Question

Do students tend to improve their Math SAT scores the second time they take the test? We take a random sample of 100 hundred students who took the test twice. The mean score and the standard deviation of these 100 students on the first try are 500 and 90 respectively; the mean score and the standard deviation of these 100 students on the second try are 530 and 92 respectively. We also examine the change in Math SAT score (second score – first score) for these 100 students and find out the mean change is 30 with standard deviation of 60. Is there evidence that students improve their Math SAT score on average? Please carefully define your parameters. State the H0 and Ha and carry out an appropriate test. Give the P-value and state your conclusion using significance level a = 0.10.

Answer 1

The test for the mean difference of the test scores. The data for first test and the second used the same students. So the samples are dependent/ related. We want to test if the students improved their score.

Since the data is dependent we will a dependent samples t-test. The claim is students improved that is if after > before: after - before > 0

We also examine the change in Math SAT score (second score – first score) for these 100 students and find out the mean change is 30 with standard deviation of 60.

d: pairwise difference of score = after - before

Is there evidence that students improve their Math SAT score on average? Please carefully define your parameters. State the H0 and Ha and carry out an appropriate test. Give the P-value and state your conclusion using significance level a = 0.10.

Test

$H_{0}:d=0$

$H_{0}:d>0$

This is one left sided test

$\bar{d}$ = 30 S_d = 60 n = 100

Test stat:

d - do SDd/n

Test Stat = 5

Since the population SD is not given and n > 30, we assume the data is normal and t-dist can be used.

p - value = 2 P( > |Test Stat |)

tn-1

=2P(t₉₉ > 5)  .............using t-dist tables

p-vlue =0.0000

Since p-value < 0.10

we reject the null hypothesis.There is sufficient evidence that students improve their Math SAT score on average.