9. Consider a multinomial experiment involving 100 trials and 4 categories (cells). The observed
frequencies resulting from the experiment are shown in the accompanying table.
Category
1
2
3
4
Frequency
18
30
25
27
Use the 5% significance level to test the hypotheses.
H0 : p1 = 0.25, p2 = 0.30, p3 = 0.20, p4 = 0.25.
H1 ;At least two proportions differ from their specified values.
10. In 2003, the student body of a university in NSW consisted of 30% first-years, 25% second-years, 27%
third-years, and 18% fourth-years. A sample of 665-6 students taken from the 2004 student body
showed that there are 138 first-years, 88 second-years, 94 third-years, and 80 fourth-years. Test with
5% significance level to determine whether the student body proportions have changed.
:
1
H
11. Consider a multinomial experiment involving 160 trials and 4 categories (cells). The observed
frequencies resulting from the experiment are shown in the following table.
Category
1
2
3
4
Frequency
53
35
30
42
Use the 10% significance level to test the hypotheses.
43210 :ppppH ===
.
:
1
H
At least two proportions differ from their specified values.
12. A statistics professor posted the following grade distribution guidelines for his elementary statistics
class: 8% A, 35% B, 40% C, 12% D, and 5% F. A sample of 100 elementary statistics grades at the
end of last semester showed 12 As, 30 Bs, 35 Cs, 15 Ds, and 8 Fs. Test at the 5% significance level to
determine whether the actual grades deviate significantly from the posted grade distribution guidelines.
13. Conduct a test to determine whether the two classifications A and B are independent, using the data in
the accompanying table and
01.=
.
1
B
2
B
1
A
42
28
2
A
23
57
14. Explain what is meant by the rule of five.
15. In 2003, computers of Brand A controlled 25% of the market, Brand B 20%, Brand C 10% and Brand
D 45%. In 2004, sample data were collected from many randomly selected stores throughout the
country. Of the 1200 computers sold, 280 were Brand A, 270 were Brand B, 90 were Brand C and 560
were Brand D. Has the market changed since 2003? Test at the 1% significance level.
16. A major insurance firm interviewed a random sample of 1500 college students to find out the type of
life insurance preferred, if any. The results follow:
Insurance Preference
Gender
Term
Whole life
No insurance
Female
170
110
470
Male
195
75
480
Is there evidence that the life insurance preference of male students is different to that of female
students? Test using the 5% level of significance.
17. The number of cars sold by three salespersons over a 3-month period are shown below:
Brand of Car
Salesperson
Brand A
Brand B
Brand C
David
7
2
6
Edward
11
4
8
Frank
8
5
3
Using the 5% level of significance, test for the independence of salesperson and type of product sold.
18. A telephone company prepared four versions of a set of instructions for placing collect calls. The
company asked a sample of 1600 people which one of the four forms was easiest to understand. In the
sample, 425 people preferred Form A, 385 preferred Form B, 375 preferred Form C, and 415 preferred
Form D. At the 5% level of significance, can one conclude that in the population there is a preferred
form?
19. Suppose that a random sample of 100 observations was drawn from a population. After calculating the
mean and standard deviation, each observation was standardised and the number of observations in
each of the intervals below was counted. Can we infer at the 10% significance level that the data were
drawn from a normal population?
Intervals
Frequency
Z
–1
12
–1 < Z
0
32
0 < Z
1
36
Z > 1
20
20. Suppose that a random sample of 150 observations was drawn from a population. After calculating the
mean and standard deviation, each observation was standardised and the number of observations in
each of the intervals below was counted. Can we infer at the 5% significance level that the data were
drawn from a normal population?
Intervals
Frequency
Z
–1.5
15
–1.5 < Z
–0.5
32
–0.5
Z
0.5
65
0.5 < Z
1.5
25
Z > 1.5
13
21. The president of a large university has been studying the relationship between male/female supervisory
structures in his institution and the level of employees’ job satisfaction. The results of a recent survey
are shown in the table below. Conduct a test at the 5% significance level to determine whether the
level of job satisfaction depends on the boss/employee gender relationship.
Boss/Employee
Level of Satisfaction
Male/Female
Female/Male
Male/Male
Female/Female
Satisfied
60
15
50
15
Neutral
27
45
48
50
Dissatisfied
13
32
12
55
22. Consumer panel preferences for three proposed fast food restaurants are as follows:
Restaurant A
Restaurant B
Restaurant C
48
62
40
Using the 0.05 level of significance, test to see if there is a preference among the three restaurants.
23. A cafeteria proposes to serve four main courses. For planning purposes, the manager expects that the
proportions of each that will be selected by his customers will be:
Selection
Proportion
Chicken
0.50
Roast Beef
0.20
Steak
0.10
Fish
0.20
Of the first 100 customers, 44 select chicken, 24 select roast beef, 13 select steak, and 10 select fish.
Should the manager revise his estimates? Use
= 0.01.
24. A large carpet store wishes to determine if the brand of carpet purchased is related to the purchaser’s
family income. As a sampling frame, they mailed a survey to people who have a store credit card. Five
hundred customers returned the survey and the results follow:
Brand of Carpet
Family Income
Brand A
Brand B
Brand C
High income
65
32
32
Middle income
80
68
104
Low income
25
35
59
At the 5% level of significance, can you conclude that the brand of carpet purchased is related to the
purchaser’s family income?
25. To determine whether a single coin is fair, the coin was tossed 200 times. The observed frequencies
with which each of the two sides of the coin turned up are recorded as 112 heads and 88 tails. Is there
sufficient evidence at the 5% significance level to allow you to conclude that the coin is not fair?
26. Consider a multinomial experiment involving n = 200 trials and k = 5 cells. The observed frequencies
resulting from the experiment are shown in the following table:
Cell
1
2
3
4
5
Frequency
16
44
56
48
36
The null hypothesis to be tested is as follows.
H0 : p1 = 0.05, p2 = 0.25, p3 = 0.35, p4 = 0.20, p5 = 0.15.
Test the hypothesis at the 5% level of significance.
27. Consider a multinomial experiment involving n = 200 trials and k = 5 cells. The observed frequencies
resulting from the experiment are shown in the following table:
Cell
1
2
3
4
5
Frequency
8
22
28
24
18
The null hypothesis to be tested is as follows.
0 1
: 0.10H p =
,
20.25,p=
30.30,p=
40.20,p=
50.15p=
50.15p=
.
28. Consider a multinomial experiment involving n = 200 trials and k = 5 cells. The observed frequencies
resulting from the experiment are shown in the following table.
Cell
1
2
3
4
5
Frequency
4
11
14
12
9
The null hypothesis to be tested is as follows.
0 1
: 0.10H p =
,
20.25,p=
30.30,p=
40.20,p=
50.15p=
50.15p=
.
29. a. Consider the data in the accompanying table with classifications A and B:
1
B
2
B
1
A
40
80
2
A
56
48
Conduct a test to determine whether the two classifications A and B are independent, using
0.05
=
.
b. What would be the effect of decreasing the sample size?
30. Consider the data in the accompanying table with classifications A and B:
1
B
2
B
1
A
20
30
2
A
28
24
Conduct a test to determine whether the two classifications A and B are independent, using
0.05
=
.
31. Insurance companies would like to know whether the proportion of their clients who submit claims for
car accidents is about the same for different age groups. In order to study this issue, a statistician took
a random sample of 350 clients of a major insurance company and prepared the following contingency
table:
Age group
25 and under
Over 25 and
under 50
50 and over
Claim
35
60
68
No claim
65
80
42
Conduct a test to determine whether the two classifications A and B are independent, using
0.05
=
.
32. A biology professor claimed that the proportions of grades in his classes are the same. A sample of 100
students showed the following frequencies.
Grade
A
B
C
D
F
Frequency
14
23
27
26
10
State the null and alternative hypotheses to be tested.
.
33. A biology professor claimed that the proportions of grades in his classes are the same. A sample of 100
students showed the following frequencies.
Grade
A
B
C
D
F
Frequency
14
23
27
26
10
Determine the rejection region at the 5% significance level.
34. A biology professor claimed that the proportions of grades in his classes are the same. A sample of 100
students showed the following frequencies.
Grade
A
B
C
D
F
Frequency
14
23
27
26
10
Compute the value of the test statistic.
35. A biology professor claimed that the proportions of grades in his classes are the same. A sample of 100
students showed the following frequencies.
Grade
A
B
C
D
F
Frequency
14
23
27
26
10
Use statistical software to compute the p-value for this test.
36. A biology professor claimed that the proportions of grades in his classes are the same. A sample of 100
students showed the following frequencies.
Grade
A
B
C
D
F
Frequency
14
23
27
26
10
Do the data provide enough evidence to support the professor’s claim?
37. A salesperson makes five calls per day. A sample of 200 days gives the frequencies of sales volumes
listed below:
Number of sales
Observed frequency (days)
0
10
1
38
2
69
3
63
4
18
5
2
Assume that the population is binomially distributed with a probability of purchase p = 0.50.
Compute the expected frequencies for x = 0, 1, 2, 3, 4 and 5 by using the binomial probability function
or the binomial tables. Combine categories if necessary to satisfy the rule of five.
38. A salesperson makes five calls per day. A sample of 200 days gives the frequencies of sales volumes
listed below:
Number of sales
Observed frequency (days)
0
10
1
38
2
69
3
63
4
18
5
2
Assume that the population is binomially distributed with a probability of purchase p = 0.50.
Should the assumption of a binomial distribution be rejected at the 5% significance level?
39. An Australian tyre manufacturer operates a plant in Melbourne and another plant in Adelaide.
Employees at each plant have been evenly divided among three issues (wages, working conditions and
super benefits) in terms of which one they feel should be the primary issue in the upcoming contract
negotiations. The secretary of the union has recently circulated pamphlets among the employees,
attempting to convince them that super benefits should be the primary issue. A subsequent survey
revealed the following breakdown of the employees according to the plant at which they worked and
the issue that they felt should be supported as the primary one.
Issues
Plant Location
Very interesting
Fairly interesting
Not interesting
Melbourne
60
62
78
Adelaide
70
56
74
Can you infer at the 5% significance level that the proportional support by the employees at both
plants for the issues has changed since the pamphlet was circulated?
40. An Australian tyre manufacturer operates a plant in Melbourne and another plant in Adelaide.
Employees at each plant have been evenly divided among three issues (wages, working conditions and
super benefits) in terms of which one they feel should be the primary issue in the upcoming contract
negotiations. The secretary of the union has recently circulated pamphlets among the employees,
attempting to convince them that super benefits should be the primary issue. A subsequent survey
revealed the following breakdown of the employees according to the plant at which they worked and
the issue that they felt should be supported as the primary one.
Issues
Plant Location
Very interesting
Fairly interesting
Not interesting
Melbourne
60
62
78
Adelaide
70
56
74
Can you infer at the 5% significance level that the proportional support by the Melbourne employees
for the three issues has changed since the pamphlet was circulated?
41. An Australian tyre manufacturer operates a plant in Melbourne and another plant in Adelaide.
Employees at each plant have been evenly divided among three issues (wages, working conditions and
super benefits) in terms of which one they feel should be the primary issue in the upcoming contract
negotiations. The secretary of the union has recently circulated pamphlets among the employees,
attempting to convince them that super benefits should be the primary issue. A subsequent survey
revealed the following breakdown of the employees according to the plant at which they worked and
the issue that they felt should be supported as the primary one.
Issues
Plant Location
Very interesting
Fairly interesting
Not interesting
Melbourne
60
62
78
Adelaide
70
56
74
Can you infer at the 5% significance level that the proportional support by the Adelaide employees for
the three issues has changed since the pamphlet was circulated?
42. An Australian tyre manufacturer operates a plant in Melbourne and another plant in Adelaide.
Employees at each plant have been evenly divided among three issues (wages, working conditions and
super benefits) in terms of which one they feel should be the primary issue in the upcoming contract
negotiations. The secretary of the union has recently circulated pamphlets among the employees,
attempting to convince them that super benefits should be the primary issue. A subsequent survey
revealed the following breakdown of the employees according to the plant at which they worked and
the issue that they felt should be supported as the primary one.
Issues
Plant Location
Very interesting
Fairly interesting
Not interesting
Melbourne
60
62
78
Adelaide
70
56
74
Do the data indicate at the 5% significance level that there are differences between the two plants
regarding which issue should be the primary one?