The degree of success at mastering a skill often depends on the method used to l
ID: 3349790 • Letter: T
Question
The degree of success at mastering a skill often depends on the method used to learn the skill. An article reported on a study involving the following four learning methods: (1) visual contact and imagery, (2) nonvisual contact and imagery, (3) visual contact, and (4) control. There were 20 subjects randomly assigned to each method. The summary information on putting performance score is listed in the following table.
(a) Is there sufficient evidence to conclude the mean putting performance score is not the same for the four methods? (Use Table 6 in Appendix A. Round all your intermediate calculations to two decimal places.)
There ---Select--- is is not sufficient evidence to conclude that the mean putting performance score is not the same for the four methods.
(b) Let 1, 2, 3, and 4 represent the true performance score for learning methods 1, 2, 3 and 4, respectively. Calculate the corresponding 95% T-K intervals. (Use Table 7 in Appendix A. Give the answers to two decimal places.)
Explanation / Answer
Let Xi = performance score for learning method i, and we assume
Xi ~ N(µi, i2), i = 1, 2, 3, 4.
We have two questions on hand – testing for equality of µi’s and developing confidence interval for pair-wise difference between these µi’s.
We will combine these two questions into one and obtain answers for both by employing confidence interval for testing of hypothesis.
Back-up Theory
Confidence Interval for (1 - 2) when 1 = 2 = but unknown and n1 = n2 = n is:
(X1bar – X2bar) ± {(t2n – 2, /2)(s)(2/n)}, where
X1bar, X2bar are respective sample means; s = pooled sample estimate of given by s = sqrt{(s12 + s22)/2}; s1, s2 are respective sample standard deviations; t2n – 2, /2 = upper (/2) percent point of t-distribution with degrees of freedom = 2n – 2; and n = common sample size.
Use of CI for testing for (1 - 2) = 0,
If CI for (1 - 2) does not contain zero, the hypothesis, (1 - 2) = 0 is rejected.
Calculations
n =
20
t38,0.025 =
2.024
(2/n) =
0.316228
Difference
Sample
Averages
Sampe
SD's
s^2
s
1 - 2
16.1
15.25
2.23
3.33
8.0309
2.8338843
1 - 3
16.1
12.05
2.23
2.91
6.7205
2.5923927
1 - 4
16.1
9.1
2.23
2.75
6.2677
2.5035375
2 - 3
15.25
12.05
3.33
2.91
9.7785
3.1270593
2 - 4
15.25
9.1
3.33
2.75
9.3257
3.0538009
3 - 4
12.05
9.1
2.91
2.75
8.0153
2.8311305
X1bar – X2bar
{(t2n – 2, /2)(s)(2/n)}
Lower Bound for CI
Upper Bound for CI
0.85
1.813813447
-0.963813447
2.663813447
4.05
1.659247993
2.390752007
5.709247993
7
1.602376653
5.397623347
8.602376653
3.2
2.001458678
1.198541322
5.201458678
6.15
1.954569999
4.195430001
8.104569999
2.95
1.812050927
1.137949073
4.762050927
Since the CI contains zero only for the CI of 1 - 2, only this difference is not significant.
All other differences are significant.
DONE
n =
20
t38,0.025 =
2.024
(2/n) =
0.316228
Difference
Sample
Averages
Sampe
SD's
s^2
s
1 - 2
16.1
15.25
2.23
3.33
8.0309
2.8338843
1 - 3
16.1
12.05
2.23
2.91
6.7205
2.5923927
1 - 4
16.1
9.1
2.23
2.75
6.2677
2.5035375
2 - 3
15.25
12.05
3.33
2.91
9.7785
3.1270593
2 - 4
15.25
9.1
3.33
2.75
9.3257
3.0538009
3 - 4
12.05
9.1
2.91
2.75
8.0153
2.8311305
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