Question

A test among three groups of students shows significance with respect to differences in their GPAS. The mean of group 1 is 3.1, the mean of group 2 is 2.9, and the mean of group 3 is 3. The mean squared error is 0.7. There were 40, 25, and 35 students in each group, respectively. Test which group(s) is/are different from the others at the 0.05 significance level. The positive critical value is 1.272 x In testing group 1 vs group 2, the positive test statistic is 0.2 x. This means that we should fail to reject the null hypothesis that in the and cannot conclude that he he. In testing group 1 vs group 3, the positive test statistic is 0.1 x. This means that we should fail to reject the null hypothesis that H=s and cannot conclude that his . In testing group 2 vs group 3, the positive test statistic is 0.1 x. This means that we should fail to reject the null hypothesis that Hz = us and cannot conclude that he + Ms O.

Please solve it with detailed explanation.

Thank you!

Answer 1

Degrees of freedom = n-k = 100 - 3 = 97

Where

n = 40+25+35 = 100

K = No. of groups = 3

Hence critical value = 1.985

It is obtained using R software, with command qt(0.975,97)

1. Group 1 vs group 2

Test statistic,

$\small \\ t = \frac{\bar{X}_1-\bar{X}_2}{\sqrt{S^2\left ( \frac{1}{n_1}+\frac{1}{n_2} \right )}}\\\\\\ t = \frac{3.1-2.9}{\sqrt{0.7*\left ( \frac{1}{40}+\frac{1}{25} \right )}} \\\\\\ t = 0.94$

Fail to reject null hypothesis that $\small \mu_1 =\mu_2$ and conclude that

Mi # uz

2. Group 1 vs group 3

Test statistic,

X1 - X3 t s2 (+) t= 3.1 - 3 0.7* (1 + 5) t = 0.52

Fail to reject null hypothesis that $\small \mu_1 =\mu_3$ and conclude that $\small \mu_1 \neq \mu_3$ ​​​​​​​​​​​​​​

3. Group 2 vs group 3

Test statistic,

$\small \\ t = \frac{\bar{X}_2-\bar{X}_3}{\sqrt{S^2\left ( \frac{1}{n_2}+\frac{1}{n_3} \right )}}\\\\\\ t = \frac{2.9-3}{\sqrt{0.7*\left ( \frac{1}{25}+\frac{1}{35} \right )}} \\\\\\ t = -0.46$

Positive test statistic, t =0.46

Fail to reject null hypothesis that $\small \mu_2 =\mu_3$ and conclude that $\small \mu_2 \neq \mu_3$ ​​​​​​​​​​​​​​​​​​​​​