10.46 To test the hypothesis that students who finish an exam first get better g

Question

10.46 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 a = .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 a = .05, showing
all steps clearly.

Explanation / Answer

Given n1=25, xbar1=77, s1=19.6
n2=24, xbar2=69.3, s2=24.9

(a) State the hypotheses for a right-tailed test.
Ho:1<=2
Ha:1>2

(b) Obtain a test statistic and p-value assuming equal variance
Interpret these results.

Z=(xbar1-xbar2)/[s1^2/n1 + s2^2/n2]

=(77-69.3)/sqrt(19.6^2/25 + 24.9^2/24)

=1.2

The p-value is P(Z>1.2)= 0.1151 (check standard normal table)

(c) Is the difference in mean scores large enough to be important?

Since the p-value is larger than 0.05, we do not reject Ho.

So we can not conclude that the difference in mean scores is large enough to be important

(d) Is it reasonable to assume equal variances?

yes.

(e) Carry out a formal test for equal variances at a = .05, showing all steps clearly

Ho:1^2=2^2

Ha:1^2 not equal to 2^2

The test statistic is

F=s1^2/s2^2

=19.6^2/24.9^2

=0.62

Given a=0.05, the critical value is F(0.025, df1=n1-1=24, df2=n2-1=23)= 0.44 (check F table)

F(0.975, df1=n1-1=24, df2=n2-1=23)=2.3 (check F table)

Since 0.44<F=0.62 < 2.3, we do not reject Ho.

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