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GLENDON COLLEGE - DEPARTMENT OF PSYCHOLOGY
Introduction to Statistics 2 – GL/PSYC 2531.03 E – Fall 2017
Problem set #1 - distributed Oct 2, 2017
Value: 25% of final grade
due: Oct 16, 2017
Answer all questions below. You should do all calculations by hand, because the point is to
make sure that you understand the operations and the steps involved. You can use a standard
calculator to help with the mechanics of calculating, but not computers. You should calculate so
that your answer is correct to 2 decimal places: that is, calculate all intermediate calculations to
3 correct decimal places (4 is even better), then round the final value that you report to 2
places. Grades focus on getting the correct answers, but you must show your calculations. If
your answer is wrong, calculations may show how errors occurred and get you partial marks.
1. A nutritionist studying weight gain in college freshmen obtains a sample of n=20
first-year students at the state college. Each student is weighed on the first day of
school and again on the last day of the semester. The scores below measure the change
in weight, in pounds, for each student. Positive scores indicate weight gain (# pounds)
during the semester and negative scores indicate weight loss.
4 pt
+5, +6, +3, +1, +8, +5, +4, +4, +3, –1, +2, +7, +1, +5, +8, 0, +4, +6, +5, +3
a. Sketch a histogram showing the distribution of weight-change scores.
b. Calculate mean, median, and mode weight-change scores for this sample.
c. Does there appear to be a consistent trend in weight change during the semester?
Explain their answer
d. What shape is the distribution and what does that tell you about weight change?
ANS:
a. histogram (1 pt) : 1 pt for correct axes, frequencies, layout, labeling
: 0.5 pt if H has errors but is “more or less” correct (your judgment)
: 0 pt if H has errors so serious that the picture is incorrect
.
5 │
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┌───┐
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3 │
┌───┬───┤
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2 │
┌───┐
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├───┐
┌───┐
f
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1 │ ┌───┬───┤
├───┤
│
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├───┤
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│ │
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└─┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───┴─ X
1 0
+1 +2 +3 +4 +5 +6 +7 +8
Weight change (pounds) during the first semester, freshman year
1
1 pt
1.5 pt
0.5 pt
1 pt
b. mean - median - mode of weight change (1.5 pt = @ 0.5 pt)
(grading: 0.5 pt if both M and calculations correct (calcs must be shown)
[0.5 pt OK if minor calculation errors, use your judgment]
Mean:
∑x/n = 79/20 = 3.950 pounds (calc) = 3.95 pounds (final ans).
In words: on average, these students gained 3.95 pounds during the semester.
NB: make sure to check calculated to 3+ decimal places. Here not so
critical, other places important.
Median: Mdn = 4 (score in the middle = 50:50 above:below)
Mode: Md = 5 (most commom score, 5 occurs 4 times; visible in histogram)
c. consistent trend in weight change (0.5 pt)? Yes, nearly all students gained weight.
d. shape & wt change pattern (1.0 pt) :
skewed negatively (0.5 pt) : i.e., longer ‘tail’ on the left = low side
wt change pattern (0.5 pt) : small wt change for most, large for few
2. For each of the following, find the exam score X that should give the better grade.
In each case, explain your answer.
a. X=56 on an exam with μ=50 and σ=4, or X=60 on an exam with μ=50 and σ=20.
b. X=40 on an exam with μ=45 and σ=2, or X=60 on an exam with μ=70 and σ=20.
c. X=62 on an exam with μ=50 and σ=8, or X=23 on an exam with μ=20 and σ=2.
3 pt
ANS: 1 pt @ question a-b-c : 0.5pt (z-calculations) + 0.5 (decision)
a. X=56 corresponds to z = 1.50, X=60 corresponds to z = 0.50. SO X=56 better grade
b. X=60 corresponds to z = –0.50, X=40 corresponds to z = –2.50. SO X =60 better grade
c. X=62 corresponds to z = 1.50, and X=23 also corresponds to z = 1.50. SO the two
scores have the same relative position & shld receive the same grade (neither better).
3. A distribution with mean μ = 56 and standard deviation of σ = 20 is transformed into
a standardized distribution with μ. = 50 and σ = 10. Find the new, standardized
score for each of the following values from the original population.
a. X = 46
b. X = 40
c. X = 80
3 pt
ANS: 1 pt @ a, b, c. Calculations : z = (X-56)/20¸then standardized X = Xs = z*10+50
a. Xs = 45 (z = –0.50) z = (46-56)/20 = -10/20 = -0.5
Xs = -0.5*10+50 = -5+50 = 45
Xs = -0.8*10+50 = -8+50 = 42
b. Xs = 42 (z = –0.80) z = (40-56)/20 = -16/20 = -0.7
c. Xs = 62 (z = 1.20)
z = (80-56)/20 = 24/20 = 1.2
Xs = 1.2*10+50 = 12+50 = 62
4. Over the past 10 years, the local school district has measured physical fitness for all
high school freshmen. During that time, the average score on a treadmill endurance
task has been μ=19.8 minutes with a standard deviation of σ=7.2 minutes. Assuming
2
3 pt
that the distribution is approximately normal, find each of the following probabilities.
a. What is the probability of randomly selecting a student with a treadmill time greater than
25 minutes? In symbols, p(X > 25) = ?
b. What is the probability of randomly selecting a student with a time greater than 30
minutes? In symbols, p(X > 30) = ?
c. If the school required a minimum time of 10 minutes for students to pass the physical
education course, what proportion of the freshmen would fail?
ANS: @ Q a,b,c = 1 pt
: z value for target X-value = 0.5 pt
: p (> or < z) + answer the Q = 0.5 pt
a. for X=25, z = 0.72, p(z > 0.72) = 0.2358
b. for X=30, z = 1.42, p(z > 1.42) = 0.0778
c. for X=10, z = -1.36, p(z < –1.36) = 0.0869
5. Government critics were concerned that the quality of the public schools in their
12 pt
community was poor, so they ran a study to compare school satisfaction of teachers
working private versus public schools in their community. The teachers’ school satisfaction
ratings they obtained are below (higher rating = greater satisfaction).
Teacher
A
B
C
D
E
Public
School
2
4
6
8
10
Private
School
7
8
10
8
12
a. Assume these ratings represent samples of 5 randomly selected teachers from
5 pt
each type of school (i.e., every score is from a different teacher). Use an appropriate statistical test to assess whether public vs. private school satisfaction
ratings differ (use α = 0.05).
i.
State the test you will use and why
State your hypotheses (H1 & H0) and critical region
ii.
iii.
Calculate the test statistic
iv.
Interpret your findings
b. Now assume these ratings represent a sample of 5 randomly selected teachers
5 pt
and each teacher is tested twice - once before and once after their school was
converted from public to private (i.e., pre-post). Use an appropriate statistical test
to assess whether public vs. private school satisfaction ratings differ (α = 0.05).
i.
State the test you will use and why
State your hypotheses (H1 & H0) and critical region
ii.
iii.
Calculate the test statistic
iv.
Interpret your findings
c. Compare your findings from these two tests (i.e., what do they allow you to
2 pt
conclude. Do they differ or not, and why? Assess whether one test is better
than the other and explain your answer.
3
ANSWERS
a. Ratings represent samples of 5 randomly selected teachers from each type of
5 pt
school (i.e., @ score - different teacher). Use an appropriate statistical test to
assess whether public vs. private school satisfaction ratings differ (use α = 0.05).
i.
State the test you will use and why:
Independent samples t-test
0.5 pt
Why: each score is from an independent ‘source’ (teacher)
0.5 pt
ii.
iii.
State your hypotheses (H1 & H0) and critical region
H1: pub-priv school satisfaction ratings differ (i.e., 2-tailed test) &
H0: pub-priv school satisfaction ratings do not differ
Critical region:
tcrit (0.05, 2-tailed, df=8) = 2.306 (t-table G&W Appendix B2)
Calculate the test statistic
0.5 pt
0.5 pt
2.0 pt
Calculations below; highlighted key values students should have obtained
Teacher
A
B
C
D
E
n
sum‐Y
sum‐Y2
M
SS
df
Y1
2
4
6
8
10
Public School
Y1^2
(Y1‐M1)^2
4
16.000
16
4.000
36
0.000
64
4.000
100
16.000
Private School
TOTALS
Y2^2 (Y2‐M2)^2
49
4.000
64
1.000
100
1.000
64
1.000
144
9.000
5
10
45
75
421
641
9.000
7.5
16.000
4
8
Y2
7
8
10
8
12
5
30
220
6.000
40.000
4
pooled s2
2
s
t
7.000
1.400
‐1.793
1.400
Grading: means (0.5), both SS (0.5), both s2 (0.5), t value (0.5)
iv.
Interpret your findings
1.0 pt
Cannot reject H0 (stats conclusion: 0.5 pt)
can’t confirm pub-priv school satisfaction differs (interpretation: 0.5 pt)
b. Now assume these ratings represent a sample of 5 randomly selected teachers
5 pt
and each teacher is tested twice - once before and once after their school was
converted from public to private (i.e., pre-post). Use an appropriate statistical test
to assess whether public vs. private school satisfaction ratings differ (α = 0.05).
i.
State the test you will use and why:
4
Repeated measures t-test
Why: each S rates both public & private school satisfaction
so each subjects is tested repeatedly (twice)
ii.
State your hypotheses (H1 & H0) and critical region
H1: pub-priv school satisfaction ratings differ (i.e., 2-tailed test)
H0: pub-priv school satisfaction ratings do not differ
Critical region:
tcrit (0.05, 2-tailed, df=n-1=4) = 2.776 (t-table G&W App’x B2)
iii. Calculate the test statistic
0.5 pt
0.5 pt
0.5 pt
0.5 pt
2.0 pt
Calculations below; highlighted key values students should have obtained
Teacher
A
B
C
D
E
n
sum‐Y
M
SS
df
Public
School
Y1
2
4
6
8
10
5
30
6.000
Private
School
Y2
7
8
10
8
12
5
45
9.000
Diff
Formulae
D
5
4
4
0
2
(D‐M)^2
4
1
1
9
1
5
5
15
3.000
** OK +3 or ‐3
16
4
s2
4.00
SS/df
s2M
0.80
0.89
3.354
s2 / N
sM
t
sqrt (above)
(M‐0)/sM
Grading: D-mean (0.5), SS (0.5), sM (0.5), t value (0.5)
iv.
Interpret your findings
Can reject H0 (statistical conclusion: 0.5 pts)
Can conclude that pub-private school satisfaction differs (0.5 pt)
c. Compare your findings from these two tests (i.e., what do they allow you to
conclude). Do they differ or not, and why? Assess whether one test is better
than the other and explain your answer.
DIFFER:
Y/N : Yes, the two tests (IM-RM) differ.
RM test : teacher satisfaction clearly differs public-private schools
IM test : teacher satisfaction does not clearly differ ….
5
1.0 pt
2 pt
0.5 pt
: variability differs (RM lower than IM) RM higher t significant
0.5 pt
IS ONE TEST BETTER? (several answers OK with suitable reasoning)
YES RM better because it is more sensitive
Why: able to detect a difference if it is possibly there
(impt where consequences are serious if you miss a difference)
IM better because it is more cautious/conservative
Why: does not identify a difference unless evidence is very strong
(impt where consequences are serious if you claim a ‘fake’ diff)
: OR not confounded by order effects whereas RM can be
NO
One isn’t better than the other, because they serve different
purposes (per the “why” above)
1.0 pt
Why
~ THE END ~
6
GLENDON COLLEGE - PSYCHOLOGY
Introduction to Statistics – GL/PSYC 2531, Fall 2017
Problem set #2
Value: 30%
due: Nov. 20, 2017
Answer all questions below, doing all calculations by hand. You can use a standard calculator to
help with calculation mechanics but not computers. You must show all your calculations so that
it is clear how you got your answers. Calculate so your final answers are correct to 2 decimal
places: Numbers used in calculating should be correct to 3-4 decimal places (e.g., OK to write
1.2 if the value is 1.2000; OK to use 0.1667 as 0.167 but not 0.17). Then round the final value to
2 decimal places when you report it, nowhere before that, and not if you are using it in further
calculations. Penalty 0.5 pt for rounding when you shouldn’t.
SECTION A. For the following questions and data, determine the appropriate statistical test,
calculate it, and use your findings to answer the researchers’ questions.
(40 pts)
1. Professors tested their students’ claim that drinking coffee while studying improves their
exam performance. They randomly selected 36 students from a statistics course, divided
them randomly into 3 groups, and gave them coffee while they studied statistics. One group
got high-caffeine coffee, one got moderate caffeine, and one got decaf. The professors then
tested them on the material studied. Grades (%) on this exam are in the table below. Answer
the questions below to test if this study confirms students’ claims.
(15 pts)
a.
b.
c.
d.
e.
f.
Subject
Decaf
1
2
3
4
5
6
68
74
59
61
65
72
ΣX =
ΣX2 =
SSWG =
815
56009
656.92
Moderate
High
Subject
Caffeine caffeine
68
72
7
80
65
8
64
68
9
65
83
10
69
79
11
79
92
12
865
62891
538.92
922
71622
781.67
Mean
s2
s
Decaf
80
58
65
60
78
75
67.92
59.72
7.73
Moderate
High
caffeine caffeine
80
69
63
74
69
78
70
83
83
88
75
71
72.08
48.99
7
76.83
71.06
8.43
What statistical test will you need to conduct? Explain why.
(1 pt)
State the null and alternative hypotheses.
(1 pt)
Using α=0.05, what is the tabled critical value to use? Explain how you determined it. (1 pt)
Graph the results for the three groups, give an eyeball assessment of findings
(2 pt)
Calculate the appropriate statistical test.
(8 pt)
Evaluate the null and alternative hypotheses based on your statistical findings
(2 pt)
1
2. Re-analyze Question 1 data, now assuming the professors used a random sample of only 12
students and manipulated each student three times. That is, they gave each student coffee
while they studied plus an exam on the material studied three times: once with high-caffeine
coffee, once with moderate caffeine coffee, and once with decaf. (The professors
counterbalanced the order in which they gave subjects each type of coffee). Answer the
questions below to test if this study confirms students’ claims.
a.
b.
c.
d.
e.
f.
What statistical test will you need to conduct? Explain why.
State the null and alternative hypotheses.
Using α=0.05, what is the tabled critical value to use? Explain how you determined it
Calculate the appropriate statistical test.
Evaluate the null and alternative hypotheses based on your statistical findings
Compare your results with those from Q #1, i.e., are they the same or not, and why?
(1 pt)
(1 pt)
(1 pt)
(8 pt)
(2 pt)
(2 pt)
3. Use a two-factor independent measures ANOVA (α=.05) to assess the significance of the
effects of drugs, hospitals, and their interaction in the following study. Researchers tested
patients suffering from depression in three different hospitals with a three different drugs that
may help control depressive symptoms. Patients in each hospital (H1-H3) were divided into
three equal groups - one group got Drug 1, one group got Drug 2, one group got Drug 3.
Drug 1
Drug 2
Drug 3
H1
H2
H3
H1
H2
H3
H1
H2
H3
10
7
11
1
6
4
3
2
5
8
4
9
2
7
3
2
1
6
7
3
10
1
6
6
3
2
4
9
2
9
4
5
4
3
3
5
1
2
3
4
6
5
These data generated the following summary table
Source
df
SS
Hospital
2
145.33
Drug
2
45.24
Interaction
4
93.33
Error
Total
??
??
1.45
41
331.90
2
a.
b.
c.
d.
What test should you use to test the three hypotheses? Explain why
(1 pt)
State the hypothesis for each of the 3 statistical tests
(1.5 pt)
Find decision criteria for each test using a = 0.05 (first calculate degrees of freedom) (2 pt)
Calculate the statistical tests by completing the summary table (fill empty cells and (3.5 pt)
?? columns where suitable)
e State your conclusion for each tests (in statistical and real terms)
(3 pt)
f. Draw a graph to show scores for both factors then, by eye, interpret the interaction
(4 pt)
SECTION B. For each question below, decide what statistical procedure is the most appropriate
for answering the researchers’ question(s) and explain your choice. You do not have to carry out
the statistical test—just figure out which test you should use.
(@ 2 pts = 10 pts)
1. Students are greatly concerned about the consistency of marking between lecturers. Lecturers
quite often gain reputations as ‘hard’ or ‘light’ markers, but there is often little to substantiate
these reputations. So, a group of students tested the consistency of marking by submitting the
same essay to four different lecturers. The mark given by each lecturer was recorded for each
of the 8 essays (table below shows each lecturer’s marks for each essay). What statistical
procedure should students use to test whether lecturers grade differently (hard - light)?
Essay
1
2
3
4
5
6
7
8
mean
Lecturer
Dr. Field
62
63
65
68
69
71
78
75
68.875
Dr. Smith
58
60
61
64
65
67
66
73
64.25
Dr. Scrote
63
68
72
58
54
65
67
75
65.25
Dr. Death
64
65
65
61
59
50
50
45
57.375
Mean
Variance
61.75
64.00
65.75
62.75
61.75
63.25
65.25
67.00
6.92
11.33
20.92
18.25
43.58
84.25
132.92
216.00
2. A professor conducted a study to test if where students sat in his classroom affected their
grades. For one class of 200 students, he divided the class into four equal-sized groups
according to where they normally sat in the classroom (front and center, far left side, far right
side, back) and recorded their grades on the final exam. He used the appropriate statistical
procedure to test his hypothesis and obtained a significant result: where students sit did
affect their grades. He then wanted to estimate how large a grade difference seating location
makes (i.e., effect size). What statistical tests should he have used (hypothesis, effect size)?
3. The US Census Bureau publishes information on the ages of married people. Suppose we
want to decide whether the mean age of married men differs from the mean age of married
women in the USA. Ten (10) married couples in the USA were selected at random; their ages
(in years) are shown in the table below. What test could you use to make this decision?
3
Couple
1
2
3
4
5
6
7
8
9
10
Husband
59
21
33
78
70
33
68
32
54
52
Wife
53
22
36
74
64
35
67
28
41
44
Difference (D)
6
-1
-3
4
6
-2
1
4
13
8
4. Red palm oil, due to its high content of vitamin A, is thought to reduce the occurrence and
severity of malaria for young children. To assess if this is true, researchers developed a food
supplement containing a high dose, a low dose, or no (placebo) red palm oil. They tested the
supplement on boys and girls (75 each), in case they differ in their exposure to malaria and
response to the red palm oil supplement, and randomly assigned 25 children of each sex to
each of the supplement levels. The study design is shown below. What statistical test should
they use to test their hypotheses about red palm oil supplements, sex, and malaria?
Red palm oil
Placebo
Low Dose
High Dose
Total
Girls
25
25
25
75
Boys
25
25
25
75
Total
50
50
50
150
5. Neurosurgeons wanted to determine whether a dynamic system (Z-plate) reduced
neurosurgery operative time relative to a static system (ALPS plate). They obtained the data
shown below on operative times (in minutes), for the two systems. What statistical procedure
and test criterion should they use to assess whether the data provide sufficient evidence to
conclude that the mean operative time is less with the dynamic than the static system?
THE END
4