Question

At the beginning of the school year, the mean score of a group of 24 students in an educational achievement test in reading was 45 with a standard deviation of 6. At the end of the school year, the mean score on an equivalent form of the same test was 50 with a standard deviation of 5. Has the class made significant progress in reading during the year? Use alpha equal to 0.05.

Answer 1

$H_0:\mu_1=\mu_2$ , mean scores at the beginning of year and at the end of year are equal

$H_a:\mu_2>\mu_1$ , mean score at the end of year is more than at the beginning of year (the class has made significant progress)

Paired T-Test:

$t=\frac{\mu_2-\mu_1}{s_d/\sqrt n}$

where standard deviation of the difference:

$s_d=\sqrt{s_1^2+s_2^2}=\sqrt{6^2+5^2}=7.81$

$t=\frac{50-45}{7.81/\sqrt { 24}}=3.136$

$df=n-1=23$

critical value:

$t_{crit}=1.714$

Since $t>t_{crit}$ we reject null hypothesis. The class has made significant progress.