Walden University Unit 6 Chi Square Goodness of Fit Test & Critical Value HW 1. Historically, the MBA program at Whatsamattu U. has about 40% of their stud

Walden University Unit 6 Chi Square Goodness of Fit Test & Critical Value HW 1. Historically, the MBA program at Whatsamattu U. has about 40% of their students choose a Leadership major, 30% choose a Finance major, 20% choose a Marketing major, and 10% choose no major. Does the most recent class of 200 MBA students fit that same pattern or has there been a shift in the choice of majors. Using the sample of 200 students (in the data file), conduct a Chi Square Goodness of Fit test to determine if the current distribution fits the historical pattern. Use a .05 significance level. 2. While job opportunities for men and women are considerably more balanced than they were 40 years ago, the career aspirations may still differ. Is there a difference in majors chosen by men and women? Using the sample of 200 MBA students (in the data file), conduct a Chi Square Test of Independence to determine if one’s choice of major is independent of their gender. Use a .05 significance level. ID
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
Gender
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
Finance
No Major
No Major
Finance
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
Finance
No Major
No Major
No Major
Finance
No Major
Finance
Finance
No Major
Finance
No Major
Finance
Finance
No Major
Finance
Finance
Employ
Unemployed
Full Time
Part Time
Full Time
Full Time
Unemployed
Full Time
Full Time
Part Time
Full Time
Part Time
Full Time
Full Time
Full Time
Part Time
Full Time
Full Time
Part Time
Full Time
Unemployed
Full Time
Part Time
Full Time
Full Time
Part Time
Full Time
Part Time
Unemployed
Part Time
Full Time
Full Time
Unemployed
Full Time
Full Time
Part Time
Part Time
Full Time
Unemployed
Full Time
Part Time
Full Time
Full Time
Full Time
Unemployed
Full Time
Age
39
55
43
56
38
54
30
37
38
42
52
35
37
53
51
40
33
53
43
35
57
32
59
48
34
53
35
38
37
46
44
31
51
47
56
42
44
54
51
42
45
55
47
43
57
MBA_GPA
2.82
4
3.45
2.61
3.5
4
3
2.5
2.84
3.72
3.21
3.44
3.65
3.02
3.03
3.8
4
3.26
3.53
3.75
3.15
3.66
3.36
3.79
2.85
3.74
3.23
3.52
3.32
2.89
2.83
2.93
3.71
3.47
3.52
2.83
3.64
2.96
3.59
3.33
3.38
3.44
3.31
3.03
3.26
BS GPA
3
4
3.5
4
3.3
3.05
4
3.6
3.05
3.7
3.5
3.55
2.78
3.3
3.25
4
3.5
3.5
3.75
3.9
3.2
3.75
3.45
2.55
3.05
3.9
4
3.7
3.45
3.1
3.05
3.1
3.8
2.6
3.8
4
3.55
3.1
3.9
3.9
3.6
3.35
3.9
3.25
3.4
Hrs_Studying
10
15
3
4
5
5
6
6
6
6
6
6
6
6
6
6
6
7
6
7
6
8
8
8
8
8
2
2
2
2
1
1
1
4
4
4
6
6
6
6
6
6
7
7
7
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
1
1
1
1
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Finance
No Major
Finance
Finance
Finance
No Major
Finance
No Major
Finance
Finance
Marketing
Marketing
Marketing
Leadership
Leadership
Marketing
Marketing
Marketing
Marketing
Marketing
No Major
No Major
No Major
No Major
Marketing
Leadership
Leadership
Leadership
Leadership
Leadership
No Major
Leadership
No Major
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
No Major
Marketing
Marketing
No Major
Finance
Finance
Finance
Finance
Full Time
Part Time
Full Time
Full Time
Full Time
Full Time
Full Time
Full Time
Unemployed
Full Time
Part Time
Full Time
Full Time
Part Time
Full Time
Full Time
Full Time
Full Time
Full Time
Full Time
Unemployed
Full Time
Part Time
Full Time
Part Time
Full Time
Part Time
Full Time
Full Time
Full Time
Full Time
Full Time
Full Time
Full Time
Unemployed
Full Time
Full Time
Full Time
Full Time
Full Time
Full Time
Full Time
Part Time
Full Time
Full Time
Full Time
Full Time
36
58
46
53
59
49
34
46
46
33
56
39
51
55
38
33
34
31
37
46
31
47
54
52
43
44
34
59
45
30
32
32
40
48
51
30
31
35
33
35
31
38
46
45
59
58
46
3.04
2.98
2.8
3.75
3.64
3.65
3.18
3.44
3.06
3.51
3.33
2.81
3.64
3.05
2.85
3.56
2.92
3.35
3.46
3.59
3.11
3.65
3.17
2.97
3.77
3.21
3.17
3.65
2.94
3.53
3.65
3.61
3.7
2.91
3.09
3.77
3.79
3.59
3.38
4
2.97
3.44
3.64
3.48
2.76
3.73
2.91
4
3.1
3.05
3.75
3.65
3.8
3.3
4
3.15
3.75
3.4
3.05
3.8
3.4
3.25
3.6
3.1
3.5
3.35
3.75
3.2
3.7
3.5
3.1
3.9
3.2
3.15
3.65
3.1
3.7
3.6
3.7
3.9
3.1
3.25
3.95
3.8
3.6
3.5
3.5
3.1
3.65
3.55
3.4
3.1
3.8
3.05
7
7
7
3
3
3
3
3
3
10
2
2
8
7
3
7
5
7
10
8
6
8
7
5
8
6
6
10
5
8
7
8
8
5
6
9
8
7
8
8
8
8
8
8
8
8
8
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Finance
Finance
Finance
Finance
Finance
Finance
Finance
Finance
No Major
Marketing
Marketing
Leadership
Leadership
No Major
Leadership
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
No Major
Leadership
Leadership
Leadership
Leadership
Finance
No Major
No Major
Finance
Finance
Finance
Finance
Finance
Finance
Finance
Finance
Finance
Leadership
Leadership
Leadership
Finance
Finance
Finance
Finance
Full Time
Part Time
Full Time
Full Time
Full Time
Full Time
Full Time
Full Time
Unemployed
Full Time
Part Time
Full Time
Full Time
Full Time
Full Time
Part Time
Full Time
Full Time
Part Time
Full Time
Full Time
Unemployed
Full Time
Full Time
Part Time
Unemployed
Full Time
Part Time
Full Time
Full Time
Part Time
Full Time
Unemployed
Part Time
Full Time
Part Time
Full Time
Unemployed
Part Time
Full Time
Full Time
Full Time
Full Time
Unemployed
Full Time
Part Time
Full Time
35
53
31
50
38
50
48
53
53
30
32
42
56
46
49
32
36
42
37
31
31
42
39
47
28
28
52
35
38
44
38
52
53
53
31
47
51
37
46
48
54
48
36
39
28
45
31
3.78
3.5
3.13
3.14
3.24
3.56
3.16
3.53
3.7
3.3
4
3.5
3.39
3.65
2.78
3.44
3.88
2.84
3.53
3.22
3.56
3.2
3.56
3.41
3.56
3.34
2.56
3.76
3.55
3.88
3.31
3.09
3.82
3.01
3.66
3.64
3.59
3.49
3.13
3.83
3.04
3.91
3.56
3.96
3.46
3.22
3.27
3.95
3.4
3.15
3.25
3.3
3.5
3.25
3.55
3.15
3.35
3.6
3.4
3.4
3.8
3.7
3.6
3.95
3.95
3.6
3.3
3.8
3.25
3.3
3.6
3.7
3.6
3.6
3.8
3.45
3.9
3.45
3.15
4
3.2
3.85
3.7
3.65
3.55
3.2
3.9
3.15
4
3.7
4
3.4
3.15
3.2
9
7
6
6
6
7
6
7
6
6
7
7
7
8
8
7
9
9
7
6
8
6
6
7
8
7
7
8
7
8
7
6
9
6
8
8
7
7
6
8
6
10
8
9
7
6
6
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
0
1
1
1
0
1
1
1
0
1
Finance
Finance
Finance
Finance
Finance
Finance
Finance
Leadership
Leadership
Leadership
Leadership
No Major
No Major
No Major
Marketing
Marketing
Marketing
Marketing
Marketing
Marketing
Marketing
Marketing
Marketing
Marketing
Marketing
Marketing
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Full Time
Part Time
Full Time
Part Time
Unemployed
Full Time
Part Time
Full Time
Unemployed
Part Time
Unemployed
Part Time
Full Time
Unemployed
Full Time
Unemployed
Full Time
Full Time
Unemployed
Unemployed
Part Time
Full Time
Unemployed
Full Time
Part Time
Unemployed
Full Time
Part Time
Full Time
Part Time
Unemployed
Full Time
Part Time
Unemployed
Part Time
Full Time
Full Time
Part Time
Full Time
Unemployed
Full Time
Full Time
Full Time
Full Time
Full Time
Unemployed
Part Time
47
35
52
52
55
52
46
31
33
45
50
33
37
33
46
55
30
51
35
40
29
52
27
51
56
35
46
39
31
52
35
32
44
43
38
54
30
38
45
48
43
34
54
36
45
55
45
3.43
3.85
3.89
3.37
3.32
3.54
3.8
3.74
3.6
2.6
3.8
2.67
3.95
3.56
3.79
3.93
3.79
3.71
3.05
3.22
3.85
3.82
3.23
3.56
3.53
3.62
3.8
3.47
3.64
3.03
3.17
3.22
3.92
3.82
3.26
3.8
3.2
3.46
3.67
4
3.66
3.96
3.75
3.83
3.55
3.36
3.21
3.45
3.95
3.9
3.45
3.3
3.55
3.9
3.85
3.45
3.55
3.3
3.45
4
3.75
3.75
4
3.85
3.85
3.35
3.2
3.95
3.95
3.95
3.65
3.65
4
3.95
3.35
3.65
3.15
3.25
3.2
4
3.95
3.55
3.85
3.2
3.35
3.75
3.4
3.85
4
3.85
3.85
3.2
3.35
3.25
7
9
8
7
6
7
8
8
7
7
6
7
9
8
8
9
8
8
6
6
9
9
9
7
7
9
9
6
7
5
6
6
10
9
7
8
6
6
8
7
8
10
8
8
6
6
6
187
188
189
190
191
192
193
194
195
196
197
198
199
200
1
0
1
1
1
1
1
1
1
1
1
1
1
1
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Leadership
Part Time
Part Time
Full Time
Full Time
Full Time
Full Time
Unemployed
Full Time
Unemployed
Unemployed
Unemployed
Unemployed
Unemployed
Full Time
34
54
36
24
34
45
33
22
27
33
36
34
55
33
2.97
3.99
3.07
3.65
3.67
3.06
3.98
3.93
3.41
3.43
3.7
3.76
3.9
3.23
3.15
4
3.15
3.65
3.85
3.35
3.7
4
3.3
3.5
3.65
3.75
3.9
3.3
5
10
6
7
8
6
8
10
6
7
7
8
8
6
Works FT
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
0
1
0
0
1
Variable descriptions
Gender = 0 (female), 1 (male)
Major = student’s major
Age = age of student in years
MBA_GPA = overall GPA in the MBA program
BS_GPA = overall GPA in the BS program
Hrs_Studying = average hours studied per week
Works FT = 0 (No), 1 (Yes)
0
0
0
1
1
1
0
1
1
0
1
0
1
0
1
1
0
1
1
1
0
1
0
1
1
1
0
0
0
1
1
1
1
1
0
1
1
)
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
0
0
1
1
1
0
1
1
1
0
1
1
1
1
1
0
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
1
1
1
0
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
0
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
0
1
0
1
Chi-Square Goodness of Fit Test (Assuming Equal Expected)
Items
Pool
No Pool
Observed Expected
62
50.00
38
50.00
–
Chi Square
2.88
2.88
–
©2007 DrJimMirabella.com
Data
Level of Significance
Degrees of Freedom
0.05
1
Results
Critical Value
3.8415
Chi-Square Test Statistic
5.76
p -Value
0.0164
Reject the null hypothesis
This tests the null hypothesis that the distribution is equal across all categories.
It also tests if there is a difference in the frequencies of the categories / items.
Rejecting the null implies a difference in the categories / items.
Here we are testing if the sample fits the distribution of having 1/2 the homes with
The p-value of .0164 is less than .05, and so we reject the null hypothesis.
Thus we conclude that the sample does not fit the expected distribution.
There are significantly more homes with a pool.
Chi-Square Goodness of Fit Test (Assuming Unequal Expected)
Items
Brick
Stucco
Wood
Observed % Expected Expected
40
30.00%
30.00
35
30.00%
30.00
25
40.00%
40.00
–
Chi Square
3.33
0.83
5.63
–
©2007 DrJimMirabella.com
Data
Level of Significance
Degrees of Freedom
0.05
2
Results
Critical Value
5.9915
Chi-Square Test Statistic
9.79
p -Value
0.0075
Reject the null hypothesis
This tests the null hypothesis that the distribution is as expected.
In other words, it tests if the results fit the expected distribution.
Rejecting the null implies that the results do not fit the distribution.
Here we are testing if the sample fits the distribution of having 30% b
The p-value of .0075 is less than .05, and so we reject the null hypoth
Thus we conclude that the sample does not fit the expected distribut
There are significantly more homes than expected which are made of
Chi-Square Test of Independence
Row variable
Brick
Stucco
Wood
Total
Row variable
Brick
Stucco
Wood
0
0
Total
Data
Level of Significance
Number of Rows
Number of Columns
Degrees of Freedom
Observed Frequencies
Column variable
Pool
No Pool
30
10
18
17
14
11
62
38
0
Expected Frequencies
Column variable
Pool
No Pool
0
24.80
15.20
0.00
21.70
13.30
0.00
15.50
9.50
0.00
0.00
0.00
0.00
0.00
0.00
0.00
62
38
0
0
0
0
0
0.00
0.00
0.00
0.00
0.00
0
Total
40
35
25
0
0
100
Use the YELLOW cells to set up the Chi Square table.
The table can handle up to 5 rows and 5 columns of values.
If fewer rows or columns are needed, leave the excess blank.
The BLUE table computes the expected frequencies needed to comp
statistic. The only values that ultimately matter to you is in the RESUL
Total
0.00
0.00
0.00
0.00
0.00
0
40
35
25
0
0
100
0.05
3
2
2
Results
Critical Value
5.991465
Chi-Square Test Statistic
4.911472
p -Value
0.0858
Do not reject the null hypothesis
This tests the null hypothesis that the row variable and column variable are independent.
Rejecting the null implies that the two variables are related (one is dependent on the other).
Here we are testing if having a pool is independent of what
The p-value of .0858 is greater than .05, and so we do not r
There is insufficient evidence to conclude a relationship be
©2007 DrJimMirabella.com
CHAPTER SIX
CHI SQUARE TESTING
C
A
L being categorical in nature so we can analyze
Most hypothesis tests involve one variable
subgroups (e.g., scores for men vs. scores V
for women, with GENDER being the categorical
variable). In these parametric tests, the other
E variable must be a scale variable (i.e., interval
or ratio) since the tests involve computingRa mean. Yet there are many occasions in which
we have only categorical variables, and the technique is quite simple. Its applications
T
range from games of chance to analyzing surveys and polls.
,
Goodness of Fit
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Let’s start with the simplest application in which we have a categorical variable in which
E
we expect all values to have equal occurrences.
When we flip a coin, we expect to get
50% heads and 50% tails. When we roll aRdie, we expect to roll a one 1/6 of the time, and
likewise for a two through six. With categorical
data, there is no averaging; we merely
R
count the frequencies and compute the percentages
(i.e., relative frequencies). So if you
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flipped 12 heads and 8 tails, you would have 60% heads and 40% tails; the question is
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whether that is significantly different from the 50/50 you expected to get. In evaluating if
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it is a significant difference and if the coin flips do not fit with the distribution of a perfectly
E things – the percent breakdown of the sample
fair coin, we are essentially looking at two
Categorical Analysis
and the size of the sample. So getting 75% heads is not significant if it is based on only 4
coin flips, and yet 55% heads would be significant
if it were based on 10,000 flips. We see
1
newspapers report poll results and tell us 8
how a candidate has a lead, but that lead may be
insignificant if the sample is not large enough.
5
The proper way to word our test is to state9whether the results fit the expected distribution.
So if we flipped a coin 100 times and got T
59 heads and 41 tails, does that outcome fit with
the distribution of a fair coin?
S
Ho: The coin flips fit the distribution of a fair coin.
Ha: The coin flips do not fit the distribution of a fair coin.
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
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Chapter Six: Chi Square Testing
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The math behind this analysis is actually quite simple. The statistics is called CHI SQUARE and
V between the observed and expected values,
the computation involves computing the differences
E
squaring those differences, and dividing by the expected
value. We had 59 heads but out of 100
coin flips we would have expected 50. The difference
R between the observed 59 and the expected
50 is 9, and 9 squared is 81, and if we divide that
T 81 by the expected value of 50 we get 1.62,
which is the value shown in the Chi Square column
, for Heads. We do the same for Tails and also
get 1.62. Adding them up we get 3.24, our Chi Square test statistic. Statistics tables tell us the
magic number above which the Chi Square statistic is considered to be significant. If our observed
values exactly matched the expected values (i.e.,Twe flipped 50 heads and 50 tails), the computed
E as it gets. So the larger this value, the more
Chi Square statistic would be zero, which is as small
that the results veer from the expected results. In R
this case we have a p-value of .0719. This means
that if you took a perfectly fair coin and flipped it R
100 times, there is a 7.19% probability of getting
a 59/41 split. Since we set a significance level of E
5%, that means that if these results could happen
by chance more than 5% of the time, we don’t draw conclusions about the coin being unfair (i.e.,
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it could have come from a fair coin). We would not reject the null hypothesis in this case.
C
E
Had there been 60
heads and 40 tails, the
p-value would drop to
1
.0455, and there is less
8
than a 5% chance of
5
getting such a result
by random chance.
9
We would then reject
T
the null hypothesis
S
and conclude that the
results are not from the distribution of a fair coin (i.e., the coin is not fair).
Be careful not to jump to conclusions merely on the percentage of heads or tails. As stated earlier,
it can be deceiving and depends on the sample size too.
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
6-2
Chapter Six: Chi Square Testing
Here you see 75% heads,
and it is not significant
since it is based on only 4
coin flips.
And here you see 51%
heads which is significant
because it is based on
10,000 flips. The closer
the results are to equality,
the larger the sample you
need to reject the null, and
vice versa. So for a small
sample to have significant
results, the deviation from
equality must be large.
With only 10 coin flips,
we got significant results
because of the 90% heads
in the sample.
Here you can see how this
test can be applied to a
6-sided die to determine if
the rolls follow a uniform
distribution in which all
six sides have an equal
likelihood of occurrence.
Copyright 2011, Savant Learning SystemsTM
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1
8
5
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Introduction to Statistics by Jim Mirabella
6-3
Chapter Six: Chi Square Testing
So Chi Square can do something simple like determining if a coin is fair or if dice are loaded by
comparing the results of a test sample to an expected distribution. And likewise you can take
polling results and compare the vote distribution of two candidates to see if there is a definitive
winner. Truly powerful and yet simple.
Now what if we don’t expect an even spread of the values? Maybe we expect one value to occur
more or less than the others (which is typically the case). We can still use the Chi Square Goodness
of Fit test, but we just need to load the expected values manually. Let’s start with another game
of chance – Roulette. On a roulette wheel there are 38 numbers (1 – 36, 0 and 00). Numbers 1
through 36 are evenly split between red and black, while the 0 and 00 are both green. So the wheel
has 18 red, 18 black and 2 green. When peopleCbet on red or black, the payoff is even money
(which is fine if the green numbers weren’t there,A
but they are what gives the house its edge). 95%
of the time, red or black wins and the house pays off, but 5% of the time, green wins and the house
L
pretty much cleans the table. The casinos have computers that monitor every game in the house,
and they essentially conduct Chi Square testing toVdetermine if a pattern doesn’t fit.
E
R
T
,
T
E
R
R
E
N where we assume Unequal Expected values.
Here we use the Chi Square Goodness of Fit test
The 200 spins show 100 red, 80 black and 20 green,
C which probably doesn’t feel all that odd. A
perfectly fair roulette wheel should have 47.37%Ered and black, and 5.26% green (18/38 red and
black, 2/38 green). According to these results, the p-value is .0039 which means that this should
occur .39% of the time with a fair game – mighty suspicious. The house should get green only
1
about 5% of the time and here they are getting green about twice as often. It is subtle enough to
8 see how with a small data set you can expose
get by the non-statistician in the crowd, but you can
corruption. It is doubtful that any major casino5 cheats like this, but they use these analytical
techniques to determine if any gamblers are cheating
9 or if their games are paying off too much, as
it is a business and they wish to make money. T
This Goodness of Fit technique is the method S
used to test if a sample comes from a Normal
distribution; essentially we expect to see a certain percentage within 1 standard deviation, within
2 standard deviations, and within 3 standard deviations, and the frequencies are compared to those
expected results. .
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
6-4
Chapter Six: Chi Square Testing
Crosstabulations
We’ve seen how Chi Square testing can be used to test for the goodness of fit of a single categorical
variable. Its other common use is to test for the relationship of two categorical variables.
With so many employee surveys and customer surveys conducted regularly, the temptation is to
analyze them by merely looking at questions individually, but that tells us less than you might
think. On employee surveys that use …
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