Assuming that the population variances are equal for Male and Female GPA’s, test the following sample data to see if Male and Female PhD candidate GPA’s (Means) are equal. Conduct a two-tail hypothesis test at á =.01 to determine whether the sample means are different. Do NOT do a confidence interval.
Male GPA’s Female GPA’s
Sample Size 12 13
Sample Mean 2.8 4.95
Sample Standard Dev .25 .8
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√(n-1)
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.
To test whether the male and female PhD candidate GPAs are significantly different, you can conduct a two-sample t-test under the assumption that the population variances are equal for both groups.
Here are the steps to calculate the t-statistic and perform the hypothesis test:
1. State the null and alternative hypotheses:
- Null hypothesis (H0): The mean GPA for male PhD candidates is equal to the mean GPA for female PhD candidates.
- Alternative hypothesis (Ha): The mean GPA for male PhD candidates is different from the mean GPA for female PhD candidates.
2. Determine the significance level:
- The significance level (α) is given as 0.01, which means we will reject the null hypothesis if the probability of obtaining the observed sample difference (or more extreme) under the assumption of the null hypothesis is less than 0.01.
3. Calculate the test statistic (t):
- The formula for the two-sample t-test is: t = (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2)), where:
- x1 and x2 are the sample means for male and female GPAs, respectively.
- s1 and s2 are the sample standard deviations for male and female GPAs, respectively.
- n1 and n2 are the sample sizes for male and female GPAs, respectively.
- Plugging in the values from the provided data, we have:
- x1 = 2.8
- x2 = 4.95
- s1 = 0.25
- s2 = 0.8
- n1 = 12
- n2 = 13
4. Determine the degrees of freedom:
- The degrees of freedom (df) for a two-sample t-test is calculated using the formula: df = (s1^2 / n1 + s2^2 / n2)^2 / ((s1^2 / n1)^2 / (n1 - 1) + (s2^2 / n2)^2 / (n2 - 1)).
- Plugging in the values, we get:
- df = (0.25^2 / 12 + 0.8^2 / 13)^2 / ((0.25^2 / 12)^2 / (12 - 1) + (0.8^2 / 13)^2 / (13 - 1))
5. Calculate the critical t-value:
- Since this is a two-tail test, we need to find the t-value that corresponds to a rejection region in both tails, with an area of 0.01 split evenly between the two tails.
- Looking up the critical t-value in a t-distribution table or using statistical software, we find that the critical t-value is approximately ±2.764 for a two-tailed test with 23 degrees of freedom.
6. Compare the calculated t-value with the critical t-value and make a decision:
- If the calculated t-value falls within the rejection region (i.e., if it is greater than the positive critical t-value or less than the negative critical t-value), we reject the null hypothesis and conclude that there is a significant difference between the mean GPAs of male and female PhD candidates. If the calculated t-value does not fall within the rejection region, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant difference.
7. Calculate the t-value:
- Plugging in the values, we calculate:
- t = (2.8 - 4.95) / sqrt((0.25^2 / 12) + (0.8^2 / 13))
8. Make the decision:
- Compare the calculated t-value with the critical t-value (-2.764 and 2.764).
- If the calculated t-value falls outside the range (-2.764, 2.764), we reject the null hypothesis.
- If the calculated t-value falls within the range (-2.764, 2.764), we fail to reject the null hypothesis.
Performing the calculations, you can compare the calculated t-value with the critical t-value to make a decision.