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Lecture

# Chapter 11 - Part Two

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Ryerson University

Quantitative Methods

QMS 202

Jason Chin- Tiong Chan

Winter

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QMS202-Business Statistics II Chapter11
Chapter11 Two-Sample Tests and
One-Way ANOVA (Part II)
Outcomes:
1. Conduct a test of hypothesis for two independent population means
σ σ
-Use z-test when 1 and 2 are known
- Use pooled-variance t test when σ 1 and σ 2 are unknown but equal
- Use separate-variance t test when σ 1 and σ 2 are unknown and
unequal
2. Conduct a test of hypothesis for paired or dependent observations,
using the paired t-test
3. Conduct a test of hypothesis for two population proportions
using the z test
4. List the characteristics of the F distributions
5. Conduct a test of hypothesis to determine whether the variances
of two populations are equal, using the F Test
6. Discuss the general idea of Analysis of Variance (ANOVA) and its
assumptions
7. Conduct the F Test when there are more than two means
8. Discuss multiple comparisons: The Tukey-Kramer procedure
9. Conduct Levene’s Test for Homogeneity of Variance
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Characteristics of F distribution
1. There is a “family” of F distributions. Each member of the family is
determined by two parameters
i. the degree of freedom in the numerator
ii. the degree of freedom in the denominator
2. The F distribution is continuous
This means that it can assume an infinite number of values between 0 and
positive infinity
3. The F distribution cannot be negative
The smallest value F can assume is 0
4. It is positively skewed
The long tail of the distribution is to the right-hand side. As the number of degrees
of freedom increases in both numerator and denominator, the distribution
approaches a normal distribution
5. It is asymptotic
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Question#1
Ryerson Limos offers limousine service from the city hall in Toronto, Ontario, to the
Toronto International Airport, John, president of the company, is considering two
routes. One is via Highway 427 and the other via Avenue. He wants to study the
time it takes to drive to the airport using route and then compare the results. He
collected the following sample data, which is reported in minutes. Using the 0.10
significance level, is there a difference in the variation in the driving time using the
two routes?
Avenue Road Highway 427
52 59
67 60
56 61
45 51
70 56
54 63
64 57
65
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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, and the critical value)
F statistic p-value =
Step6 (Statistical Decision and Business Conclusion)
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Question#2
Ryerson Research Associates conducted a study of the iPod listening habits of men
and women. One facet of the study involved the mean listening time. It was
discovered that the mean listening time for men was 35 minutes per day. The
standard deviation of the sample of the 10 men studied was 10 minutes per day. The
mean listening time for the 12 women studied was also 35 minutes, but the standard
deviation of the sample was 12 minutes. At the 0.10 significance level, can we
conclude that there is a difference in the variation in the listening times for men and
women.
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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, and the critical value)
F = p-value =
statistic
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Step6 (Statistical Decision and Business Conclusion)
Analysis of Variance (ANOVA)
1. Another use of the F distribution is the analysis of variance (ANOVA) technique in
which we compare three or more population means to determine whether they could
be equal.
2. To use ANOVA, we assume the following:
i. The populations follow a normal distribution.
ii. The populations have equal standard deviations
iii. The populations are independent
Among-Group Variation in one-way ANOVA
c
2
SSA = ∑j=n j(X j X )
where
c = number of group
n j = number of values in group j
X j = sample mean of group j
X = grand mean
Within-Group Variation in One-Way ANOVA
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n 2
c j
SSW = ∑∑ (X ij− Xj )
j=1i 1
where
X = ith value in group j
ij
X j = sample mean of group j
Computing The Mean Squares in One-Way Anova
SSA
MSA =
c − 1
SSW
MSW =
n − c
SST
MST =
n − 1
One-Way Anova F-Test Statistic
MSA
F = MSW
SPSS output
ANOVA
Model Degree Sum of Mean Square
Of Freedom Square F
Among c-1 SSA MSA = SSA F = MSA
groups c − 1 MSW
SSW
MSW =
Within n-c SSW n − c
groups
Total n-1 SST
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Question#3
Joyce manages a regional financial centre. She wishes to compare the productivity,
as measured by the number of customers served, among three employees. Four
days are randomly selected and the number of customers served by each employee
is recorded. The results are:
Andy Ben Chris
55 66 47
54 76 51
59 67 46
56 71 48
Is there a difference in the mean number of customers served? Test at 0.05 level of
significance
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