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Lecture

# Chapter 12

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Ryerson University

Quantitative Methods

QMS 202

Jason Chin- Tiong Chan

Winter

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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 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)
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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
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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
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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
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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
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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”
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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
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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)
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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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