Question

1. Can taking chess lessons and playing chess improve memory? The online article “The USA Junior Chess Olympics Research: Developing Memory and Verbal Reasoning” (New Horizons for Learning, April 2001) described a study in which sixth-grade students who had not previously played chess participated in a program in which they took chess lessons and played chess daily for 9 months. Each student took a memory test (the Test of Cognitive Skills) before starting the chess program and again at the end of the 9-month period. The table below shows the results for 12 randomly selected students.

Student	1	2	3	4	5	6	7	8	9	10	11	12
Pre-chess memory score	510	610	640	675	600	550	610	625	450	720	575	675
Post-chess memory score	850	790	850	775	700	775	700	850	690	775	540	680

Test the claim at the α = 0.01 level of significance that students who participated in the chess program achieve higher memory scores after completion of the program.

1. State the null and alternate hypotheses [3]

2. Check the assumptions [4]

3. Give the test statistic [2]

4. Give the P-value and the conclusion reached about the null hypothesis based on the P-value. [2]

5. Summarize the final conclusion in the context of the claim. [2]

Answer 1

Here we are comparing two data sets where the sample subjects are same. So this is a dependent samples t-test. We will use t-dist to compare the mean of the difference of scores

d = Pre -post

We want to test whether the post memoery is higher or not. So in terms of 'd' we are testing whether 'd <0' or not.

	Pre-chess memory score	Post-chess memory score	d	d^2
1	510	850	-340	115600
2	610	790	-180	32400
3	640	850	-210	44100
4	675	775	-100	10000
5	600	700	-100	10000
6	550	775	-225	50625
7	610	700	-90	8100
8	625	850	-225	50625
9	450	690	-240	57600
10	720	775	-55	3025
11	575	540	35	1225
12	675	680	-5	25
Total			-1735	383325
Mean			-144.583
SD			109.7406

Mean =

SD =

State the null and alternate hypotheses [3]

: d = 0 .(There is no change in the scores)

: d < 0 .(The scores are higher after the completion of the program.)

Check the assumptions [4]

1. The sample is randomly selected
2. the samples are independent of each other
3. The data comes from a normal population.

Give the test statistic [2]

Test Stat =

null difference = 0

n = 12

Test Stat = -4.564

Give the P-value and the conclusion reached about the null hypothesis based on the P-value. [2]

p- value =P( > |test stat| ) ..........this is one tailed so we check only on one side

= P( > 4.56)

p-value = 0.00041 .............using t-dist tables

Summarize the final conclusion in the context of the claim. [2]

Since p-value < 0.01

We reject the null hypothesis at 0.01 level.

We conclude that the chess program significantly improved the memory of the students.