The effect of 4 types of exercise on psychological well-being
Q1 [24 points]
An investigator studied the effect of 4 types of exercise on psychological well-being by
placing five volunteers on an aerobic exercise regimen and having them respond to a
measure of psychological well-being (y) after each exercise regimen. The four types of
exercise differ in terms of the duration of training, with B4 having the longest duration,
followed by B3, B2 and B1. Each participant followed a particular exercise regimen for
one week before switching to another exercise regime. All participants followed the same
exercise regimen order: B1 in week 1, B2 in week 2, B3 in week 3 and B4 in week 4.
Scores on this measure could range from 1 to 70, with higher values indicating greater
psychological well-being. All participants were of normal weight and health, and none was
presently involved in a physical fitness program. The well-being scores for the four
assessments are listed in the table below. One-way repeated measures ANOVA was used to
test for a relationship between the amount of exercise and psychological well-being. Unless
otherwise specified, use α=0.05 for all hypothesis testing below. Show your calculations.
State clearly the hypotheses, decision rule, decision and your conclusion. Table below
shows data from subject 1 and subject 5.
Subject ( i )
Types of Exercise (B)
B1
(week 1)
B2
(week 2)
B3
(week 3)
B4
(week 4)
1 60 59 63 68
… … … … ….
5 63 62 66 67
Column mean 56 57.4 61.2 63
The data is analyzed by one-way repeated measures ANOVA which is given by:
,
()()( or ),
ik kiT ik
ik T TS TB ik TBS
Y r
Y Y i k i k
+++=
+−−+−+−+=
βπµ
µµµµµ µµµ
where Yik is the psychological well-being response from subject i after exercise type k.
Sample grand mean is 59.4.
(a) Plot out the average psychological well-being assessment at the four measured period
listed in the table. Based on the plot, what can you comment about the relationship
psychological well-being and the amount of exercise? [3 points]
(b) Decompose the score Y52 into the grand mean, “subject” effect, treatment effect and
the residual. [2 points]
(c) From SPSS/Jamovi output, it is found that SStreatment =158.80, SSsubject = 956.80 and
SSresidual = 31.20. Based on the given information, construct the ANOVA summary
table (No need to compute Fobs here, you will be asked to do so in (d)). [4 points]
4
(d) Test if there is significant main effect of this treatment factor. [3 points]
(e) Based on the design and results, can we conclude that the longer the duration of
exercise, the better its effect on psychological well-being. Is there any
confounding variable in this study? If so, what is it and what would you suggest to
control this confounding variable? [4 points]
(f) If the experiment followed a between-subjects design, and the dataset was analyzed as a
one-way between-subjects ANOVA. Construct the ANOVA summary table for this
re-analysis. Re-test the hypothesis in (d) and does the conclusion differ from that in (d)?
What are the major differences between the F-tests in (d) and (f) for testing the
significance of treatment effect?
[8 points]
Q2 [30 points]
A researcher wished to determine whether the regression of achievement (Y) on
achievement motivation (X) is the same for males and females. For a sample of 10 males
and 10 females, she obtained measures of X and Y. Part of the data is shown in the
following table.
Male Female
subject (i) Mot. (X) Ach. (Y) subject (i) Mot. (X) Ach. (Y)
1 2 12 11 1 12
2 2 14 12 1 14
… … … … … …
10 10 23 20 10 18
Given: ;114 ;318 ;1942 ;838 5218
20
1
2
20
1
2
20
1
20
1
20
1
= = ∑∑∑ = = ∑∑ = = = = = i=
i
i
i i
i
i
i
i
i
Xi Y YX X Y
Correlation between Mot. (X) and Ach. (Y) = 0.742.
Furthermore, for male group:
� 𝑋𝑋𝑖𝑖
10
𝑖𝑖=1
= 58;� 𝑌𝑌𝑖𝑖
10
𝑖𝑖=1
= 168;� 𝑋𝑋𝑖𝑖
10
𝑖𝑖=1
𝑌𝑌𝑖𝑖 = 1056;� 𝑋𝑋𝑖𝑖
2
10
𝑖𝑖=1
= 418;� 𝑌𝑌𝑖𝑖
2
10
𝑖𝑖=1
= 2938
(a) The researcher first fitted a regression to predict Ach. with gender as a dummy-coded
predictor (d1, where females are assigned 1 and males are assigned 0) as:
M1: Y = β0+ β1 d1 + ε.
Find the estimate for β0 and β1. [3 points]
(b) If the gender variable is effect-coded instead, what should the regression model
be (i.e. determine the regression coefficients in M2)?
M2: Y = β0+ β1 f1 + ε, where f1 = 1 for females and f1 = -1 for males. [3 points]
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(c) The researcher further fits a model by regressing achievement (Y) on achievement
motivation (X) to the whole dataset, i.e. M3: Y = β0+ β2 X + ε.
Find the regression coefficient estimate in M3. [4 points]
(d) The researcher fitted another two models (M4 and M5) and below shows the
summary information.
M4: Y = β0 + β1 f1 + β2 X + ε 0.635 ˆ 2 Ry•12 =
M5: Y = β0 + β1 f1 + β2 X + β3 (f1*X) + ε 0.727 ˆ 2 Ry•123 = ,
where f1 = 1 for females and f1 = -1 for males.
Test the following hypotheses. Use α = 0.05. Show your calculations. State clearly the
critical value(s), decision rule and decision.
(i) H0: β3 = 0 (vs. H1: β3 ≠ 0). [4 points]
(ii) H0: β1 = β2 = β3 = 0 (vs. H1: not H0). [4 points]
(e) The fitted regression for M5: Y = β0 + β1 f1 + β2 X + β3 (f1*X) + ε is:
Y = 11.789 + 0.789 f1 + 0.716 X - 0.284 (f1*X) + e, where
f1 = 1 for females and f1 = -1 for males.
(i) Based on the given information, obtain the fitted regression lines (regressing Y on X)
for male and female groups. [3 points]
(ii) Based on (i), graph the two lines with the Achievement score on Y-axis and the
Motivation score on X-axis (keep the range of Motivation score from 0 to 10 in your
plot). Also, describe the similarity and difference in the relationship between the
motivation and achievement between males and females? [4 points]
(iii) How can we test if there is statistical significant difference between the two regression
lines in part (i) above (i.e. do the two regression lines overlap on top of each other)?
Show your calculations.