Question

2. Suppose a randomly chosen group of 150 high school juniors and seniors who took the SAT twice over a period of six months showed an average improvement on the second SAT of 25 points. The standard deviation of the difference in the scores between the first and second SAT was 20 points.

a. What is the appropriate design for this situation?

b. Set up an appropriate hypotheses to test the claim that the score on the second SAT is, on average at least 20 points higher than on the first SAT.

c. Test the hypotheses stated in part b. What distribution should be used? What assumptions, if any, should be checked? Explain. Make a decision and give a conclusion.

Answer 1

2 a) The sample is a paired observation , that is pair of SAT scores of 150 students

So this is matched experiment deign . In this design , there are two treatments (two SAT exams over a period of six months ) and two treatments are assigned to each subject and the paired observations are compared.

So paired sample t test is appropriate test for this design

b) The null and alternative hypotheses are

where ,d = second SAT score - first SAT score

is the population mean difference

c) To test the hypotheses stated above , we use t distribution

Assumptions for using t distribution

i) The sample observation ( test scores) should be numeric . Here the sample observation :test scores are numeric data .

ii) The samples are drawn with random sampling . Here the sample is randomly chosen 150 students .

iii) The sample is drawn from a Normal population . But as the sample size is quite large (150) , we can proceed without normality check.

Test statistic

given ,

Thus

degrees of freedom = n-1 =149

We find one tailed P value ( as the test is one tailed) using excel "=T.DIST.RT(3.06,149)"

P value =0.0013

As P value < 0.05

We reject The null hypothesis

There is sufficient evidence to conclude that scores on second test is on an average at least 20 points higher than on the first test .