Soc2205a/b Final Review
Healey 1e 8.2, 2/3e 7.2
• Problem information:
 = 3.3
X = 3.8
s = .53
n = 117
• Use the 5-step method….
• Note:
– 1 sample, Interval-ratio
– Sample is large n ≥ 100
• z-test
– Question asks “Is there a significant difference?”
• 2-tailed test
Step 4: Calculations
X 
Z (obtained ) 
s
N 1
3.8  3.3
Z (obtained ) 
.53
117  1
.5
Z (obtained ) 
.53
116
.5
Z (obtained ) 
.53 10.77
.5
Z (obtained ) 
.0492
Z (obtained )  10.16
Step 5: Interpretation
• α = .05
• Zcr = ± 1.96
• Reject Ho
• Sociology majors are significantly different
(Z=10.16, α = .05)
Healey 1e 8.11, 2/3e 7.11
• Problem information:
Pu = .10
N = 527
X
Ps = .14
• Use the 5-step method….
• Note for Steps 1 - 3:
– 1 sample, Nominal
– Sample is large n ≥ 100
• z-test
– Question asks “Are older people more likely…?”
• 1-tailed test (Note: question says do a 2 tailed test also)
Step 4: Calculations
Z (obtained ) 
Ps  Pu
Pu (1  Pu ) N
Z (obtained ) 
.14  .10
(.10)(.90) / 527
.04
Z (obtained ) 
.09 / 527
.04
Z (obtained ) 
.00017
.04
Z (obtained ) 
.013
Z (obtained )  3.06
Step 5: Interpretation
• α = .05
• Zcr = +1.65 (for 1-tailed)
• Reject Ho
• Older people are more likely to be
victimized. (Z=3.06, α = .05)
X2
Healey 1e 9.3a, 2/3e 8.3a
• Problem information:
Hockey
X 1 = 460
s1 = 92
n1 = 102
Football
X 2 = 442
s2 = 57
n2 = 117
• Use the 5-step method….
• Note:
– 2 samples, Interval-ratio
– Sample is large n ≥ 100
• z-test
– Question asks “Is there a significant difference?”
• 2-tailed test
Step 4: Calculations


X  X
X  X

s1 2
s22

N 1 1 N 2 1

(92) 2
(57) 2

102  1 117  1

X  X

8464 3249

101
116

X  X

83.80  28.01
X  X

111.81
X  X
 10.57


Step 4: Calculations (cont.)
• Z
( X 1  X 2) (460  442) 18



 1.70
X  X
10.57
10.57
Step 5: Interpretation
• α = .05
• Zcr = ± 1.96
• Fail to reject Ho
• Hockey players are not significantly
different from football players.
• What if the question had asked “Do hockey
players have a higher aptitude score…?”
• Try conducting the significance test again!
X2
Healey 1e 9.12a, 2/3e 8.12a
• Problem information:
Special
Regular
Ps1 = .53
Ps2 = .59
n1 = 78
n2 = 82
Use the 5-step method….
• Use the 5 step method…
• Note:
– 2 samples, Nominal
– Sample is large n ≥100
• z-test
– Question asks “Did the new program work? (i.e. is it
better”
• 1-tailed test
Step 4: Calculations
N 1 Ps1  N 2 Ps 2 (78)(.53)  (82)(.59) 41.34  48.38
Pu 


 0.56
N1  N 2
78  82
160
N1  N 2
78  82
160
 p  p  Pu (1  Pu )
 (.56)(.44)
 .2464
 (.4964)(.1582)  .079
N 1N 2
(78)(82)
(6396)
Step 4: Calculations (cont.)
• Z
( Ps1  Ps 2)

p  p
.53  .59 .06


 0.76
.079
.079
Step 5: Interpretation
• α = .05
• Zcr = - 1.65
• Fail to reject Ho
• The new program did not work.
X2
Healey 1e 10.8a, 2/3e 9.8a
• Problem information:
– Occupational Prestige Scores for 3 Groups (Urban,
Suburban, Rural)
• Use the 5 step method…
• Note:
– 3 samples, Interval-ratio
• F-test, One-way ANOVA
– Question asks “Are there differences by place of
residence (Urban, Suburban, Rural)
– dfw = N - k = 30 - 3 = 27
– dfb = k - 1 = 3 - 1 = 2
– Fcr = 3.35
Step 4: Make Computational Table
Urban
Suburban
Rural
∑Xi
∑X2
Group
Means
Grand Mean=
(include n-sizes too)
Step 4: Calculations (cont.)
SSB   Nk  Xk  X 
2
SSB  10(49.5  52)  10(59.3  52)  10(47.2  52)
2
2
SSB  10(2.5)  10(7.3)  10(4.8)
2
2
2
SSB  10(6.25)  10(53.29)  10(23.04)
SSB 62.25  532.9  230.4
SSB  825.8
2
Step 4: Calculations (cont.)
SST   X  NX
2
2
SST  84710  (30)(52)
SST  84710  81120
SST  3590
SSW = SST - SSB
SSW = 3590 – 825.8
SSW = 2764.2
2
Step 4: Calculations (cont.)
SSW
2764.2
Within estimate (MSW) 

 102.38
dfw
27
Between estimate (MSB)  SSB  825.8  412.9
dfb
2
F = Between estimate (MSB) / within estimate (MSW)
= 412.9 / 102.38 = 4.03
Step 5: Interpretation
• α = .05
• Fcr = 3.32
• Reject Ho
• At least one of the groups (urban,
suburban, rural) is significantly different.
• (F = 4.03, df = 2, 27, α = .05)
X2
Healey 1e 11.5, 2/3e 10.5
• Problem information:
Salary
Union
Non-union
High
21
29
Low
14
36
Total
35
65
Total
50
50
100
• Is there a relationship? Answer the 3 questions…
• Use the 5 step method for hypothesis test.
• Note: Tabular Data, Nominal x Ordinal
– Df= (rows-1 x columns-1) = 1
– α=.05, X2cr = 3.841
Step 4: Expected Frequencies
Top left cell:
(50)(35) 1750
fe 

 17.50
100
100
Top right cell:
(50)(65) 3250
fe 

 32.50
100
100
Bottom left cell:
fe 
(50)(35) 1750

 17.50
100
100
(50)(65) 3250

 32.50
Bottom right cell: fe 
100
100
Step 4: Computational Table
fo
fe
fo–fe
21 17.50 3.50
29 32.50 -3.50
14 17.50 -3.50
36 32.50 3.50
N=100
0.00
(fo - fe)2
12.25
12.25
12.25
12.25
(fo - fe)2/fe
0.70
0.38
0.70
0.38
χ 2(obt.) = 2.16
X2
% and Strength of Association
Salary
High
Low
Total
Union
60%
40%
100%
Non-union
44.6%
55.4%
100%
• Max. difference: 15.4%
• Strength: Phi =
2

n
• Weak association.
2.16
 .147
100
Step 5: Decision and Interpretation
• Fail to reject Ho
• There is no significant relationship
between salary levels and unionization.
• Three questions:
– Association?
Not significant
– Strength?
Weak, Phi = .147
– Pattern?
Union members more likely
to make high salary while non-union more
likely to make low salary.
X2
Healey 1e 14.8, 2/3e 12.8
Problem information: Authoritarianism
Depression
Low Moderate High Total
Few
7
8
9
24
Some
15
10
18
43
Many
8
12
3
23
Total
30
30
30
90
• Is there a relationship? Answer the 3 questions…
• Note: Tabular Data, Ordinal x Ordinal, Gamma
• Use the 5 step method for hypothesis test.
– α=.05, Zcr = ±1.96
Step 4: Calculations
Ns:
7 (10+18+12+3) = 7 (43) = 301
8 (18+3) = 8 (21) = 168
15 (12+3) = 15 (15) = 225
10 (3) = 30
• Total Ns = 724
Nd:
9 (15+10+8+12) = 9 (45) = 405
8 (15+8) = 8 (23) = 184
18 (8+12) = 18 (20) = 360
10 (8) = 80
• Total Nd = 1029
Step 4: Calculations (cont.)
n s  nd
724  1029
304
G


 0.174
n s  nd
724  1029
1753
There is a weak, negative relationship between parenting
style and depression.
z G
n s  nd

N 1 G
2

 .174
724  1029

90 1  (.174 2

1753
 .174
 .78
87 .275
Zobt<Zcrit. Fail to reject Ho. The association is not
significant (Note: Hypothesis test. Use 5 step model)
Step 5 Interpretation
• Answering the 3 questions….
• Association?
Not significant
• Strength?
Weak, G = -.174
• Pattern/Direction?
Negative, parents
who are higher in authoritarianism have
children with fewer depression symptoms.*
– *calculate % also.
Healey 1e 15.3, 2/3e 13.3
• Is there a relationship?
–
–
–
–
Draw scattergram
Find r and r2
Find regression line
Calculate predicted visitors
for activity = 5 and 18
• Answer the 3 questions…
• Note: Interval-ratio data,
regression and correlation
• Use the 5 step method for
hypothesis test…
– α=.05, df=n-2, tcr = ±2.306
Problem information:
Case Activity Visitors
X
Y
A
10
14
B
11
12
C
12
10
D
10
9
E
15
8
F
9
7
G
7
10
H
3
15
I
10
12
J
9
2
Scattergram
Make A Computational Table
Case
A
B
C
D
E
F
G
H
I
J
Totals
X
10
11
12
10
15
9
7
3
10
9
ΣX
Y
14
12
10
9
8
7
10
15
12
2
ΣY
X2
Y2
XY
ΣX2
ΣY2
ΣXY
X
Y
Totals of Computational Table
•
•
•
•
•
•
•
X = 96
Y = 99
X²= 1010
Y²= 1107
XY= 918
X  9.6
Y  9.9
Slope (b)
N XY  (X )(Y )
b
2
2
N X  (X )
(10)(918)  (96)(99)
b
2
(10)(1010)  (96)
9180  9504
b
10100  9216
324
b
884
b  0.37
* 3 decimals
b = -.367
Y-intercept (a)
a  Y  bX
a  9.9  (.367 )(9.6)
a  13 .42
Pearson’s r
r
r
r
N XY  (X )(Y )
[ N X 2  (X ) 2 ][ N Y 2  (Y ) 2 ]
(10)(918)  (96)(99)
[(10)(1010)  (96) 2 ][(10)(1107)  (99) 2 ]
324
(884)(1269)
324
r
1121796
324
r
1059.15
r  .31
* 3 decimals
r = -.306
Coefficient of Determination (r2)
and Hypothesis (t) test
• Coefficient of Determination:
• r2 = (r)2 = (-.306)2 = .094
• 9.4% of variation in visitors is explained by
activity level
• Hypothesis test:
• Fail to reject Ho (t obs = -.91 < tcr = ±2.306)
Predictions* for Activity Level
• For X = 5
– Ŷ = a + bX = 13.42 + (-.367)(5) = 11.6 visitors
• For X = 18
– Ŷ = a + bX = 13.42 + (-.367)(18) = 6.8 visitors
• *use the calculated prediction values to draw
actual regression line on the scattergram
Summary
• r = -.306
• r2 = .094
• There is a weak, negative relationship
between # of visitors and activity levels for
seniors. As activity levels go down, # of
visitors increases. The relationship is not
significant. Activity levels explain 9.4% of
the variation in # of visitors.