This preview shows pages 1–5. Sign up to view the full content.
1
Let us look at another tool that will help as compare population
proportions for more than two populations.
Chi Square Distribution Introduction and Applications
Atul N Roy
The Chi Square distribution is a probability distribution, where the
random variable assumes only non negative real values. It is skewed
to the right. Its shape depends on a characteristic called the degrees
of freedom. Here are a few examples:
1.
Chi Square
()
2
also expressed by
χ
with 2 degrees of freedom
2.
2
χ
with 3 degrees of freedom looks like the following
3.
2
χ
with 11 degrees of freedom looks like
0
0.1
0.2
0.3
0.4
0.5
24681
0
x
0
0.05
0.1
0.15
0.2
0
x
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 2
4.
2
χ
with 31 degrees of freedom looks like
Note that the distribution has become quite symmetric with the
increase in the degrees of freedom.
Fact:
If the degrees of freedom of a
2
χ
distribution is n, then its mean is n
and the standard deviation is
2
n.
We shall either use the tables or a computing device to use the
probabilities involving the
2
χ
distribution.
I shall write the description of the computer usage later in this section,
first let us look at a table that is typically found in most statisticstext
books.
0
0.02
0.04
0.06
0.08
5
1
01
52
02
53
0
x
0
0.01
0.02
0.03
0.04
0.05
10
20
30
40
50
60
x
3
The values in the top row are the right tail probabilities and the values
in the first column are the degrees of freedom.
Following is the way to interpret this table.
Note that
005
6
12.591
.
↓
→
means that the area under the graph of the density function
to the right of
2
χ
=12.591 at 6 degrees of freedom is approximately
0.05 as shown in the following graph.
0.5
0.1
0.05
0.025
0.01
0.005
2 1.386294 4.605176 5.991476 7.377779 9.210351 10.59653
3 2.365973 6.251394 7.814725 9.348404 11.34488 12.83807
4 3.356695 7.779434 9.487728 11.14326
13.2767 14.86017
5 4.351459 9.236349 11.07048 12.83249 15.08632 16.74965
6 5.348119 10.64464 12.59158 14.44935 16.81187 18.54751
7 6.345809 12.01703 14.06713 16.01277 18.47532 20.27774
8
7.34412 13.36156 15.50731 17.53454 20.09016 21.95486
9 8.342832 14.68366 16.91896 19.02278 21.66605 23.58927
10 9.341816 15.98717 18.30703
20.4832 23.20929 25.18805
11
10.341 17.27501 19.67515 21.92002 24.72502 26.75686
12 11.34032 18.54934 21.02606 23.33666 26.21696 28.29966
13 12.33975 19.81193 22.36203 24.73558 27.68818 29.81932
14 13.33927 21.06414 23.68478 26.11893 29.14116 31.31943
15 14.33886 22.30712
24.9958 27.48836 30.57795 32.80149
16
15.3385 23.54182 26.29622 28.84532 31.99986 34.26705
17 16.33818 24.76903
27.5871 30.19098 33.40872 35.71838
18
17.3379 25.98942 28.86932 31.52641 34.80524 37.15639
19 18.33765 27.20356 30.14351 32.85234 36.19077 38.58212
20 19.33743 28.41197 31.41042 34.16958 37.56627 39.99686
21 20.33723 29.61509 32.67056 35.47886 38.93223 41.40094
22 21.33704 30.81329 33.92446 36.78068 40.28945 42.79566
23 22.33688 32.00689 35.17246 38.07561 41.63833 44.18139
24 23.33673 33.19624 36.41503 39.36406 42.97978 45.55836
25 24.33658 34.38158 37.65249
40.6465 44.31401 46.92797
26 25.33646 35.56316 38.88513 41.92314 45.64164 48.28978
27 26.33634 36.74123 40.11327 43.19452 46.96284 49.64504
28 27.33623 37.91591 41.33715 44.46079 48.27817 50.99356
29 28.33613 39.08748 42.55695 45.72228 49.58783
52.3355
30 29.33603 40.25602 43.77295 46.97922 50.89218 53.67187
Area=0.05
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 4
This distribution has many uses in statistical analyses, we shall use it
for comparing population proportions.
Example 1.
The following data is based on the information provided by a heartburn
medication called Aciphex regarding relief from heartburn symptoms
among the patients who took placebo and the patients who took
different dosage of Aciphex.
The table shows the number that maintained the healing from ulcers
verses the number that did not under two treatment groups and the
placebo group 52 weeks after the treatment.
Maintained Healing
YES
NO
Placebo
49
120
ACIPHEX 10 mg
119
40
ACIPHEX 20 mg
139
21
We call such a table a 3
×
2 table that is a table with 3 rows and 2
columns and having 6 cells overall.
Note that the following shows us the proportion of the subjects who
maintained healing for these three groups.
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 09/22/2011 for the course STAT 101 taught by Professor Unknown during the Fall '08 term at Alabama State University.
 Fall '08
 unknown
 Probability

Click to edit the document details