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Homework answers / question archive / SanfordBrown College  MATH CHAPTER 14 SECTION 1: ANALYSIS OF VARIANCE 1)The Ftest used in oneway ANOVA is an extension of the ttest of 12
SanfordBrown College  MATH
CHAPTER 14 SECTION 1: ANALYSIS OF VARIANCE
1)The Ftest used in oneway ANOVA is an extension of the ttest of _{12}.
2. We use the analysis of variance (ANOVA) technique to compare two or more population means.
3. The Ftest in ANOVA tests whether or not the population variances are equal.
4. The sum of squares for treatments, SST, achieves its smallest value (zero) when all the sample means are equal.
5. The analysis of variance (ANOVA) technique analyzes the variance of the data to determine whether differences exist between the population means.
6. Conducting ttests for each pair or population means is statistically equivalent to conducting one Ftest comparing all the population means.
7. The sum of squares for error is also known as the betweentreatments variation.
8. In oneway ANOVA, the total variation SS(Total) is partitioned into two sources of variation: the sum of squares for treatments (SST) and the sum of squares for error (SSE).
9. In ANOVA, a criterion by which the populations are classified is called a factor.
10. In oneway ANOVA, the test statistic is defined as the ratio of the mean square for error (MSE) and the mean square for treatments (MST), namely, F = MSE / MST.
11. The Fstatistic in a oneway ANOVA represents the variation within the treatments divided by the variation between the treatments.
12. The distribution of the test statistic for analysis of variance is the Fdistribution.
13. The sum of squares for treatments (SST) is the variation attributed to the differences between the treatment means, while the sum of squares for error (SSE) measures the withintreatment variation.
14. If the numerator (MST) degrees of freedom is 3 and the denominator (MSE) degrees of freedom is 18, the total number of observations must equal 21.
15. The sum of squares for error (SSE) measures the amount of variation that is explained by the ANOVA model, while the sum of squares for treatments (SST) measures the amount of variation that remains unexplained.
16. The analysis of variance (ANOVA) tests hypotheses about population variances and requires all the population means to be equal.
17. The Ftest in ANOVA is an expansion of the ttest for two independent population means.
18. When the Ftest is used for ANOVA, the rejection region is always in the right tail.
19. The withintreatments variation provides a measure of the amount of variation in the response variables that is caused by the treatments.
20. We can use the Ftest to determine whether _{1} is greater than _{2}.
MULTIPLE CHOICE
21. The test statistic of the singlefactor ANOVA equals:
a. 
sum of squares for treatments / sum of squares for error. 
b. 
sum of squares for error / sum of squares for treatments. 
c. 
mean square for treatments / mean square for error. 
d. 
mean square for error / mean square for treatments. 
22. In a singlefactor analysis of variance, MST is the mean square for treatments and MSE is the mean square for error. The null hypothesis of equal population means is rejected if:
a. 
MST is much larger than MSE. 
b. 
MST is much smaller than MSE. 
c. 
MST is equal to MSE. 
d. 
None of these choices. 
23. In oneway ANOVA, the amount of total variation that is unexplained is measured by the:
a. 
sum of squares for treatments. 
b. 
degrees of freedom. 
c. 
total sum of squares. 
d. 
sum of squares for error. 
24. In a oneway ANOVA, error variability is computed as the sum of the squared errors, SSE, for all values of the response variable. This variability is the:
a. 
the total variation. 
b. 
withintreatments variation. 
c. 
betweentreatments variation. 
d. 
None of these choices. 
25. Which of the following is not a required condition for oneway ANOVA?
a. 
The sample sizes must be equal. 
b. 
The populations must all be normally distributed. 
c. 
The population variances must be equal. 
d. 
The samples for each treatment must be selected randomly and independently. 
26. The analysis of variance is a procedure that allows statisticians to compare two or more population:
a. 
proportions. 
b. 
means. 
c. 
variances. 
d. 
standard deviations. 
27. The distribution of the test statistic for analysis of variance is the:
a. 
normal distribution. 
b. 
Student tdistribution. 
c. 
Fdistribution. 
d. 
None of these choices. 
28. In the oneway ANOVA where there are k treatments and n observations, the degrees of freedom for the Fstatistic are equal to, respectively:
a. 
n and k. 
b. 
k and n. 
c. 
nk and k1. 
d. 
k1 and nk. 
29. In the oneway ANOVA where k is the number of treatments and n is the number of observations in all samples, the degrees of freedom for treatments is given by:
a. 
nk 
b. 
k1 
c. 
n1 
d. 
nk + 1 
30. In ANOVA, the Ftest is the ratio of two sample variances. In the oneway ANOVA (completely randomized design), the variance used as a numerator of the ratio is:
a. 
mean square for treatments. 
b. 
mean square for error. 
c. 
total sum of squares. 
d. 
None of these choices. 
31. In a completely randomized design for ANOVA, the numerator and denominator degrees of freedom are 4 and 25, respectively. The total number of observations must equal:
a. 
24 
b. 
25 
c. 
29 
d. 
30 
32. The number of degrees of freedom for the denominator in oneway ANOVA test involving 4 population means with 15 observations sampled from each population is:
a. 
60 
b. 
19 
c. 
56 
d. 
45 
33. The value of the test statistic in a completely randomized design for ANOVA is F = 6.29. The degrees of freedom for the numerator and denominator are 5 and 10, respectively. Using an F table, the most accurate statements to be made about the pvalue is that it is:
a. 
greater than 0.05 
b. 
between 0.001 and 0.010. 
c. 
between 0.010 and 0.025. 
d. 
between 0.025 and 0.050. 
34. In oneway ANOVA, the term
refers to the:
a. 
weighted average of the sample means. 
b. 
sum of the sample means divided by the total number of observations. 
c. 
sum of the population means. 
d. 
sum of the sample means. 
35. For which of the following is not a required condition for ANOVA?
a. 
The populations are normally distributed. 
b. 
The population variances are equal. 
c. 
The samples are independent. 
d. 
All of these choices are required conditions for ANOVA. 
36. In the oneway ANOVA where k is the number of treatments and n is the number of observations in all samples, the number of degrees of freedom for error is:
a. 
k1 
b. 
n1 
c. 
nk 
d. 
nk + 1 
37. Oneway ANOVA is applied to independent samples taken from three normally distributed populations with equal variances. Which of the following is the null hypothesis for this procedure?
a. 
_{1} + _{2} + _{3} = 0 
b. 
_{1} + _{2} + _{3} 0 
c. 
_{1} = _{2} = _{3} = 0 
d. 
_{1} = _{2} = _{3} 
38. How does conducting multiple ttests compare to conducting a single Ftest?
a. 
Multiple ttests increases the chance of a Type I error. 
b. 
Multiple ttests decreases the chance of a Type I error. 
c. 
Multiple ttests does not affect the chance of a Type I error. 
d. 
This comparison cannot be made without knowing the number of populations. 
39. In oneway analysis of variance, betweentreatments variation is measured by the:
a. 
SSE 
b. 
SS(Total) 
c. 
SST 
d. 
standard deviation 
40. Oneway ANOVA is applied to independent samples taken from four normally distributed populations with equal variances. If the null hypothesis is rejected, then we can infer that
a. 
all population means are equal. 
b. 
all population means differ. 
c. 
at least two population means are equal. 
d. 
at least two population means differ. 
41. Consider the following partial ANOVA table:
Source of Variation 
SS 
df 
MS 
F 
Treatments 
75 
* 
25 
6.67 
Error 
60 
* 
3.75 

Total 
135 
19 


The numerator and denominator degrees of freedom for the Ftest (identified by asterisks) are
a. 
4 and 15 
b. 
3 and 16 
c. 
15 and 4 
d. 
16 and 3 
42. Consider the following ANOVA table:
Source of Variation 
SS 
df 
MS 
F 
Treatments 
4 
2 
2.0 
0.80 
Error 
30 
12 
2.5 

Total 
34 
14 


The number of treatments is
a. 
13 
b. 
5 
c. 
3 
d. 
12 
43. In oneway analysis of variance, withintreatments variation is measured by:
a. 
sum of squares for error. 
b. 
sum of squares for treatments. 
c. 
total sum of squares. 
d. 
standard deviation. 
44. Consider the following ANOVA table:
Source of Variation 
SS 
df 
MS 
F 
Treatments 
128 
4 
32 
2.963 
Error 
270 
25 
10.8 

Total 
398 
29 


The total number of observations is:
a. 
25 
b. 
29 
c. 
30 
d. 
32 
45. In oneway analysis of variance, if all the sample means are equal, then the:
a. 
total sum of squares is zero. 
b. 
sum of squares for treatments is zero. 
c. 
sum of squares for error is zero. 
d. 
sum of squares for error equals sum of squares for treatments. 
46. Which of the following components in an ANOVA table is not additive?
a. 
Sum of squares 
b. 
Degrees of freedom 
c. 
Mean squares 
d. 
All of these choices are additive. 
47. In which case can an Ftest be used to compare two population means?
a. 
For one tail tests only. 
b. 
For two tail tests only. 
c. 
For either one or two tail tests. 
d. 
None of these choices. 
48. The Ftest statistic in a oneway ANOVA is equal to:
a. 
MST/MSE 
b. 
SST/SSE 
c. 
MSE/MST 
d. 
SSE/SST 
49. The numerator and denominator degrees of freedom for the Ftest in a oneway ANOVA are, respectively,
a. 
(nk) and (k1) 
b. 
(k1) and (nk) 
c. 
(kn) and (n1) 
d. 
(n1) and (kn) 
50. Which of the following statements is false?
a. 
F = t^{2} 
b. 
The Ftest can be used instead of a two tail ttest when you compare two population means. 
c. 
Doing three ttests is statistically equivalent to doing one Ftest when you compare three population means. 
d. 
All of these choices are true. 
COMPLETION
51. The ANOVA procedure tests to determine whether differences exist between two or more population ____________________.
52. The null hypothesis of ANOVA is that all the population means are ____________________.
53. The alternative hypothesis of ANOVA is that ____________________ population means are different.
54. In ANOVA the populations are classified according to one or more criterion, called ____________________.
55. SST measures the variation ____________________ treatments.
56. SSE measures the variation ____________________ treatments.
57. The Ftest statistic in ANOVA is equal to MS____________________ divided by MS____________________ and H_{0} is rejected for ____________________ values of F.
58. If SST explains a significant portion of the total variation, we conclude that the population means ____________________ (do/do not) differ.
59. The Ftest in ANOVA requires that the random variable be ____________________ distributed with equal ____________________.
60. If we square the tstatistic for two means, the result is the ____________________statistic.
SHORT ANSWER
TV News Viewing Habits
A statistician employed by a television rating service wanted to determine if there were differences in television viewing habits among three different cities in New York. She took a random sample of five adults in each of the cities and asked each to report the number of hours spent watching television in the previous week. The results are shown below. (Assume normal distributions with equal variances.)
Hours Spent Watching News on Television 

Albany 
Syracuse 
Utica 
25 
28 
23 
31 
33 
18 
18 
35 
21 
23 
29 
17 
27 
36 
15 
61. {TV News Viewing Habits Narrative} Set up the ANOVA Table. Use ? = 0.05 to determine the critical value.
62. {TV News Viewing Habits Narrative} Can she infer at the 5% significance level that differences in hours of television watching exist among the three cities?
Arthritis Pain Formulas
A pharmaceutical manufacturer has been researching new medications formulas to provide quicker relief of arthritis pain. Their laboratories have produced three different medications and they want to determine if the different medications produce different responses. Fifteen people who complained of arthritis pains were recruited for an experiment; five were randomly assigned to each medication. Each person was asked to take the medicine and report the length of time until some relief was felt (minutes). The results are shown below. (Assume normal distributions with equal variances.)
Time in Minutes Until Relief Is Felt (min) 

Medication 1 
Medication 2 
Medication 3 
4 
2 
6 
8 
5 
7 
6 
3 
7 
9 
7 
8 
8 
1 
6 
63. {Arthritis Pain Formulas Narrative} Set up the ANOVA Table. Use ? = 0.05 to determine the critical value.
64. {Arthritis Pain Formulas Narrative} Do these data provide sufficient evidence to indicate that differences in the average time of relief exist among the three medications? Use ? = 0.05.
Sub Sandwich Customers
The marketing manager of a Sub Shop chain is in the process of examining some of the demographic characteristics of her customers. In particular, she would like to investigate the belief that the ages of the customers of Sub Shops, hamburger emporiums, and fastfood chicken restaurants are different. The ages of eight randomly selected customers of each of the restaurants are recorded and listed below. From previous analyses we know that the ages are normally distributed with equal variances for each group.
Customers' Ages 

Subs 
Hamburger 
Chicken 
23 
26 
25 
19 
20 
28 
25 
18 
36 
17 
35 
23 
36 
33 
39 
25 
25 
27 
28 
19 
38 
31 
17 
31 
65. {Sub Sandwich Customers Narrative} Set up the ANOVA Table. Use ? = 0.05 to determine the critical value.
66. {Sub Sandwich Customers Narrative} Do these data provide enough evidence at the 5% significance level to infer that there are differences in ages among the customers of the three restaurants?
GMAT Scores
A recent college graduate is in the process of deciding which one of three graduate schools he should apply to. He decides to judge the quality of the schools on the basis of the Graduate Management Admission Test (GMAT) scores of those who are accepted into the school. A random sample of six students in each school produced the following GMAT scores. Assume that the data are normally distributed with equal variances for each school.
GMAT Scores 

School 1 
School 2 
School 3 
650 
105 
590 
620 
550 
510 
630 
700 
520 
580 
630 
500 
710 
600 
490 
690 
650 
530 
67. {GMAT Scores Narrative} Set up the ANOVA Table. Use ? = 0.05 to determine the critical value.
68. {GMAT Scores Narrative} Can he infer at the 10% significance level that the GMAT scores differ among the three schools?
69. In a completely randomized design, 15 experimental units were assigned to each of four treatments. Fill in the blanks (identified by asterisks) in the partial ANOVA table shown below.
Source of Variation 
SS 
df 
MS 
F 
Treatments 
* 
* 
240 
* 
Error 
* 
* 
* 

Total 
2512 
* 


70. In a completely randomized design, 12 experimental units were assigned to the first treatment, 15 units to the second treatment, and 18 units to the third treatment. A partial ANOVA table is shown below:
Source of Variation 
SS 
df 
MS 
F 
Treatments 
* 
* 
* 
9 
Error 
* 
* 
35 

Total 
* 
* 


a. 
Fill in the blanks (identified by asterisks) in the above ANOVA table. 
b. 
Test at the 5% significance level to determine if differences exist among the three treatment means. 
Gold Funds
An investor studied the percentage rates of return of three different gold funds. Random samples of percentage rates of return for four periods were taken from each fund. The results appear in the table below:
Gold Funds Percentage Rates 

Fund 1 
Fund 2 
Fund 3 
12 
4 
9 
15 
8 
3 
13 
6 
5 
14 
5 
7 
17 
4 
4 
71. {Gold Funds Narrative} Set up the ANOVA Table. Use ? = 0.05 to determine the critical value.
72. {Gold Funds Narrative} Test at the 5% significance level to determine whether the mean percentage rates for the three funds differ.
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