Ch8 Anova To Compare Means Test Bank Docx

Statistics - Unlocking the Power of Data, 3e (Lock)

Chapter 8 ANOVA to Compare Means

8.1 Analysis of Variance

Use the following to answer the questions below:

Two sets of sample data, A and B, are given. Without doing any calculations, indicate in which set of sample data, A or B, there is likely to be stronger evidence of a difference in the population means.

1)

Dataset A	Dataset B
Group 1 Group 2	Group 1 Group 2
19 23	15 27
21 24	17 28
20 22	16 26
20 24	16 28
21 22	17 26
21 23	17 27
19 24	15 28
19 22	15 26
= 20.0 = 23.0	= 16.0 = 27.0

A) Dataset A

B) Dataset B

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2)

Dataset A	Dataset B
Group 1 Group 2 Group 3	Group 1 Group 2 Group 3
12 25 8	12 19 12
8 15 10	11 18 13
15 12 16	12 18 14
9 9 17	12 18 13
17 28 20	12 17 13
11 19 7	13 18 13
= 12.0 = 18.0 = 33.0	= 12.0 = 18.0 = 33.0

A) Dataset A

B) Dataset B

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3)

A) Dataset A

B) Dataset B

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4)

A) Dataset A

B) Dataset B

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5)

Dataset A	Dataset B
Group 1 Group 2	Group 1 Group 2
42 48	41 50
42 47	43 46
41 48	39 44
43 48	45 52
42 48	37 45
42 49	47 51
= 42 = 48	= 42 = 48

A) Dataset A

B) Dataset B

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6)

A) Dataset A

B) Dataset B

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7) Analysis of variance is used to test for significant differences among

A) means.

B) variances.

C) standard deviations.

D) proportions.

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8) Some computer output from an analysis of variance is provided.

How many groups are there?

A) 3

B) 4

C) 5

D) 6

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9) Some computer output from an analysis of variance is provided.

What is the overall sample size?

A) 125

B) 124

C) 123

D) 121

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10) A scientist performing analysis of variance has the following null hypothesis:

: = = = .

What is the appropriate alternative hypothesis for his analysis?

A) : ≠ ≠ ≠ .

B) : > > > .

C) : < < < .

D) : At least one ≠

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11) True or False: SSE = SSTotal + SSG

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Use the following to answer the questions below:

The sample sizes for the groups in a dataset and an outline of an analysis of variance table with partial information are provided. Fill in the missing parts of the table. Round decimal answers to two decimal places.

12) Three groups with = 10, = 10, and = 10.

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13) Four groups with = 6, = 5, = 5, and = 4.

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14) Three groups with = 8, = 7, and = 5.

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8.2 Pairwise Comparisons and Inference after ANOVA

Use the following to answer the questions below:

A small university is concerned with monitoring its electricity usage in its Student Center. Specifically, its officials want to know if the amount of electricity used differs by day of the week. They collected data for nearly a year, and the relevant summary statistics are provided. Note that electricity usage is measured in kilowatt hours.

Day of Week	n		s
Sunday	45	86.48	34.89
Monday	45	109.29	27.37
Tuesday	45	110.96	28.64
Wednesday	44	115.03	31.68
Thursday	44	114.97	33.26
Friday	45	108.58	32.22
Saturday	45	87.07	38.56
	313	104.56	34.25

1) State the appropriate null and alternative hypotheses for this test.

A) : = = = = = =

: At least one ≠

B) : = = = = = =

: At least one ≠

C) : At least one ≠

: = = = = = =

D) : At least one ≠

: = = = = = =

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2) Are the conditions for using ANOVA reasonably satisfied?

A) Yes

B) No

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3) Complete the ANOVA table below for doing this test using the template started below. Use two decimal places in the F statistic.

A)

B)

C)

D)

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4) Use the F-distribution to find the p-value for the test. Using α = 0.05, does the mean electricity usage differ significantly by day of the week? Make a conclusion in context.

A) There is very strong evidence that mean electricity usage differs significantly by day of the week (i.e., some days of the week use more electricity than others).

B) There is not enough evidence to conclude that mean electricity usage differs significantly by day of the week (i.e., some days of the week use more electricity than others).

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5) Using the results from the ANOVA analysis, at an α = 0.05 level of confidence, what is the conclusion of the test, in context?

A) There is very strong evidence that mean electricity usage differs significantly by day of the week (i.e., some days of the week use more electricity than others).

B) There is not enough evidence to conclude that mean electricity usage differs significantly by day of the week (i.e., some days of the week use more electricity than others).

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6) Use the data and ANOVA results to construct a 95% confidence interval for the difference in mean electricity use between Saturdays and Sundays. Round the margin of error to two decimal places. Does your interval suggest a significant difference in mean electricity use for these two days?

A) -12.92 to 14.10

There is no evidence that electricity use differs significantly on Saturdays and Sundays.

B) -12.92 to 14.10

There is strong evidence that electricity use differs significantly on Saturdays and Sundays.

C) -11.09 to 12.27

There is no evidence that electricity use differs significantly on Saturdays and Sundays.

D) -11.09 to 12/27

There is strong evidence that electricity use differs significantly on Saturdays and Sundays.

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7) Based on the ANOVA results, test at the 5% level whether the data provide evidence of a difference in mean electricity use on Sundays and Mondays. Use three decimal places in the test statistic.

: =

: ≠

t = = -3.323

p-value = 0.001 (two-tail in t-distribution with df = 306, using Statkey)

There is very strong evidence that electricity use differs significantly on Sundays and Mondays.

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8) Computer output provides the following grouping information:

Means that do not share a letter are significantly different.

Use the output to make a statement about how electricity usage differs significantly by day of the week.

A) Significantly less electricity is used on weekends than on weekdays.

B) There is not enough evidence to conclude that electricity usage on weekends is different than electricity usage on weekdays.

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Use the following to answer the questions below:

Penalties in ice hockey occur when a player breaks one of the rules of the game. In most cases, when a penalty occurs, the offending player is placed in the penalty box (the length of time spent in the penalty box depends on the severity of the penalty), and the team has to play with fewer people on the ice, which can result in an advantage for the opposing team. The number of penalties per game for several randomly selected games are displayed for three college men's ice hockey teams.

Team	Penalties	n		s
A	9 9 5 11 9	5	8.6	2.191
B	7 3 5 1 5	5	4.2	2.280
C	3 7 2 4 8	5	4.8	2.588
Overall		15	5.87	2.973

9) Use the summary information to compute the three sums of squares needed for using ANOVA to test for a difference in mean number of penalties among these three teams. Round each to two decimal places.

A) SSG = 56.93

SSE = 66.79

SSTotal = 123.74

B) SSG = 40.28

SSE = 83.46

SSTotal = 123.74

C) SSG = 65.79

SSE = 66.79

SSTotal = 132.58

D) SSG = 49.12

SSE = 83.46

SSTotal = 132.58

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10) Construct the ANOVA table and test, at the 5% significance level, for a difference in mean number of penalties among these three hockey teams. Use two decimal places when rounding decimal values. Include the details of your test.

: = =

Ha: At least one ≠

With F = 5.11, the p-value = 0.025 (from F-distribution with 2 and 12 degrees of freedom, using Statkey).

We have evidence to reject Ho and thus have evidence of a significance difference in mean number of penalties among the three teams.

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11) ANOVA output gives a p-value of 0.025 for the difference in mean number of penalties among the three teams. Using α = 0.05, what is the conclusion of the test in context?

A) There is evidence of a significance difference in mean number of penalties among the three teams.

B) There is not enough evidence to conclude that there is a significance difference in mean number of penalties among the three teams.

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(a) Team A and Team B

(b) Team A and Team C

(c) Team B and Team C

In each case, round the margin of error to two decimal places. Based on your work, which teams have significantly different means? Briefly justify your answer.

(a) (8.6 - 4.2) ± 2.179

4.4 ± 2.179(1.492649)

4.4 ± 3.25

1.15 to 7.65

(b) (8.6 - 4.8) ± 2.179

3.8 ± 3.25

0.55 to 7.05

(c) (4.2 - 4.8) ± 2.179

-0.6 ± 3.25

-3.85 to 2.65

Because the first two intervals do not contain 0, they provide evidence that those pairs of means are significantly different. In both cases, since the interval encompasses entirely positive values, there is evidence that Team A has the larger mean.

There is evidence that Team A earns significantly more penalties than either Team B or Team C. There is no evidence that the mean number of penalties for Teams B and C differs.

Note that the results here are different from those in the computer output problems because these intervals have not been adjusted for multiple comparisons.

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13) Computer output provides the following information about the pairwise differences:

Based on this output, which teams have significantly different means?

A) Teams A and B

B) Teams A and B; Teams A and C

C) Teams A and C

D) All three teams have significantly different means

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14) Computer output provides the following grouping information:

Means that do not share a letter are significantly different.

Based on this output, which teams have significantly different means?

A) Teams A and B

B) Teams A and B; Teams A and C

C) Teams A and C

D) All three teams have significantly different means

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Use the following to answer the questions below:

Breakfast is often considered to be the most important meal of the day. Data on the number of calories per serving for randomly selected cereals from three different brands (General Mills, Kellogg's, and Kashi) are summarized in the provided plot and table.

Brand	n		s
General Mills	26	125.77	41.10
Kashi	16	178.13	39.02
Kellogg's	33	141.52	43.24
Overall	75	143.87	45.38

15) State the appropriate null and alternative hypotheses for testing if the mean calories per serving differs among the three brands.

A) : = =

: At least one ≠

B) : At least one ≠

: = =

C) : = =

: > >

D) : > >

: = =

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16) Are the conditions for using ANOVA reasonably satisfied?

A) Yes

B) No

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17) Computer analysis gives a p-value of 0.001. Using α = 0.05, what is the conclusion of the test, in context?

A) There is very strong evidence that the mean calories per serving is not the same for all three brands.

B) There is not enough evidence to conclude that the mean calories per serving is not the same for all three brands.

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18) Use the summary information and the fact that the sums of squares for groups is SSG = 27,476 and for error is SSTotal = 152,379 to complete an ANOVA table and find the F-statistic. Round decimal answers to two decimal places.

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19) Use the summary information and the fact that the sums of squares for groups is SSG = 27,476 and for error is SSTotal = 152,379 to complete an ANOVA table and find the F-statistic. Use the F-distribution to find the p-value and state the conclusion of the test in context (using ).

A) p-value = 0.00078

There is very strong evidence that the mean calories per serving differs significantly among the three brands.

B) p-value = 0.00078

There is not enough evidence to conclude that the mean calories per serving differs significantly among the three brands.

C) p-value = 0.078

There is very strong evidence that the mean calories per serving differs significantly among the three brands.

D) p-value = 0.078

There is not enough evidence to conclude that the mean calories per serving differs significantly among the three brands.

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20) Computer output from Minitab is provides the following information about the pairwise differences:

Brand = General Mills subtracted from:

Brand = Kashi subtracted from:

Based on this output, which brands have significantly different means?

A) Kashi and General Mills have significantly different means.

Kashi and Kellogg's have significantly different means.

Kellogg's and General Mills are not significantly different.

B) Kashi and General Mills have significantly different means.

Kashi and Kellogg's are not significantly different.

Kellogg's and General Mills are not significantly different.

C) Kashi and General Mills are not significantly different.

Kashi and Kellogg's are not significantly different.

Kellogg's and General Mills are not significantly different.

D) Kashi and General Mills are not significantly different.

Kashi and Kellogg's have significantly different means.

Kellogg's and General Mills have significantly different means.

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L.O.: 8.2.0

21) Computer output provides the following grouping information:

Means that do not share a letter are significantly different.

Based on this output, which brands have significantly different means? Briefly justify your answer.

Kashi and Kellogg's have significantly different means because they do not share a letter. Kashi cereals have significantly more calories per serving than General Mills cereals.

Kellogg's and General Mills are not significantly different because they share a letter (B).

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(a) General Mills and Kashi

(b) General Mills and Kellogg's

(c) Kashi and Kellogg's

In each case, round the test statistic to three decimal places. Based on your work, which brands have significantly different means? Briefly justify your answer.

(a) General Mills and Kashi

: =

: ≠

t = = -3.956

p-value ≈ 0 (two-tail probability, df = 72, using Statkey)

Very strong evidence that the mean calories per serving differs significantly for General Mills and Kashi cereals (Kashi has the larger mean).

(b) General Mills and Kellogg's

: =

: ≠

t = = -1.442

p-value = 0.154 (two-tail probability, df = 72, using Statkey)

No evidence that the mean calories per serving differs significantly for General Mills and Kellogg's cereals.

(c) Kashi and Kellogg's

: =

: ≠

t = = 2.885

p-value = 0.005 (two-tail probability, df = 72, using Statkey)

Very strong evidence that the mean calories per serving differs significantly for Kashi and Kellogg's cereals (Kashi has the larger mean).

Diff: 2 Type: ES Var: 1

L.O.: 8.2.2

Use the following to answer the questions below:

Breakfast is often considered to be the most important meal of the day. Data on the amount of sugar (g) per serving for randomly selected cereals from three different brands (General Mills, Kellogg's, and Kashi) are summarized in the provided plot and table.

Brand	n		s
General Mills	26	8.538	4.492
Kashi	16	8.500	3.183
Kellogg's	33	10.636	3.516
Overall	75	9.453	3.916

23) Are the conditions for using ANOVA reasonably satisfied?

A) Yes

B) No

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L.O.: 8.1.3

24) Computer output from the analysis gives a p-value of 0.066. Test, at the 5% level, if there is evidence that the average amount of sugar per serving differs significantly among the three brands.

A) There is no evidence that the average amount of sugar per serving differs significantly among the three brands.

B) The average amount of sugar per serving differs significantly between Kellogg's and Kashi.

C) Kellogg's average amount of sugar per serving id significantly greater than both Kashi and General Mills.

D) The average amount of sugar per serving differs significantly among the three brands.

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25) Use the summary information to compute the three sums of squares needed for using ANOVA to test for a difference in the mean amount of calories per serving among the three brands. Round each to two decimal places.

SSTotal = (75 - 1) ∙ = 1,134.79

SSE = (26 - 1) ∙ + (16 - 1) ∙ + (33 - 1) ∙ = 1,052.02

SSG = SSTotal - SSE = 1,134.79 - 1,052.02 = 82.77

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L.O.: 8.1.1

26) Construct the ANOVA table and test, at the 5% significance level, for a difference in mean amount of sugar among the three brands. Use two decimal places in all decimal values. Is there enough evidence to conclude that the average amount of sugar per serving differs significantly among the three brands.

A) Yes

B) No

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L.O.: 8.1.1

27) Should you conduct inference after the ANOVA to investigate differences among the pairs of means in this situation? Briefly explain why or why not.

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L.O.: 8.2.0

Use the following to answer the questions below:

An environmental studies student working on an independent research project was investigating metal contamination in the St. Lawrence River. The metals can accumulate in organisms that live in the river (known as bioaccumulation). He collected samples of Quagga mussels at three sites in the St. Lawrence River and measured the concentration of copper (in micrograms per gram, μg/g or mcg/g) in the mussels. His data are summarized in the provided table and plot. He wants to know if there are any significant differences in mean copper concentration among the three sites.

Site	Copper Concentration (μg/ g)	n s
1 2 3	19.9 23.4 17.5 25.4 20.5 13.0 18.8 18.4 16.1 18.4 13.8 7.0 11.4 15.2	5 21.34 3.092 4 16.60 2.687 5 13.16 4.274
Overall		14 17.06 4.82

28) Are the conditions for using ANOVA reasonably satisfied?

A) Yes

B) No

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29) Use the summary information to compute the three sums of squares needed for using ANOVA to test for a difference in mean copper concentration among the three sites. Round each to two decimal places.

SSTotal = (14 - 1) ∙ = 302.02

SSE = (5 - 1) ∙ + (4 - 1) ∙ + (5 - 1) ∙ = 132.97

SSG = SSTotal - SSE = 169.05

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30) Construct the ANOVA table and test, using α = 0.05, for a difference in mean copper concentration among the three sites. Round decimal values to two decimal places. Include all details of the test.

: = =

: At least one ≠

F = 6.99

p-value = 0.011 (right-tail probability in the F distribution with 2 and 11 degrees of freedom, using Statkey)

There is evidence of a difference in mean copper concentration among the three sites.

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31) Computer output from the analysis gives a p-value of 0.011. Test, using α = 0.05, for a difference in mean copper concentration among the three sites. Include all details of the test.

: = =

: At least one ≠

F = 6.97 and p-value = 0.011

There is evidence of a difference in mean copper concentration among the three sites.

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L.O.: 8.1.1

(a) Site 1 and Site 2

(b) Site 1 and Site 3

(c) Site 2 and Site 3

In each case, round the margin of error to two decimal places. Based on your work, which sites have significantly different means? Briefly justify your answer.

(a) Site 1 and Site 2

(21.34 - 16.6) ± 2.201

4.74 ± 2.201(2.332488)

4.74 ± 5.13

-0.39 to 9.87

Because this confidence interval contains 0, there is no evidence that Sites 1 and 2 have significantly different mean concentrations of copper.

(b) Site 1 and Site 3

(21.34 - 13.16) ± 2.201

8.18 ± 2.201(2.199091)

8.18 ± 4.84

3.34 to 13.02

The confidence interval does not contain 0, thus there is evidence that Sites 1 and 3 have significantly different mean concentrations of copper. Because the interval only contains positive values, there is evidence that Site 1 has a significantly higher mean concentration of copper than Site 3.

(c) Site 2 and Site 3

(16.6 - 13.16) ± 2.201

3.44 ± 2.201(2.332488)

3.44 ± 5.13

-1.69 to 8.57

Because this confidence interval contains 0, there is no evidence that Sites 2 and 3 have significantly different mean concentrations of copper.

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L.O.: 8.2.1

33) Computer output from the analysis provides the following information about the pairwise differences:

Site = 1 subtracted from:

Site = 2 subtracted from:

Based on this output, which sites have significantly different means?

A) Only Sites 1 and 3 have significantly different means.

B) Only Sites 1 and 2 have significantly different means.

C) Only Sites 2 and 3 have significantly different means.

D) None of the sites have significantly different means.

Diff: 2 Type: BI Var: 1

L.O.: 8.2.0

34) Computer output from the analysis provides the following grouping information:

Means that do not share a letter are significantly different.

Based on this output, which sites have significantly different means? Briefly justify your answer.

A) Only Sites 1 and 3 have significantly different means.

B) Only Sites 1 and 2 have significantly different means.

C) Only Sites 2 and 3 have significantly different means.

D) None of the sites have significantly different means.

Diff: 3 Type: BI Var: 1

L.O.: 8.2.0

Use the following to answer the questions below:

Summary statistics from a dataset and the corresponding computer analysis of variance output are provided.

35) What is the pooled standard deviation?

A) 24.9

B) 4.99

C) 16.90

D) 42.34

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36) What degrees of freedom are used in doing inferences for these means and differences in means after ANOVA?

A) 2

B) 3

C) 72

D) 74

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L.O.: 8.2.0

37) Find a 90% confidence interval for the mean of population A. Round the margin of error to three decimal places.

A) 35.040 to 38.366

B) 35.061 to 38.345

C) 34.747 to 38.660

D) 34.937 to 38.470

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L.O.: 8.2.1

38) Find a 95% confidence interval for the difference in the means of Populations A and B. Round the margin of error to three decimal places.

A) 4.333 to 9.035

B) 4.362 to 9.006

C) 3.918 to 9.405

D) 4.191 to 9.177

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L.O.: 8.2.1

39) Test for a difference in population means between groups A and C. Use α = 0.05 and show all details of the test. Round the test statistic to two decimal places.

: =

: ≠

t = = 2.99

p-value = 0.004

There is very strong evidence that the means of populations A and C are significantly different (and that population A has a significantly larger mean that Population C).

Diff: 2 Type: ES Var: 1

L.O.: 8.2.2

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