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CHAPTER 11 TEST FORM A First classify the problem as one of the following: Chi-square test of independence Chi-square goodness of fit Chi-square for testing or estimating σ 2 or σ F test for two variances One-way ANOVA Two-way ANOVA Then, in each of the problems when a test is to be performed, do the following: Give the value of the level of significance. State the null and alternate hypotheses. Find the sample test statistic and degrees of freedom. Find or estimate the P-value of the sample test statistic. Conclude the test. Interpret the conclusion in the context of the application. In the case of one-way ANOVA, make a summary table. 1. How old are college students? The national age distributions for college students is shown below. National Age Distribution for College Students Age Clients
Under 26
26−35
36−45
46−55
Over 55
39%
25%
16%
12%
8%
The Western Association of Mountain Colleges took a random sample of 212 students and obtained the following sample distribution. Sample Distribution, Western Association of Mountain Colleges Age Number of Students
Under 26
26−35
36−45
46−55
Over 55
65
73
41
21
12
Is the sample age distribution for the Western Association of Mountain Colleges a good fit to the national distribution? Use α = 0.05.
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1.______________________________
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Instructor’s Resource Guide Understandable Statistics, 8th Edition
CHAPTER 11, FORM A, PAGE 2 2. Are teacher evaluations independent of grades? After midterm, a random sample of 284 students were asked to evaluate teacher performance. The students were also asked to supply their midterm grade in the course being evaluated. In this study, only students with a passing grade (A, B, or C) were included in the summary table. Mid Term Grade Teacher Evaluation
A
B
C
Row Total
Positive
35
33
28
96
Neutral
25
46
35
106
Negative
20
22
40
82
Column Total
80
101
103
284
Use a 5% level of significance to test the claim that teacher evaluations are independent of midterm grades. 3. If we have a normal population with variance σ2 and a random sample of n measurements taken from this population, what probability distribution do we use to test claims about the variance?
2. _____________________________
3. _____________________________
4. A technician tested 25 motors for toy electric trains and found the sample standard deviation of electrical current to be s = 4.9 amperes. (a) Find a 95% confidence interval for σ, the population standard deviation of electric current in all such toy trains. 4. (a) __________________________ (b) If the manufacturer specifies that σ = 4.1 amperes, does the sample data indicate that σ is larger than 4.1? Use a 1% level of significance. 5. Two methods of manufacturing large roller bearings are under study. For Method I, a random sample of n1 = 16 bearings had sample standard deviation of diameters s1 = 2.9 mm. For Method II, a random sample of 18 bearings had sample standard deviation of diameters s2 = 1.2 mm. Assume the diameters follow a normal distribution. Test the claim that σ 12 > σ 22 using a 1% level of significance.
(b) __________________________
5. _____________________________
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Part III: Sample Tests, Chapter 11
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CHAPTER 11, FORM A, PAGE 3 6. Sasha has decided to buy one of four different cars. She is interested in whether there is a difference in the gas mileage of the cars. In a study, miles per gallon were obtained for a random sample for each of the four cars. The results of a one-way ANOVA test are summarized below. ANOVA for mpg Source
SS
d.f.
MS
F
P-value
Between Groups
65.43
3
21.81
2.42
0.091
Within Groups
216.29
24
9.01
Number of Students
281.71
27
Use a 5% level of significance to test whether there is a difference among the population means.
6. _____________________________
7. A study to determine if management style affects the number of sick leave days taken by employees in a department was conducted. Three departments with the same number of employees were studied. The management style used in one department was top down with employees having little input into decisions; in another department, quality control experts made recommendations; in the last department the management gathered input informally from the employees. The total number of sick leave days taken per month by all of the employees in the department was recorded. For a random sample of 3 months, the numbers follow: Top Down Management:
19
34
28
Quality Teams:
16
21
15
Informal Input:
15
12
14
Use a one-way ANOVA to test if the mean number of sick leave days for departments managed in the various styles are different. Use α = 0.05.
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7. _____________________________
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Instructor’s Resource Guide Understandable Statistics, 8th Edition
CHAPTER 11, FORM A, PAGE 4 8. Will students perform better if they can choose the section of a course in which they enroll? Does the class status of the student make a difference? A researcher is studying this question. Four blocks of students are formed according to class status: freshman, sophomore, junior, senior. Each of the students must enroll in the course Spanish I. The researcher selects a random sample of 10 students from each of the blocks and allows them to enroll in the section of their choice. Another random sample of 10 students from each block are assigned a section of Spanish I. At the end of the semester, all students take the same final exam. The researcher records the scores and compares the scores for all the students participating in the study. (a) Draw a flowchart showing the design of this experiment.
8. (a) __________________________
(b) Does the design fit the model for a two-way ANOVA randomized block design? Explain.
(b) __________________________
9. James drives to work each morning during rush-hour. Does commute time depend on route? Does it depend on time of departure? In a study, the times (in minutes) were gathered for random samples. There were four different routes and three different departure times. The results of a two-way ANOVA test are summarized below. ANOVA Source
SS
d.f.
MS
F
P-value
Route
613.4
3
204.5
5.00
0.008
19.1
2
9.5
0.23
0.796
Interaction
357.2
6
59.5
1.45
0.237
Error
982.7
24
40.9
Total
1972.4
35
Departure Time
(a) Test to see if there is any evidence of interaction between the two factors at a level of significance of 0.01. (b) If there is no evidence of interaction, test to see if there is a difference in mean time based on route. Use α = 0.01.
9. (a) __________________________ (b) __________________________
(c) If there is no evidence of interaction, test to see if there is a difference in mean time based on departure time. Use α = 0.01. (c) __________________________
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Part III: Sample Tests, Chapter 11
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CHAPTER 11 TEST FORM B First classify the problem as one of the following: Chi-square test of independence Chi-square goodness of fit Chi-square for testing or estimating σ 2 or σ F test for two variances One-way ANOVA Two-way ANOVA Then, in each of the problems when a test is to be performed, do the following: Give the value of the level of significance. State the null and alternate hypotheses. Find the sample test statistic and degrees of freedom. Find or estimate the P-value of the sample test statistic. Conclude the test. Interpret the conclusion in the context of the application. In the case of one-way ANOVA, make a summary table. 1. The Fish and Game Department in Wisconsin stocked a new lake with the following distribution of game fish. Initial Stocking Distribution Fish
Pike
Trout
Perch
Bass
Bluegill
Percent
10%
15%
20%
25%
30%
After six years a random sample of 197 fish from the lake were netted, identified, and released. The sample distribution is shown next. Sample Distribution after Six Years Fish Number
Pike
Trout
Perch
Bass
Bluegill
35
17
33
55
57
Is the sample distribution of fish in the lake after six years a good fit to the initial stocking distribution? Use a 5% level of significance. 1.______________________________
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CHAPTER 11, FORM B, PAGE 2 2. Is the choice of college major independent of grade average? A random sample of 445 students were surveyed by the Registrar’s Office regarding major field of study and grade average. In this study, only students with passing grades (A, B, or C) were included in the survey. Grade averages were rounded to the nearest letter grade (e.g. 3.6 grade point average was rounded to 4.0 or A). Grade Average Major
A
B
C
Row Total
Science
38
49
63
150
Business
41
42
59
142
Humanities
32
53
68
153
Column Total
111
144
190
445
Use a 1% level of significance to test the claim that choice of major field is independent of grade average.
2. _____________________________
3. If we have two normal populations with equal variances and random samples n1 and n2 are taken from these populations, what probability distribution do we use to test claims about the variances? 3. _____________________________ 4. An automobile service station times the Quick Lube service for a random sample of 22 customers. The sample standard deviation of times was s = 7.2 minutes. (a) Find a 90% confidence interval for σ, the population standard deviation of Quick Lube times. 4. (a) __________________________ (b) The service manager specifies that σ be 6.0 minutes. Does the sample data indicate that σ is different from 6.0? Use a 1% level of significance. (b) __________________________ 5. A large national chain of department stores has two basic inventories. Variation of cash flow for the two types of inventories is under study. A random sample of n1 = 9 stores with Inventory I had sample standard deviation of daily cash flow s1 = $3115. Another random sample of n2 = 11 stores with Inventory II had sample standard deviation of daily cash flow s2 = $2719. Assume daily cash flow follows a normal distribution. Test the claim that the population variances of two inventories are different. Use a 5% level of significance. 5. _____________________________
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CHAPTER 11, FORM B, PAGE 3 6. Elizabeth watches the sodium content of foods because she has high blood pressure. The sodium content was measured for random samples from each of four different brands of tuna. The results of a one-way ANOVA test are summarized below. ANOVA for sodium content Source
SS
d.f.
MS
F
P-value
Between Groups
11,786
3
3929
4.37
0.016
Within Groups
17,979
20
899
Total
29,766
23
Use a 5% level of significance to test whether there is a difference among the population means.
6. _____________________________
7. A study of depression and exercise was conducted. Three groups were used: those in a designed exercise program; a group that is sedentary; and a group of runners. A depression rating (higher scores meaning more depression) was given to the participants in each group. Small random samples from each group provided the following data on the depression rating. Treatment Group:
63
58
61
Sedentary Group:
71
64
68
Runners:
49
52
47
Use a one-way ANOVA to test if the mean depression ratings for the three groups are different. Use α = 0.05.
7. _____________________________
8. A study was conducted to measure sales volume of a grocery store item. The study looked at sales volume for the product placed in 3 different shelf locations: eye level, low, special display. In addition, the study looked at sales volume for the item when it was advertised in two different ways: on TV or with newspaper coupons. A two-way ANOVA test was used to determine if there was any difference in mean sales volume according to shelf location or advertising method. (a) List the factors and the levels of each factor for this study. (b) Explain what it means to have interaction between the factors. State the null and the alternate hypotheses used to test for interaction.
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8. (a) __________________________
(b) __________________________
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CHAPTER 11, FORM B, PAGE 4 9. A travel agent primarily reserves flights with four major airlines. The agent would like to know if the price depends on the airline or if the price depends on the destination. In a study, the prices were gathered for random samples. There were four different airlines and four different destinations. The results of a two-way ANOVA test are summarized below. ANOVA Source
SS
d.f.
MS
F
P-value
Airline
19,080
3
6,360
0.66
0.585
Destination
326,793
2
163,397
17.05
0.000
Interaction
49,168
6
8,195
0.85
0.545
Error
230,050
24
9,585
Total
625,091
35
(a) Test to see if there is any evidence of interaction between the two factors at a level of significance of 0.05.
9. (a) __________________________
(b) If there is no evidence of interaction, test to see if there is a difference in mean price based on airline. Use α = 0.05.
(b) __________________________
(c) If there is no evidence of interaction, test to see if there is a difference in mean price based on destination. Use α = 0.05.
(c) __________________________
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Part III: Sample Tests, Chapter 11
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CHAPTER 11 TEST FORM C Write the letter of the response that best answers each problem. 1. A market research study was conducted to compare three different brands of antiperspirant. The results of the study are summarized below. Use a 5% level of significant to test the claim that opinion is independent of brank. Brand Opinion
A
B
C
Total
Excellent
29
37
50
116
Satisfactory
83
65
83
231
Unsatisfactory
18
9
6
33
Total
130
111
139
380
A. State the null and alternate hypotheses.
1. A. __________
(a) H0: Opinion and brand are dependent; H1: Opinion and brand are independent (b) H0: µA = µB = µC ; H1: Not all µ1, µ2, µ3, are equal. (c) H0: Opinion and brand are independent; H1: Opinion and brand are not independent (d) H0: σ12 = σ 22 ; H1: µ1 = µ2 (e) H0: The distributions are normal; H1: The distributions are not normal
B. What is the value of the sample test statistic? (a) χ2 = 19.00
(b) χ2 = 9.49
(c) d.f. = 4
(d) F = 0.10
B. __________
(e) χ2 = 11.9
C. What is the P-value of the sample test statistic? (a) 0.100 < P-value < 0.900
(b) 0.05 < P-value < 0.010
(c) P-value < 0.005
(d) 0.010 < P-value < 0.025
D. What is your conclusion? (a) Reject H0
(b) Do not reject H0
C. __________
(e) 0.025 < P-value < 0.050
D. __________ (c) Cannot determine
(d) Brand A is the best. (e) All of the brands work equally well.
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CHAPTER 11, FORM C, PAGE 2 2. How much do second graders weigh? A county hospital found the weight distribution shown below. Hospital Weight Distribution Weight (lb)
Under 45
45−59
60−74
75−89
Over 89
7%
21%
41%
19%
12%
Percent
An elementary school within the county took a random sample of 125 second graders and obtained the following sample distribution.
Sample Distribution, Elementary School Weight (lb)
Under 45
45−59
60−74
75−89
Over 89
6
29
50
30
10
Number of Second Graders
Is the sample weight distribution for the elementary school a good fit to the hospital distribution? Use α = 0.05.
A. State the null and alternate hypotheses.
2. A. __________
(a) H0: µ1 = µ2 = µ3 = µ4 = µ5; H1: Not all µ1, µ2, µ3, are equal. (b) H0: Weight and percent are independent; H1: Weight and percent are dependent (c) H0: σ 12 = σ 22 ; H1: µ1 = µ2 (d) H0: The distributions are the same; H1: The distributions are different. (e) H0: The distributions are the same; H1: The distributions for elementary school are higher.
B. What is the value of the sample test statistic? (a) χ2 = 11.07
(b) χ2 = 5.35
(c) χ2 = 4.49
(d) t = 7.27
B. __________
(e) χ2 = 9.49
C. What is the P-value of the sample test statistic? (a) 0.100 < P-value < 0.900
(b) 0.050 < P-value < 0.100
(c) P-value > 0.900
(d) 0.050 < P-value < 0.010
D. What is your conclusion? (a) Reject H0
(b) Do not reject H0
C. __________
(e) P-value < 0.005
D. __________ (c) Cannot determine
(d) The elementary school has heavier second graders. (e) The means are the same.
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CHAPTER 11, FORM C, PAGE 3 3. Find the P-value for each situation. A. Test statistic χ 2 = 30 for a right-tailed test. d.f. = 18. (a) P-value < 0.01
(b) P-value > 0.05
(c) 0.01 < P-value < 0.025
(d) P-value = 0.05
3. A. __________
(e) 0.025 < P-value < 0.05
B. Test statistic χ 2 = 3.5 for a left-tailed test. d.f. = 15. (a) 0.005 < P-value < 0.010
(b) P-value > 0.005
(c) P-value < 0.005
(d) P-value = 0.01
B. __________
(e) 0.025 < P-value < 0.05
C. Test statistic χ 2 = 25 for a right-tailed test. n = 20. (a) P-value < 0.100
(b) P-value < 0.001
(c) 0.05 < P-value < 0.10
(d) P-value > 0.100
C. __________
(e) 0.025 < P-value < 0.05
4. A salesperson tested 30 sport utility vehicles for gas mileage (in miles per gallon) and found the sample standard deviation to be s = 4.7 mpg. A. Find a 95% confidence interval for σ 2 , the population variance of mileage for all such sport utility vehicles. (a) 14.01 < σ2 < 39.91
(b) 15.05 < σ2 < 36.17
(c) 2.98 < σ2 < 8.49
(d) 14.49 < σ2 < 41.29
4. A. __________
(e) 13.64 < σ2 < 36.65
B. If the manufacturer specifies that σ = 4.0 mpg, does the sample data indicate that σ is larger than 4.0? Use α = 0.01 and find the value of the sample test statistic and the P-value.
B. __________
(a) χ2 = 49.59, 0.005 < P-value < 0.010
(b) χ2 = 40.04, 0.100 < P-value < 0.200
(c) χ2 = 41.42, 0.050 < P-value < 0.100
(d) χ2 = 41.42, 0.005 < P-value < 0.010
(e) χ2 = 40.04, 0.050 < P-value < 0.100
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CHAPTER 11, FORM C, PAGE 4 C. What is your conclusion for the test in Part B?
C. __________
(a) There is sufficient evidence to conclude that the standard deviation is larger than 4.0 mpg. (b) The distributions are different. (c) The variances are unequal. (d) The mpg for the sample is larger than 4.0 mpg. (e) There is insufficient evidence to conclude that the standard deviation is larger than 4.0 mpg.
5. Two printing machines are under study. For Machine I, a random sample of n1 = 10 newspapers had a sample standard deviation of time s1 = 1.4 minute. For Machine II, a random sample of n2 = 12 newspapers had a sample standard deviation of s2 = 0.8 minutes. Assume the times follow a normal distribution. Test the claim that σ 12 > σ 22 using a 1% level of significance. A. State the null and alternate hypotheses.
5. A. ___________
(a) H0: σ 12 > σ 22 ; H1: σ 12 = σ 22
(b) H0: σ12 = 0; H1: σ 22 > 0
(c) H0: σ 12 = σ 22 ; H1: σ 12 ≠ σ 22
(d) H0: σ 12 = σ 22 ; H1: σ 12 < σ 22
(e) H0: σ 12 = σ 22 ; H1: σ 12 > σ 22
B. What is the value of the sample test statistic? (a) F = 0.33
(b) F = 1.75
(c) F = 3.06
(d) t = 3.06
B. __________
(e) F = 0.57
C. What is the P-value?
C. __________
(a) 0.001 < P-value < 0.010
(b) 0.025 < P-value < 0.050
(c) > 0.100
(d) < 0.100
(e) < 0.001
D. What is your conclusion? (a) Reject H0
(b) Do not reject H0
D. __________ (c) Cannot determine
(d) The variances are different. (e) The distributions are different.
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Part III: Sample Tests, Chapter 11
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CHAPTER 11, FORM C, PAGE 5 6. Do cough medicines differ in the length of time of relief? The hours of relief was recorded for random samples of volunteers from each of four different brands of cough medicine. The results of ANOVA test are summarized below. ANOVA Source
SS
d.f.
MS
F
P-value
Between Groups
6.17
3
2.056
2.11
0.122
Within Groups
27.26
28
0.974
Total
33.43
31
Use a 5% level of significance to test whether there is a difference among the population means.
A. What are the null and alternate hypotheses?
6. A. __________
(a) H0: µ1 = µ2 = µ3 = µ4; H1: Not all of the means are equal. (b) H0: µ1 = µ2 = µ3 = µ4; H1: µ1 > µ2 > µ3 > µ4 (c) H0: µ1 = µ2 = µ3; H1: Not all of the means are equal. (d) H0: σ 12 = σ 22 = σ 32 = σ 42 ; H1: σ 12 ≠ σ 22 ≠ σ 32 ≠ σ 42 (e) H0: µ1 ≠ µ2 ≠ µ3 ≠ µ4; H1: µ1 = µ2 = µ3 = µ4
B. What is the value of the sample test statistic? (a) F = 6.85
(b) F = 2.056
(c) F = 0.974
(d) F = 0.122
B. __________
(e) F = 2.11
C. What is the P-value? (a) 0.974
(b) 2.95
(c) 2.056
(d) 0.122
C. __________
(e) 2.11
D. What is your conclusion? (a) Reject H0
(b) Do not reject H0
D. __________ (c) Cannot determine
(d) The distributions are the same. (e) The variances are unequal.
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CHAPTER 11, FORM C, PAGE 6 7. An ornithologist is studying the length (in seconds) of bird calls. Random samples from three different breeds yielded the following results. Breed 1
2.3
1.9
2.1
Breed 2
0.8
1.2
1.1
Breed 3
2.0
1.4
1.7
Use one-way ANOVA to test if the mean call length differs for the three breeds. Use α = 0.05.
A. What are the null and alternate hypotheses?
7. A. __________
(a) H0: µ1 ≠ µ2 ≠ µ3; H1: µ1 = µ2 = µ3 (b) H0: σ 12 = σ 22 = σ 32 ; H1: σ 12 ≠ σ 22 ≠ σ 32 (c) H0: µ1 = µ2 = µ3; H1: Not all of µ1, µ2, µ3 are equal. (d) H0: µ1 ≠ µ2 ≠ µ3; H1: µ1 > µ2 > µ3 (e) H0: The distributions are the same; H1: The distributions are not the same.
B. What is the value of the sample test statistic? (a) F = 0.005
(b) F = 19.33
(c) χ2 = 15.08
(d) F = 15.08
B. __________
(e) F = 5.14
C. What is the P-value? (a) 0.871
(b) 0.05
(c) 0.010
(d) 0.005
C. __________
(e) 15.08
D. What is your conclusion? (a) Reject H0
(b) Do not reject H0
D. __________ (c) Cannot determine
(d) The distributions are the same. (e) The variances are the same.
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Part III: Sample Tests, Chapter 11
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CHAPTER 11, FORM C, PAGE 7 8. When performing a two-way ANOVA test, which of the following is not a required assumption?
8. __________
(a) There are the same number of measurements in each cell. (b) The measurements in each cell of a two-way ANOVA model are assumed to come from distributions with approximately the same variance. (c) There are the same number of levels for each factor. (d) The measurements in each cell come from independent random samples. (e) The measurements in each cell of a two-way ANOVA model are assumed to be drawn from a population with a normal distribution.
9. Are differences in test scores due to schools or are they due to exam forms? Three different, although supposedly equivalent, forms of a standardized achievement exam were given to a random sample of students in each of four different schools. Their scores were recorded. The results of a two-way ANOVA test are summarized below. ANOVA Source
SS
d.f.
MS
F
P-value
Exam Form
3532
2
1766
8.61
0.002
School
749
3
250
1.22
0.324
Interaction
768
6
128
0.62
0.712
Error
4917
24
205
Total
9965
35
A. When conducting a test to see if there is evidence of interaction between the factors, what are the value of the test statistic and the P-value? (a) F = 8.61; P-value = 0.002
(b) F = 1.22; P-value = 0.324
(c) F = 0.62; P-value = 0.712
(d) F = 128; P-value = 0.05
9. A. __________
(e) F = 8.62; P-value = 0.002
B. What is your conclusion from the test in Part A? Use α = 0.05. (a) Do not reject H0; There is evidence of interaction. (b) Do not reject H0; There is no evidence of interaction. (c) Reject H0; There is evidence of interaction. (d) Reject H0; There is no evidence of interaction. (e) Cannot determine.
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B. __________
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CHAPTER 11, FORM C, PAGE 8 C. Assume that there is no evidence of interaction between the factors. When conducting a test to see if there is a difference in mean scores based on an exam form, what are the critical F value and the value of the test statistic? (a) F = 8.61, P-value = 0.002
(b) F = 1.22; P-value = 0.324
(c) F = 0.62; P-value = 0.712
(d) F = 3.40; P-value = 0.002
C. __________
(e) F = 8.61; P-value = 0.712
D. Assume that there is no evidence of interaction between the factors. When conducting a test to see if there is a difference in mean scores based on school, what are the value of the test statistic and the P-value? (a) F = 8.62; P-value = 0.002
(b) F = 0.62; P-value = 0.712
(c) F = 1.22; P-value = 0.002
(d) F = 1.22; P-value = 0.324
D. __________
(e) F = 3.01; P-value = 0.05
E. What are your conclusions from the tests in Part C and D? Use α = 0.05.
E. __________
(a) Cannot determine. (b) Do not reject H0 from C; Do not reject H0 from D. (c) Do not reject H0 from C; Reject H0 from D. (d) Reject H0 from C; Reject H0 from D. (e) Reject H0 from C; Do not reject H0 from D.
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