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Lecture

Chapter 12

18 Pages
340 Views

Department
Quantitative Methods
Course Code
QMS 202
Professor
Jason Chin- Tiong Chan

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QMS202-Business Statistics II Chapter12
Chapter 12 Chi-Square Tests
Outcomes:
1.Conduct a Chi-Square Test for the difference between two
proportions
2.Conduct a Chi-Square Test for the difference among more
than two proportions
3.Conduct a Chi-Square Test of Independence
Characteristics of the Chi-Square Distribution
1. It is denoted by the symbol
2
χ
2. It is positively skewed
3. It is always non-negative value
4. It has only one parameter, called the degrees of freedom. The
shape of a Chi-Square distribution curve is skewed to the right
for small df and becomes symmetric for large df. (There is a
family of Chi-Square distributions)
Winter2011 1
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QMS202-Business Statistics II Chapter12
Chi-Square Test
1.This test is available in which there is no assumption regarding
the shape of the population.
2.This test is for qualitative data. Examples include gender,
province of birth, or brand of peanut butter purchased.
Chi-Square Test for the difference between two proportions
(Independent Sample)
1. A two-way cross-classification table (contingency table) is
developed for comparing the count of categorical responses between
two independent groups.
2. The following contingency table has two rows and two columns
and is called a 2X2 table. The cells in the table indicate the frequency
for each row and column combination.
Column Variable (Group)
Row Variable 1 2 Totals
Items of interest X1 X2 X
Items not of interest n 1 - X 1 n 2 X 2 n-X
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No
Yes
QMS202-Business Statistics II Chapter12
Total n1 n2 n
Where
1
X
= number of items of interest in group 1
2
X
= number of items of interest in group 2
11
Xn
= number of items that are not interest in group 1
22
Xn
= number of items that are not interest in group 2
X
=
21
XX
+
is the total number of items of interest
n - X =
)()(
2211
XnXn
+
, the total number of items that are
not
interest
1
n
= sample size in group 1
2
n
= sample size in group 2
= total sample size
3.
2
χ
Test for the different between two proportions
( )
=
cellsall e
e
f
ff
2
0
2
χ
where
o
f
= observed frequency in a particular cell of a contingency table
e
f
= expected frequency in a particular cell of a contingency table
Winter2011 3
test of the equality of two proportions,
H0: 1 = 2
H1: 1 2
5
Advanced
Statistics
-test, using the test statistic
{always a right tail test}
Where
1
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Description
QMS202-Business Statistics II Chapter12 Chapter 12 Chi-Square Tests Outcomes: 1. Conduct a Chi-Square Test for the difference between two proportions 2. Conduct a Chi-Square Test for the difference among more than two proportions 3. Conduct a Chi-Square Test of Independence Characteristics of the Chi-Square Distribution 1. It is denoted by the symbolχ 2 2. It is positively skewed 3. It is always non-negative value 4. It has only one parameter, called the degrees of freedom. The shape of a Chi-Square distribution curve is skewed to the right for small df and becomes symmetric for large df. (There is a family of Chi-Square distributions) Winter2011 1 www.notesolution.com QMS202-Business Statistics II Chapter12 Chi-Square Test 1. This test is available in which there is no assumption regarding the shape of the population. 2. This test is for qualitative data. Examples include gender, province of birth, or brand of peanut butter purchased. Chi-Square Test for the difference between two proportions (Independent Sample) 1. A two-way cross-classification table (contingency table) is developed for comparing the count of categorical responses between two independent groups. 2. The following contingency table has two rows and two columns and is called a 2X2 table. The cells in the table indicate the frequency for each row and column combination. Column Variable (Group) Row Variable 1 2 Totals Items of interest X1 X 2 X Items not of interest n 1- X1 n2 – X2 n-X Winter2011 2 www.notesolution.com QMS202-Business Statistics II Chapter12 Total n 1 n 2 n Where X 1 = number of items of interest in group 1 X 2 = number of items of interest in group 2 n − X = number of items that are not interest in group 1 1 1 n2− X 2 = number of items that are not interest in group 2 X = X 1 X 2is the total number of items of interest n - X = (n1− X )1+ (n −2X ) 2 , the total number of items that are not interest n 1 = sample size in group 1 n = sample size in group 2 2 n 1 n 2= total sample size 3. χ 2 Test for the different between two proportions 2 2 (f0− fe ) χ = ∑ all cells fe where f o = observed frequency in a particular cell of a contingency table fe = expected frequency in a particular cell of a contingency table test of the equality of two proportions , H 0 1 2 H 1 1 2 No Advanced ≥ Statistics 5 Yes -test, using the test statistic {always a right tail test } Where 1 Winter2011 3 www.notesolution.com QMS202-Business Statistics II Chapter12 Example1 The following contingency table has two rows, indicating whether the guests would return to the hotel or would not return to the hotel, and two columns, one for each hotel. Hotel Choose Hotel Again Hilton Ryerson Totals Yes 180 190 370 No 80 50 130 Total 260 240 500 At 5% level of significance, is there evidence of a significance difference between the population proportion of guests who return to Winter2011 4 www.notesolution.com QMS202-Business Statistics II Chapter12 Hilton Hotel and the population proportion of guests who would return to Ryerson Hotel? Calculator Output χ 2Test 2 χ = p = d.f = What type of parameter is being tested here? a.μ b. μˆ c. x d. x e. σ f.σˆ g. s p pˆ 2 h. sˆ i.π j.πˆ k. l. m. χ Step1 Define the parameter(s) Step 2 State the null and alternative hypothesis Winter2011 5 www.notesolution.com QMS202-Business Statistics II Chapter12 Step3 Level of significance = Step 4 Test statistic Step5 (Determine the test statistic, the p-value, degree of freedom, and the critical value) Step6 (Statistical Decision and Business Conclusion) Example2 A sample of 2000 shoppers was conducted in the GTA to determine the consumer behavior. Among the questions asked was ”Do you enjoy on-line shopping” Winter2011 6 www.notesolution.com QMS202-Business Statistics II Chapter12 Enjoy on-line Gender shopping Male Female Yes 544 896 No 416 144 Is there evidence of a significance difference between the proportion of males and females who enjoy on-line shopping? Test at 0.01 level of significance. Calculator Output χ 2 Test 2 χ = p = d.f = What type of parameter is being tested here? a. μ b. μˆ c. x d. x e. σ f.σˆ g. s h. s i.π j.πˆ k. p l. p m. χ 2 Winter2011 7 www.notesolution.com QMS202-Business Statistics II Chapter12 Step1 Define the parameter(s) Step 2 State the null and alternative hypothesis Step3 Level of significance = Step 4 Test statistic Step5 (Determine the test statistic, the p-value, the degree of freedom, and the critical value) Step6 (Statistical Decision and Business Conclusion) Winter2011 8 www.notesolution.com QMS202-Business Statistics II Chapter12 Chi-Square Test for the difference among more than two proportions (Independent Sample) (f − f 2 ) χ = ∑ 0 e all cellsfe where f o = observed frequency in a p
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