Question

3. Can taking chess lessons and playing chess improve memory? The online article 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	410	610	640	675	500	780	610	625	450	720	675	670
Post-chess Memory Score	550	670	850	650	710	795	700	795	690	710	785	675
Memory Score Difference

Test the claim at the α = 0.05 level of significance that students who participated in the chess program achieve higher memory scores after completion of the program.

1. Fill in the last row of the table with the differences in memory.

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u_d = 0

Alternative hypothesis: u_d < 0

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05.

Analyze sample data. Using sample data, we compute the standard deviation of the differences (s), the standard error (SE) of the mean difference, the degrees of freedom (DF), and the t statistic test statistic (t).

Student 1 2 3 4 5 6 7 8 9 10 11 Pre-chess Post-chess d=B-A (d-dbar)^2 410 550 - 140 1501.5625 610 670 -60 1701.5625 640 850 -210 11826.5625 675 650 25 15939.0625 500 7101 -210 11826.5625 7801 795 - 15 7439.0625 610 700 -90 126.5625 625 795 -170 4726.5625 450 690 -240 19251.5625 720 710 10 12376.5625 675 785 -110 76.5625 670 675 -5 9264.0625 7365 8580 -1215 96056.25 613.75 715 -101.25 8004.6875 12 Sum Mean

S - Σ(d d)2 η –1

s = 93.4472

SE = s / sqrt(n)

S.E = 26.9759

DF = n - 1 = 12 -1

D.F = 11

t = [ (x₁ - x₂) - D ] / SE

t = - 3.75

where d_i is the observed difference for pair i, d is mean difference between sample pairs, D is the hypothesized mean difference between population pairs, and n is the number of pairs.

Since we have a one-tailed test, the P-value is the probability that a t statistic having 11 degrees of freedom is less than - 3.75.

P-value = P(t < - 3.75)

Use the t-value calculator for finding p-values.

P-value = 0.002

Interpret results. Since the P-value (0.002) is less than the significance level (0.05), we have to reject the null hypothesis.

Reject H0. From the above test we have sufficient evidence in the favor of the claim that students who participated in the chess program achieve higher memory scores after completion of the program.