1. The MBA department is concerned that dual degree students may be receiving lower grades
than the regular MBA students. Two independent random samples have been selected 350
observations from population 1 (dual degree students) and 310 from population 2 (MBA
students). The sample means obtained are X1(bar)=84 and X2(bar)=87. It is known from
previous studies that the population variances are 4.6 and 5.0 respectively. Using a level of
significance of .05, is there evidence that the dual degree students are receiving lower grades?
To determine if there is evidence that dual degree students are receiving lower grades, we can conduct a hypothesis test.
Step 1: State the null hypothesis (H0) and the alternative hypothesis (Ha):
- Null hypothesis (H0): The mean grade of dual degree students is not significantly lower than the mean grade of MBA students.
- Alternative hypothesis (Ha): The mean grade of dual degree students is significantly lower than the mean grade of MBA students.
Step 2: Select the appropriate test statistic:
Since we are comparing the means of two independent samples with known population variances, we can use a two-sample z-test.
Step 3: Set the significance level (α):
In this case, the level of significance is given as .05.
Step 4: Calculate the test statistic:
The formula for the two-sample z-test is:
z = (X1(bar) - X2(bar)) / sqrt(V1/n1 + V2/n2)
where X1(bar) and X2(bar) are the sample means, V1 and V2 are the population variances, and n1 and n2 are the sample sizes.
Given:
X1(bar) = 84, X2(bar) = 87,
V1 = 4.6, V2 = 5.0,
n1 = 350, n2 = 310.
Substituting these values into the formula, we get:
z = (84 - 87) / sqrt(4.6/350 + 5.0/310)
Step 5: Determine the test statistic's critical value(s) or p-value:
To find the critical value(s), we need the z-value(s) corresponding to the significance level of .05. We can look up these values in the standard normal distribution table.
Step 6: Make a decision and interpret the result:
- If the calculated test statistic falls within the critical region (outside the critical value(s)), we reject the null hypothesis and conclude that there is evidence that dual degree students are receiving lower grades.
- If the calculated test statistic falls outside the critical region, we fail to reject the null hypothesis and cannot conclude that there is evidence that dual degree students are receiving lower grades.
Note: Since the calculations for the test statistic and critical value(s) involve standard normal distribution and are relatively lengthy and complex, you may use statistical software or calculators with the capability to perform z-tests to get more accurate results.
By following these steps, you can determine if there is evidence that dual degree students are receiving lower grades or not.