Question

ACT scores of a sample of UTC students are shown below. Student ACT Score A 22 B 28 C 20 D 21 E 28 F 23 G 26 a. Compute the mean and the variance. B. At 95% confidence, test to determine whether or not the variance of the ACT scores of the population of UTC students is significantly more than 8.

Answer 1

Solution :

a). Let  x denotes ACT scores.

ACT scores = $\{ 22,28,20,21,28,23,26 \} ; n = 7$

Mean, $(\overline x)=\frac{\sum x_i}{n}$

                $=\frac{168}{7}=24$

Sample variance, $S^2=\frac{1}{n-1}\left(\sum x_i^2-n\overline x^2 \right)$

                                  $=\frac{1}{6}(4098-7 \times 24^2)$

                                   $=\frac{66}{6}$

                                   = 11

b). To test whether or not variance of ACT scores of population (say $\sigma^2$) of the UTC students is significantly more than 8.

Consider the hypothesis :

$H_0: \sigma^2 \leq8$  vs $H_a: \sigma^2 >8$

It is a right tailed test and α = 0.05

We have

$x^2_{n-1} = \frac{(n-1)s^2}{\sigma^2}$

So test statics is

$x^2_7=\frac{(7-1)11}{8}$

   $=\frac{6 \times 11}{8}$

   = 8.25

Since our $\text{test statistics is less than the critical value }$and it falls in a acceptation region, hence we fail to reject $H_0$ and conclude that variance is not greater than 8 significantly.