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CHI Square
is used when we have categorical (nominal) rather
than interval / ratio data
can also be used for measurement data, is less
powerful and than typical tests such as means
CHI Square for a multicategory
case
Observed
Expected
Good Bad Medium Total
26
40
15
81
27
27
27
81
CHI square or X2 =   (Observed  Expected)2
Expected

(26  27)
27
X2 = 11.63
2

(40  27)
27
2

(15  27)
27
2
SPSS output
NPar Tests
Chi-Square Test
Frequencies
PAGEQUAL
1.00
2.00
3.00
Total
Observed N
26
40
15
81
Expected N
27.0
27.0
27.0
Residual
-1.0
13.0
-12.0
Te st Statistics
PAGEQUAL
11.630
df
2
As ymp. Sig.
.003
Chi-Squarea
a. 0 c ells (.0% ) have expected frequencies les s than
5. The minimum ex pec ted cell frequenc y is 27.0.
CHI Square for a Contingency
Table Analysis
(when there is more than one variable)
Finance
Newspaper
Good Bad Medium
26 (27) 40 (27) 15 (27)
21 (30) 27 (30) 42 (30)
Total
81
90
The table shows that webpages in the Finance category were more
were more likely to be good than were webpages in the Newspaper
condition. Thus, the column a webpage is in (Good, Bad, or Medium)
graduate) is contingent upon (depends on) the row the webpage is in
(Finance or newspaper category)
SPSS:using the Non Parametric tool
NPar Tests
Chi-Square Test
Frequencies
PAGEQUAL
CATEGORY
1.00 good
2.00 bad
3.00 medium
Total
Observed N
47
67
58
172
Expected N
57.3
57.3
57.3
Residual
-10.3
9.7
.7
1.00 finance
2.00 newspapers
Total
Observed N
81
91
172
Test Statistics
Chi-Square a,b
df
As ymp. Sig.
PAGEQUAL
3.500
2
.174
CATEGORY
.581
1
.446
a. 0 cells (.0%) have expected frequencies less than
5. The minimum expected cell frequency is 57.3.
b. 0 cells (.0%) have expected frequencies less than
5. The minimum expected cell frequency is 86.0.
Expected N
86.0
86.0
Residual
-5.0
5.0
SPSS: using cross tabs
Case Processing Summary
Crosstabs
Cases
Mis sing
N
Percent
0
.0%
Valid
N
PAGEQUAL * CATEGORY
172
Percent
100.0%
Count
PAGEQUAL
1.00 good
2.00 bad
3.00 medium
Total
Chi-Square Te sts
Pearson Chi-Square
Lik elihood Ratio
Linear-by-Linear
As soc iation
N of Valid Cases
Value
16.044 a
16.588
10.016
2
2
As ymp. Sig.
(2-sided)
.000
.000
1
.002
df
172
a. 0 c ells (.0% ) have expected count less than 5. The
minimum expected count is 22. 13.
N
Percent
Mean = 393.2
172
100.0%
PAGEQUAL * CATEGORY Crosstabulation
CATEGORY
2.00
1.00 finance
newspapers
26
21
40
27
15
43
81
91
Total
Total
47
67
58
172
Confidence Limits on Mean
• Sample mean is a point estimate (estimate is in
form of a single number)
• We want interval estimate (a range of numbers),
and be able to specify with 95% confidence that
estimate will lie in that range
– Probability that interval computed this way includes m
= 0.95
CI .95  X  t.025 s X
For Darts Data
CI
 mean  criticalt * std .error
.95
CI
 X t
sX
.95
.025
 4.5  1.98 1.94
 4.5  3.8
 .66  m  8.33
Displaying Confidence Intervals
Comparing Darts and Dow (Means with 95% confidence
intervals)
10
8
6
4
2
0
Darts
Dow
What would 99% confidence intervals look like?
Margin of Error
Generally computed for 95% confidence
Computed for Proportions
Simple Formula
(for estimating sample size before starting study)
Margin = + 1/ sqrt(N)
For sample size 100 = 1/sqrt(100) = .1 or 10%
For sample size 400 = 1/sqrt(400) = .05 or 5%
Formula for calculating Margin of Errors after
gathering data
Probability Yes = .55, No = .45
N = 200
Margin = 1.96 * sqrt((p*(1-p))/N)
= 1.96 * sqrt((.25)/200)
= 1.96 *.0012