Soc2205a/b Final Review
Healey 1e 8.2, 2/3e 7.2
• Problem information: = 3.3 = 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
X
Step 4: Calculations( )
1
3.8 3.3( )
.53 117 1
.5( )
.53 116
.5( )
.53 10.77
.5( )
.0492( ) 10.16
XZ obtained
s N
Z obtained
Z obtained
Z obtained
Z obtained
Z obtained
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 Ps = .14N = 527
• 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)
X
Step 4: Calculations
( )(1 )
.14 .10( )
(.10)(.90) / 527
.04( )
.09 / 527.04
( ).00017.04
( ).013
( ) 3.06
s u
u u
P PZ obtained
P P N
Z obtained
Z obtained
Z obtained
Z obtained
Z obtained
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)
Healey 1e 9.3a, 2/3e 8.3a• Problem information:
Hockey Football
= 460 = 442
s1 = 92 s2 = 57
n1 = 102 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
2X
2X 2X
1X
Step 4: Calculations1 2
1 2
2 2
2 2
1 1
(92) (57)
102 1 117 1
8464 3249
101 116
83.80 28.01
111.81
10.57
X X
X X
X X
X X
X X
X X
s s
N N
Step 4: Calculations (cont.)
• Z1 2( ) (460 442) 18
1.7010.57 10.57X X
X X
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!
Healey 1e 9.12a, 2/3e 8.12a• Problem information:
Special Regular
Ps1 = .53 Ps2 = .59
n1 = 78 n2 = 82Use 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
2X 2X
Step 4: Calculations
1 1 22
1 2
(78)(.53) (82)(.59) 41.34 48.380.56
78 82 160u
s sN P N PP
N N
1 2
1 2
78 82 160(1 ) (.56)(.44) .2464 (.4964)(.1582) .079
(78)(82) (6396)p p u u
N NP P
N N
Step 4: Calculations (cont.)
• Z1 2( ) .53 .59 .06
0.76.079 .079p p
s sP P
Step 5: Interpretation
• α = .05
• Zcr = - 1.65
• Fail to reject Ho
• The new program did not work.
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
2X 2X
Step 4: Make Computational Table
Grand Mean= (include n-sizes too)
Urban Suburban Rural
∑Xi
∑X2
Group Means
Step 4: Calculations (cont.)
22 2 2
2 2 2
10(49.5 52) 10(59.3 52) 10(47.2 52)
10( 2.5) 10(7.3) 10( 4.8)
10(6.25) 10(53.29) 10(23.04)
62.25 532.9 230.4
825.8
k kSSB N X X
SSB
SSB
SSB
SSB
SSB
Step 4: Calculations (cont.)
SSW = SST - SSB
SSW = 3590 – 825.8
SSW = 2764.2
2 2
284710 (30)(52)
84710 81120
3590
SST X NX
SST
SST
SST
Step 4: Calculations (cont.)
Within estimate (MSW)
Between estimate (MSB)
F = Between estimate (MSB) / within estimate (MSW) = 412.9 / 102.38 = 4.03
2764.2102.38
27
SSW
dfw
825.8412.9
2
SSB
dfb
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)
Healey 1e 11.5, 2/3e 10.5• Problem information:Salary Union Non-union TotalHigh 21 29 50Low 14 36 50Total 35 65 100
• 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, X2
cr = 3.841
2X 2X
Step 4: Expected Frequencies
Top left cell:
Top right cell:
Bottom left cell:
Bottom right cell:
(50)(35) 175017.50
100 100fe
(50)(65) 325032.50
100 100fe
(50)(35) 175017.50
100 100fe
(50)(65) 325032.50
100 100fe
Step 4: Computational Table
fo fe fo–fe (fo - fe)2 (fo - fe)2/fe
21 17.50 3.50 12.25 0.70
29 32.50 -3.50 12.25 0.38
14 17.50 -3.50 12.25 0.70
36 32.50 3.50 12.25 0.38
N=100 0.00 χ 2(obt.) = 2.16
% and Strength of Association
Salary Union Non-unionHigh 60% 44.6%Low 40% 55.4%Total 100% 100%
• Max. difference: 15.4%
• Strength: Phi =
• Weak association.
2X 2X
147.100
16.22
n
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.
Healey 1e 14.8, 2/3e 12.8Problem information: AuthoritarianismDepression Low Moderate High TotalFew 7 8 9 24Some 15 10 18 43Many 8 12 3 23Total 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
2X 2X
Step 4: CalculationsNs: 7 (10+18+12+3) = 7 (43) = 301 8 (18+3) = 8 (21) = 168
15 (12+3) = 15 (15) = 22510 (3) = 30
• Total Ns = 724
Nd: 9 (15+10+8+12) = 9 (45) = 4058 (15+8) = 8 (23) = 18418 (8+12) = 18 (20) = 36010 (8) = 80
• Total Nd = 1029
Step 4: Calculations (cont.)
There is a weak, negative relationship between parenting style and depression.
Zobt<Zcrit. Fail to reject Ho. The association is not significant (Note: Hypothesis test. Use 5 step model)
174.01753
304
1029724
1029724
ds
ds
nn
nnG
78.275.87
1753174.
174.(190
1029724174.
1 22
GN
nnGz ds
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 14B 11 12C 12 10D 10 9E 15 8F 9 7G 7 10H 3 15I 10 12J 9 2
Scattergram
Y=a+bX
Make A Computational Table
Case X Y X2 Y2 XY 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 2Totals ΣX ΣY ΣX2 ΣY2 ΣXY X Y
Totals of Computational Table
X = 96 Y = 99 X²= 1010 Y²= 1107 XY= 918• 9.6• 9.9
XY
Slope (b)
* 3 decimals
b = -.367
2 2
2
( )( )
( )
(10)(918) (96)(99)
(10)(1010) (96)
9180 9504
10100 9216324
8840.37
N XY X Yb
N X X
b
b
b
b
Y-intercept (a)
42.13
)6.9)(367.(9.9
a
a
XbYa
Pearson’s r
* 3 decimals
r = -.306
2 2 2 2
2 2
( )( )
[ ( ) ][ ( ) ]
(10)(918) (96)(99)
[(10)(1010) (96) ][(10)(1107) (99) ]
324
(884)(1269)
324
1121796324
1059.15.31
N XY X Yr
N X X N Y Y
r
r
r
r
r
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.