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Lecture

Chapter 11 - Part Two

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

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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
- Use z-test when
1
σ
and
2
σ
are known
- Use pooled-variance t test when
1
σ
and
2
σ
are unknown but equal
- Use separate-variance t test when
1
σ
and
2
σ
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
Winter2011 1
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QMS202-Business Statistics II Chapter11
Characteristics of F distribution
1. There is a “familyof 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
Winter2011 2
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QMS202-Business Statistics II Chapter11
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
Winter2011 3
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Description
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 Winter2011 1 www.notesolution.com QMS202-Business Statistics II Chapter11 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 Winter2011 2 www.notesolution.com QMS202-Business Statistics II Chapter11 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 Winter2011 3 www.notesolution.com QMS202-Business Statistics II Chapter11 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) Winter2011 4 www.notesolution.com QMS202-Business Statistics II Chapter11 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. Winter2011 5 www.notesolution.com QMS202-Business Statistics II Chapter11 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 Winter2011 6 www.notesolution.com QMS202-Business Statistics II Chapter11 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 Winter2011 7 www.notesolution.com QMS202-Business Statistics II Chapter11 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 Winter2011 8 www.notesolution.com QMS202-Business Statistics II Chapter11 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 Winter2011 9 www.notesolution.com QMS202-Business Statist
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