Q. 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. 100 observations from population 1 (dual degree students) and 100 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 4.0 and 5.0 respectively. Using a level of significance of .10, is there evidence that the dual degree students are receiving lower grades? Fully explain your answer.
Ans.
To Test
Ho : μ1= μ2 Vs H1 : μ1< μ2 (one tailed test)
Test Statistics: As n is large >30 so we use Normal Test
follows Standard Normal distribution N(0,1)
= -10
Putting
=84 =87 n1=100 n2=100 =4 =5 we get
P-value = P (z < -10) = 0
Since the P-value 0 < 0.1, we reject H0. It is statistically significant.
Conclusion
At the 10% level of significance, the data provides enough evidence to reject the null hypothesis. Thus we conclude at 0.1 level of significance that the dual degree students are receiving lower grades
To explain the answer, we first set up the hypothesis test. The null hypothesis (Ho) states that the mean grade of the dual degree students (population 1) is equal to the mean grade of the regular MBA students (population 2). The alternative hypothesis (H1) states that the mean grade of the dual degree students is less than the mean grade of the regular MBA students.
Next, we calculate the test statistic. Since the sample sizes are large (both n1 and n2 are 100), we can use the standard normal distribution to calculate the test statistic. The formula for the test statistic is (X1(bar) - X2(bar)) / √(σ1^2/n1 + σ2^2/n2), where X1(bar) and X2(bar) are the sample means, σ1^2 and σ2^2 are the population variances, and n1 and n2 are the sample sizes.
In this case, X1(bar) is 84, X2(bar) is 87, σ1^2 is 4, σ2^2 is 5, and n1 = 100, n2 = 100. Plugging these values into the formula, we calculate the test statistic as -10.
Now we need to find the p-value. The p-value is the probability of observing a test statistic as extreme as the one calculated (or even more extreme) under the assumption that the null hypothesis is true. In this case, we want to find the probability that a standard normal random variable is less than -10. By looking up the value in the standard normal distribution table, we find that the p-value is approximately 0.
Finally, we compare the p-value to the significance level of 0.10. Since the p-value of 0 is less than 0.10, we reject the null hypothesis. This means that we have enough evidence to conclude, at the 10% level of significance, that the dual degree students are receiving lower grades than the regular MBA students.
Therefore, based on the data, we can say that there is evidence to support the concern of the MBA department that dual degree students may be receiving lower grades.