is observed that $|t| =0.6975>0.6938= t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for two-tailed, $df=13$ degrees of freedom, $t=0.6975$ is $p= 0.497776,$ and since $p= 0.497776<0.5=\alpha,$ it is concluded that the null hypothes is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu_1$ is different than $\mu_2,$ at the $\alpha = 0.5$ significance level.