To test the hypothesis that students who finish an exam first get better grades, Professor Hardtack

kept track of the order in which papers were handed in. The first 25 papers showed a mean score of
77.1 with a standard deviation of 19.6, while the last 24 papers handed in showed a mean score of
69.3 with a standard deviation of 24.9. Is this a significant difference at α = .05? (a) State the
hypotheses for a right-tailed test. (b) Obtain a test statistic and p-value assuming equal variances.
Interpret these results. (c) Is the difference in mean scores large enough to be important? (d) Is it reasonable
to assume equal variances? (e) Carry out a formal test for equal variances at α = .05, showing
all steps clearly.

You're more likely to get a response if you post your school subject so that everybody knows what it is.

Ms. Sue is right. Knowing the school subject would help.

(a) The hypotheses for a right-tailed test are as follows:

Null hypothesis (H0): There is no significant difference in the mean scores between students who finish the exam first and those who finish last.
Alternative hypothesis (Ha): Students who finish the exam first have better grades than those who finish last.

(b) To obtain a test statistic and p-value assuming equal variances, we can use the two-sample t-test. The formula for the test statistic for two independent samples assuming equal variances is:

t = (mean1 - mean2) / sqrt((s1^2 / n1) + (s2^2 / n2))

where mean1 and mean2 are the sample means, s1 and s2 are the sample standard deviations, n1 and n2 are the sample sizes.

In this case:
mean1 = 77.1, s1 = 19.6, n1 = 25
mean2 = 69.3, s2 = 24.9, n2 = 24

Using these values, we can calculate the test statistic:

t = (77.1 - 69.3) / sqrt((19.6^2 / 25) + (24.9^2 / 24))

Once we have the test statistic, we can find the p-value associated with it using a t-distribution table or a statistical software.

Interpreting the results: If the p-value is less than our significance level (α = 0.05), we reject the null hypothesis and conclude that there is a significant difference in the mean scores between students who finish the exam first and those who finish last.

(c) To determine if the difference in mean scores is large enough to be important, we can calculate the effect size. One commonly used effect size measure is Cohen's d, which is calculated as:

d = (mean1 - mean2) / pooled standard deviation

where the pooled standard deviation is calculated as:

pooled standard deviation = sqrt(((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2))

Calculating these values will give us a measure of the standardized difference between the two means.

(d) To determine if it is reasonable to assume equal variances, we can conduct a formal test for equal variances. One commonly used test is the F-test.

(e) To carry out a formal test for equal variances, we can use the F-test. The F-test compares the variances of two samples and calculates an F-ratio. If the F-ratio is significantly different from 1, then we reject the assumption of equal variances.

To perform the F-test, we follow these steps:
1. Calculate the F-ratio: F = s1^2 / s2^2
2. Obtain the critical F-value from an F-distribution table based on the significance level (α = 0.05) and the degrees of freedom (df1 = n1 - 1, df2 = n2 - 1).
3. Compare the calculated F-ratio with the critical F-value. If the calculated F-ratio is larger than the critical F-value, we reject the assumption of equal variances.

By following these steps, we can determine if it is reasonable to assume equal variances.