Chapter11 Two-Sample Tests and One-Way ANOVA
QMS202-Business Statistics II
Chapter11
Chapter11 Two-Sample Tests and One-Way ANOVA (Part I) Outcomes: 1. Conduct a test of hypothesis for two independent population means - Use z-test when σ1 and σ2 are known - Use pooled-variance t test when σ1 and σ2 are unknown but equal - Use separate-variance t test when σ1 and σ2 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
Example1 Ryerson Car magazine is comparing the total repair costs incurred during the first three years on two sport cars, The R123 and the S456. Random samples of 45 R123 cars are $5300 for the first three years. For the 50 S456 cars, the mean is $5760. Assume that the standard deviations for the two populations are $1120 and $1350, respectively. Using the 5% significance level, can we conclude that such mean repair costs are different for these two types of cars? Calculator Output
2-Sample z Test µ1 ≠ µ2
z
= -1.8137025 p = 0.06972353 x1 = 5300 x 2 = 5760 n1 = 45 n 2 = 50
Winter2011
2
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QMS202-Business Statistics II
Chapter11
Step1 Let µ1 be the population mean repair cost for sport car R123 Let µ2 be the population mean repair cost for sport car S456 Step2 H o : µ1 = µ2
H A : µ1 ≠ µ2
Step3 Level of significance = 0.05/2=0.025 Step 4 2-sample mean z test Step5 = -1.8137 0.06972353 z statistic
z critical
=
p-value =
Step6 Since the p-value > 0.05, do not reject the null hypothesis. There is not enough evidence to conclude that such mean repair costs are different for these two types of cars
Winter2011
3
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QMS202-Business Statistics II
Chapter11
Example2 Mark is the owner of the Appliance Patch. Mark observed a difference in the dollars value of sales between the men and women he employed as sales associates. A sample of 50 days revealed the men sold a mean of $1560 worth of appliances per day. For a sample of 60 days, the women sold a mean of $1650 worth appliances per day. Assume the population standard deviation for men is $204 and for women $259. At the 0.05 significance level, can Mark conclude that the mean amount sold per day is larger for women? Calculator Output 2-Sample z Test µ1 µ2 z = p = x1 = x2 = n1 = n2 =
Winter2011
4
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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) z statistic
=
z critical
=
Winter2011
p-value =
5
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QMS202-Business Statistics II
Chapter11
Step6 (Statistical Decision and Business Conclusion)
Example3 The operations manager of Ryerson Light Bulb factory wants to determine if there is any difference in the average life expectancy of bulbs manufactured on two different types of machine. The process population standard deviation of machine A is 112 hours and of machine B is 127 hours. A random sample of 35 light bulbs obtained from machine A indicates a sample mean of 379 hours, and a similar sample of 35 from machine B indicates a sample mean of 368. Using the 1% significance level.
Calculator Output 2-Sample z Test µ1 µ2 z = p = x1 = x2 = n1 = n2 =
Winter2011
6
www.notesolution.com
QMS202-Business Statistics II
Chapter11
Step1 Define the parameter(s)
Step2 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) z statistic
=
z critical
=
p-value =
Winter2011
7
www.notesolution.com
QMS202-Business Statistics II
Chapter11
Step6 (Statistical Decision and Business Conclusion)
Example4 Ryerson Computer Manufacturer offers a help line that purchasers call for help 24 hours per day, 7 days a week. Clearing these calls for help in a timely fashion is important to the company’s image. After telling the caller that resolution of the problem is important the caller is asked whether the issue is “software” or “hardware” related. A mean time takes a technician to resolve a software issue is 18 minutes with a standard deviation of 4.2 minutes. This information was obtained from a sample of 35 monitored calls. For a study of 45 hardware issues, the mean time for a technician to resolve a problem was 15.5 minutes with a standard deviation of 3.9 minutes. This information was also obtained from monitored calls. Assume both population standard deviations are the same. At the 0.05 significance level is it reasonable to conclude that it takes longer to resolve software issue? Winter2011
8
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QMS202-Business Statistics II
Chapter11
Calculator Output
2-Sample t Test
µ1 > µ2 t = 2.75012089 p = 3.7008E-03
d. f
= 78 = 18
x1 x 2 = 15.5 sx 1 = 4.2 sx 2 = 3.9
sp = 4.0335 n1 = 35 n 2 = 45
Step1 Let µ1 be the population mean time for resolving software issues Let µ2 be the population mean time for resolving hardware issues Step2
H o : µ1 ≤ µ2
H A : µ1 > µ2
Step3 Level of significance = 0.05
Winter2011
9
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QMS202-Business Statistics II
Chapter11
Step 4 2-sample mean t test- pooled variance “on”
Step5 t statistic
= 2.7501
degree of freedom =
p-value = 0.0037 t critical
=
Step6 Since the p-value µ2 t = 4.28210973 p = 2.5201E-04 d . f = 17 x1 = 10.375 x 2 = 5.6363 s1 = 2.2638 s 2 = 2.4605 sp = 2.3815 n1 = 8 n 2 = 11
Winter2011
11
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QMS202-Business Statistics II
Chapter11
Step1 Let µ1 be the population mean number of time using ATM machine for younger adults (under 25) Let µ2 be the population mean number of time using ATM machine for senior (over 60) Step2
H o : µ1 ≤ µ2
H A : µ1 > µ2
Step3 Level of significance = 0.05
Step 4 2-sample mean t test
Winter2011
12
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QMS202-Business Statistics II
Chapter11
Step5 t statistic
= 4.2821
p-value = 0.000252
degree of freedom =
t critical
=
Step6 Since the p-value 0 t = 3.0338 p = 7.0785E-03 x =3 s = 3.1269 n = 10
Winter2011
16
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QMS202-Business Statistics II
Chapter11
Step1 d = before – after
Let µd be the population mean difference in the number of days absent in the last twelve months before the yoga program had begun and in the next twelve months after the yoga program had begun. H o : µd ≤ 0
Step2
H A : µd > 0
Step3 Level of significance = 0.05 Step 4 Paired observation mean t-test Step5 t statistic
= 3.0338
degree of freedom =
p-value = 0.0070785 t critical
=
Step6
Winter2011
17
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QMS202-Business Statistics II
Chapter11
Since the p-value
Chapter11
Chapter11 Two-Sample Tests and One-Way ANOVA (Part I) Outcomes: 1. Conduct a test of hypothesis for two independent population means - Use z-test when σ1 and σ2 are known - Use pooled-variance t test when σ1 and σ2 are unknown but equal - Use separate-variance t test when σ1 and σ2 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
Example1 Ryerson Car magazine is comparing the total repair costs incurred during the first three years on two sport cars, The R123 and the S456. Random samples of 45 R123 cars are $5300 for the first three years. For the 50 S456 cars, the mean is $5760. Assume that the standard deviations for the two populations are $1120 and $1350, respectively. Using the 5% significance level, can we conclude that such mean repair costs are different for these two types of cars? Calculator Output
2-Sample z Test µ1 ≠ µ2
z
= -1.8137025 p = 0.06972353 x1 = 5300 x 2 = 5760 n1 = 45 n 2 = 50
Winter2011
2
www.notesolution.com
QMS202-Business Statistics II
Chapter11
Step1 Let µ1 be the population mean repair cost for sport car R123 Let µ2 be the population mean repair cost for sport car S456 Step2 H o : µ1 = µ2
H A : µ1 ≠ µ2
Step3 Level of significance = 0.05/2=0.025 Step 4 2-sample mean z test Step5 = -1.8137 0.06972353 z statistic
z critical
=
p-value =
Step6 Since the p-value > 0.05, do not reject the null hypothesis. There is not enough evidence to conclude that such mean repair costs are different for these two types of cars
Winter2011
3
www.notesolution.com
QMS202-Business Statistics II
Chapter11
Example2 Mark is the owner of the Appliance Patch. Mark observed a difference in the dollars value of sales between the men and women he employed as sales associates. A sample of 50 days revealed the men sold a mean of $1560 worth of appliances per day. For a sample of 60 days, the women sold a mean of $1650 worth appliances per day. Assume the population standard deviation for men is $204 and for women $259. At the 0.05 significance level, can Mark conclude that the mean amount sold per day is larger for women? Calculator Output 2-Sample z Test µ1 µ2 z = p = x1 = x2 = n1 = n2 =
Winter2011
4
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) z statistic
=
z critical
=
Winter2011
p-value =
5
www.notesolution.com
QMS202-Business Statistics II
Chapter11
Step6 (Statistical Decision and Business Conclusion)
Example3 The operations manager of Ryerson Light Bulb factory wants to determine if there is any difference in the average life expectancy of bulbs manufactured on two different types of machine. The process population standard deviation of machine A is 112 hours and of machine B is 127 hours. A random sample of 35 light bulbs obtained from machine A indicates a sample mean of 379 hours, and a similar sample of 35 from machine B indicates a sample mean of 368. Using the 1% significance level.
Calculator Output 2-Sample z Test µ1 µ2 z = p = x1 = x2 = n1 = n2 =
Winter2011
6
www.notesolution.com
QMS202-Business Statistics II
Chapter11
Step1 Define the parameter(s)
Step2 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) z statistic
=
z critical
=
p-value =
Winter2011
7
www.notesolution.com
QMS202-Business Statistics II
Chapter11
Step6 (Statistical Decision and Business Conclusion)
Example4 Ryerson Computer Manufacturer offers a help line that purchasers call for help 24 hours per day, 7 days a week. Clearing these calls for help in a timely fashion is important to the company’s image. After telling the caller that resolution of the problem is important the caller is asked whether the issue is “software” or “hardware” related. A mean time takes a technician to resolve a software issue is 18 minutes with a standard deviation of 4.2 minutes. This information was obtained from a sample of 35 monitored calls. For a study of 45 hardware issues, the mean time for a technician to resolve a problem was 15.5 minutes with a standard deviation of 3.9 minutes. This information was also obtained from monitored calls. Assume both population standard deviations are the same. At the 0.05 significance level is it reasonable to conclude that it takes longer to resolve software issue? Winter2011
8
www.notesolution.com
QMS202-Business Statistics II
Chapter11
Calculator Output
2-Sample t Test
µ1 > µ2 t = 2.75012089 p = 3.7008E-03
d. f
= 78 = 18
x1 x 2 = 15.5 sx 1 = 4.2 sx 2 = 3.9
sp = 4.0335 n1 = 35 n 2 = 45
Step1 Let µ1 be the population mean time for resolving software issues Let µ2 be the population mean time for resolving hardware issues Step2
H o : µ1 ≤ µ2
H A : µ1 > µ2
Step3 Level of significance = 0.05
Winter2011
9
www.notesolution.com
QMS202-Business Statistics II
Chapter11
Step 4 2-sample mean t test- pooled variance “on”
Step5 t statistic
= 2.7501
degree of freedom =
p-value = 0.0037 t critical
=
Step6 Since the p-value µ2 t = 4.28210973 p = 2.5201E-04 d . f = 17 x1 = 10.375 x 2 = 5.6363 s1 = 2.2638 s 2 = 2.4605 sp = 2.3815 n1 = 8 n 2 = 11
Winter2011
11
www.notesolution.com
QMS202-Business Statistics II
Chapter11
Step1 Let µ1 be the population mean number of time using ATM machine for younger adults (under 25) Let µ2 be the population mean number of time using ATM machine for senior (over 60) Step2
H o : µ1 ≤ µ2
H A : µ1 > µ2
Step3 Level of significance = 0.05
Step 4 2-sample mean t test
Winter2011
12
www.notesolution.com
QMS202-Business Statistics II
Chapter11
Step5 t statistic
= 4.2821
p-value = 0.000252
degree of freedom =
t critical
=
Step6 Since the p-value 0 t = 3.0338 p = 7.0785E-03 x =3 s = 3.1269 n = 10
Winter2011
16
www.notesolution.com
QMS202-Business Statistics II
Chapter11
Step1 d = before – after
Let µd be the population mean difference in the number of days absent in the last twelve months before the yoga program had begun and in the next twelve months after the yoga program had begun. H o : µd ≤ 0
Step2
H A : µd > 0
Step3 Level of significance = 0.05 Step 4 Paired observation mean t-test Step5 t statistic
= 3.0338
degree of freedom =
p-value = 0.0070785 t critical
=
Step6
Winter2011
17
www.notesolution.com
QMS202-Business Statistics II
Chapter11
Since the p-value
