This exercise is based on summary statistics rather than raw data. This informat

Question

This exercise is based on summary statistics rather than raw data. This information is typically all that is presented in published reports. You can perform inference procedures by hand from the summaries. Use the conservative Option 2 (degrees of freedom the smaller of

n1 1 and n2 1)

for two-sample t confidence intervals and P-values. You must trust that the authors understood the conditions for inference and verified that they apply. This isn't always true.

Coaching companies claim that their courses can raise the SAT scores of high school students. Of course, students who retake the SAT without paying for coaching generally raise their scores, too. A random sample of students who took the SAT twice found 427 who were coached and 2733 who were uncoached. Starting with their Verbal scores on the first and second tries, we have these summary statistics.

The summary statistics for Gain are based on the changes in the scores of the individual students. Let's first ask if students who are coached significantly increased their scores.

(a) You could use the information given to carry out either a two-sample t test comparing Try 1 with Try 2 for coached students, or a matched pairs t test using Gain. Which is the correct test? Why?

Two-sample test, because we are given data for the two samples which is more accurate than the data only on the gain.

Two-sample test, because matched pairs test can be performed only if we have information on every individual.

Matched pairs test, because we are interested in the difference in mean between the two groups not the gain that each student has achieved.

Matched pairs test, because we are not interested in the difference in mean between the two groups, but the gain that each student has achieved.

(b) Carry out the proper test. What do you conclude? (Use = 0.05.)

There is sufficient evidence that coached student achieved a better score on the second try.

There is no evidence that coached student achieved a better score on the second try.

There is insufficient evidence that coached student achieved a better score on the second try.

(c) Give a 95% confidence interval for the mean gain of all students who are coached. (Use conservative degrees of freedom.)

23.06 to 34.94

20 to 38    

23.34 to 34.66

21.5 to 36.5

Explanation / Answer

a) Matched pairs test, because we are interested in the difference in mean between the two groups not the gain that each student has achieved. The paired t-test provides a hypothesis test for the difference between the two means(Gains) for a pair of random samples try 1 and try 2 taken from the same population with identical sizes.

null hypothesis: mean gain<=0 ;

alternate:mean gain>0.

Here the samples for Try1 and try 2 are taken from the same population and have the same size therefore pairs test is more appropriate.

b)

t-stat = 29/(59/sqrt(427)) = 10.157

t-crit for (df=426, = 0.05)=1.648

since t-stat =10.157>1.648 we reject the null that mean gain<=0 therefore we accept the alternative that mean gain> 0 that is There is sufficient evidence that coached student achieved a better score on the second try.

c) 95% confidence interval for mean gain for all students who are

t-crit for (df=426, = 0.05)=1.97

lower limit=29-1.97*(59/sqrt(427))=23.38~23.34

upper limit=29+1.97*(59/sqrt(427))=34.63~34.66

Thus the 95% confidence interval for the mean gain of all students who are coached is 23.34 to 34.66.

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