A professor believes that all Thursday sections are cursed and will always score lower on tests compared to Monday sections. To tests this he collects samples of test scores from these two sections. For Monday's section, the mean test score was 60, the estimated population standard deviation was 5, and the sample size was 20. For Thursday's section the mean test score was 55, the estimated population standard deviation was 7 and the sample size was 20. With alpha equal to 0.05.
What is the appropriate statistical test? I'm not sure if it would be a z-test or an Independent Samples t-test.
I believe it would also be a one tailed test and I have no idea how to get the obtained value. Can anyone help? Sorry for the super long question.
Ho: M = T
Ha: M > T (one-tailed)
I do not want to do an F-test to find out if the SDs are equal.
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability
related to the Z score. Is it ≤ .05?
The appropriate statistical test in this case would be an Independent Samples t-test. This test is used to determine if there is a significant difference between the means of two independent groups.
Since the professor is comparing the mean test scores of Monday and Thursday sections, where each section is considered as a separate group, an Independent Samples t-test is the correct choice.
To perform the t-test, we need to calculate the obtained t-value and compare it to the critical t-value to determine if there is a significant difference between the means.
Here's how we can calculate the obtained t-value:
Step 1: State the null hypothesis (H0) and alternative hypothesis (Ha):
- Null Hypothesis (H0): There is no significant difference between the mean test scores of the Monday and Thursday sections.
- Alternative Hypothesis (Ha): There is a significant difference between the mean test scores of the Monday and Thursday sections.
Step 2: Set the significance level (alpha):
- In this case, alpha is given as 0.05.
Step 3: Calculate the standard error of the difference between means:
- The formula for calculating the standard error is:
SE = sqrt((s1^2/n1) + (s2^2/n2)), where s1 and s2 are the estimated population standard deviations, and n1 and n2 are the sample sizes.
- Using the given values: SE = sqrt((5^2/20) + (7^2/20)) = sqrt(1.25 + 1.225) ≈ 1.583.
Step 4: Calculate the obtained t-value:
- The formula for calculating the t-value is:
t = (mean1 - mean2) / SE, where mean1 and mean2 are the sample means.
- Using the given values: t = (60 - 55) / 1.583 ≈ 3.16.
Step 5: Determine the degrees of freedom:
- In an Independent Samples t-test, the degrees of freedom are calculated using the formula:
df = (n1 + n2) - 2, where n1 and n2 are the sample sizes.
- Using the given sample sizes: df = (20 + 20) - 2 = 38.
Step 6: Compare obtained t-value with critical t-value:
- Since this is a one-tailed test, we need to determine the critical t-value at a 0.05 significance level and 38 degrees of freedom.
- Looking up the critical t-value using a t-table, the critical t-value should be around 1.686 (one-tailed test, 0.05 significance level, 38 degrees of freedom).
Step 7: Make a decision:
- If the obtained t-value is greater than the critical t-value, we reject the null hypothesis and conclude that there is a significant difference between the means.
- If the obtained t-value is less than or equal to the critical t-value, we fail to reject the null hypothesis and conclude that there is no significant difference between the means.
To summarize, the obtained t-value is approximately 3.16. To make a final decision, we need to compare this value to the critical t-value of 1.686. If the obtained t-value is greater than the critical t-value, we can reject the null hypothesis and conclude that there is a significant difference between the mean test scores of the Monday and Thursday sections.