Question

in certain college in metro manila, a study was conducted to determine wheather the IQ scores of students who came from provincial high schools differ significantly from those of students who came from city high school. an IQ test was given to 200 studnets (100 from each group) college freshmen and the results are as follows.

students from province X1=99 SD= 5

Students from city X2= 102 SD=8

set up and test the appropriate statistical hypothesis

Answer 1

$H_0: \mu_1 = \mu_2 \\ H_1: \mu_1 ≠ \mu_2 \\ n_1=n_2=100 \\ \bar{x_1} = 99 \\ s=5 \\ \bar{x_2} = 102 \\ s= 8$

When $\sigma_1$ and $\sigma_2$ are unknown, we have to use the two-sample t test for independent random samples from two normal populations having the same unknown variance.

Test-statistic:

$t= \frac{\bar{x_1} - \bar{x_2}}{s_p \sqrt{(1/n_1) + (1/n_2)}} \\ s^2_p= \frac{(n_1-1)s^2_1 +(n_2-1)s^2_2}{n_1+n_2-2} \\ s^2_p= \frac{(100-1)(5)^2 +(100-1)(8)^2}{100+100-2} \\ = \frac{2475 + 6336 }{198} \\ = 44.5 \\ t= \frac{99- 102}{44.5 \sqrt{(1/100) + (1/100)}} \\ = \frac{-3}{6.293} \\ = -0.476$

Two-tailed test. Reject H₀ if $t≤ -t_{crit}$ or $t ≥ t_{crit}.$

Let use α=0.05

For two-teiled test and degree of freedom $df = n_1+n_2-2= 198 \; t_{crit} = 1.972$

$t= -0.4776 > -t_{crit}= -1.697$

Accept H₀ at 0.05 significance level.

The IQ scores of students who came from provincial high schools do NOT differ significantly from those of students who came from city high school at 0.05 level of significance.