Question

Arandom sample of n1 =12 students majoring in accounting in a college of business has a mean grade-point average of 2.70 (where A = 4.0) with a sample standard deviation of 0.40. For the students majoring in computer information systems, a random sample of n2 = 10 students has a mean grade-point average of 2.90 with a standard deviation of 0.30. The grade-point values are assumed to be normally distributed. Test the null hypothesis that the mean-grade point average for the two categories of students is not different, using the 5 percent level of significance.

Answer 1

Answer:

We accept the null hypothesis that the mean gpa's are equal, with the 5 percent level of significance.

Step-by-step explanation:

We have these following hypothesis:

Null

Equal means

So

$\mu_{1} = \mu_{2}$

Alternative

Different means

So

$\mu_{1} \neq \mu_{2}$

Our test statistic is:

$\frac{\overline{Y_{1}} - \overline{Y_{2}}}{\sqrt{\frac{s_{1}^{2}}{N_{1}} + \frac{s_{2}^{2}}{N_{2}}}}$

In which $\overline{Y_{1}}, \overline{Y_{2}}$ are the sample means, $N_{1}, N_{2}$ are the sample sizes and $s_{1}, s_{2}$ are the standard deviations of the sample.

In this problem, we have that:

$\overline{Y_{1}} = 2.7, s_{1} = 0.4, N_{1} = 12, \overline{Y_{2}} = 2.9, s_{2} = 0.3, N_{2} = 10$

So

$T = \frac{2.7 - 2.9}{\sqrt{\frac{0.4}^{2}{12} + \frac{0.3}^{2}{10}}} = -1.3383$

What to do with the null hypothesis?

We will reject the null hypothesis, that is, that the means are equal, with a significante level of $\alpha$ if

$|T| > t_{1-\frac{\alpha}{2},v}$

In which v is the number of degrees of freedom, given by

$v = \frac{(\frac{s_{1}^{2}}{N_{1}} + \frac{s_{2}^{2}}{N_{2}})^{2}}{\frac{\frac{s_{1}^{2}}{N_{1}}}{N_{1}-1} + \frac{\frac{s_{2}^{2}}{N_{2}}}{N_{2} - 1}}$

Applying the formula in this problem, we have that:

$v = 20$

So, applying t at the t-table at a level of 0.975, with 20 degrees of freedom, we find that

$t = 2.086$

We have that

$|T| = 1.3383$

Which is lesser than t.

So we accept the null hypothesis that the mean gpa's are equal, with the 5 percent level of significance.