8.1 An analyst studies two types of advertising methods (A and B) tried by retailers. The variable is the sum of the amount spent on advertising over the past year. The following is the sample statistics extracted independently from retailers of each type. (Unit million USD)
Type A: \(\small n_1 = 60, \; {\overline x}_1 = 14.8, \; s_1^2 = 0.180\)
Type B: \(\small n_1 = 70, \; {\overline x}_2 = 14.5, \; s_2^2 = 0.133\)
From these data, can you conclude that type A retailers have invested more in advertising than type B retailers? (Significance level = 0.05)
8.2 Paper making plants are looking to buy one of two forests. The followings are diameters of 50 trees sampled from each forest. From these data, test at the significance level of 0.05 whether the trees in area B are on average smaller than those in area A. What is the p-value of this test?
Area A: \(\small {\overline x}_1 = 28.25, \; s_1^2 = 25\)
Area B: \(\small {\overline x}_2 = 22.50, \; s_2^2 = 16\)
8.3 In order to check the period of residence at the current house in region A and B, the following statistics were examined from simple random samples of 100 households in A and 150 households in B. From this data, can households in A area live shorter on average than those in B? (Significance level = 0.05)
Region A: \(\small {\overline x}_1 = 33 months, \; s_1^2 = 900\)
Region B: \(\small {\overline x}_2 = 49 months, \; s_2^2 = 1050\)
8.4 An advertising analyst surveyed how much working men and housewives were exposed to advertisements on radio, TV, newspaper and magazines. The survey item was the number of advertisements that each group encountered in a particular week and the sample mean and standard deviation of each group are as shown in the table below. From these data, can you say that housewives are exposed to more advertisements on average than working men? (Significance level = 0.05)
Working Men: \(\small \,n_1 = 100, \; {\overline x}_1 = 200, \; s_1 = 50\)
Housewives: \(\small \;\;\; n_2 = 144, \; {\overline x}_2 = 225, \; s_2 = 60\)
8.5 One company wants to test whether a female employee uses the phone longer than a male employee. A sample survey of 10 males and 10 females for one-day call time measurement are as follows. Is there a difference in the average call time between male and female? Use the 5% significance level.
Male:\(\quad\) 8 6 4 6 2 2 4 8 10 10
Female: 4 4 10 2 8 4 10 8 13 14 (unit minutes)
8.6 One factory tries to compare the adhesion of motor oil from two companies. Among the products of each company, 32 products were randomly selected and tested as follows: Based on these data, can you conclude that the adhesion means of the two company products are different? (Significance level = 0.05.)
Company A: 13 21 60 35 38 10 36 24 35 35 45 19 42 11 35 39 25 17 51 25 52 25 11 11 55 44 25 41 16 47 50 18
Company B: 46 52 66 65 71 67 47 48 58 42 66 69 60 80 45 47 69 75 43 46 74 73 43 70 51 72 65 45 76 48 56 64
8.7 An industrial psychologist thinks that the big factor that workers change jobs is self-esteem to workers' individual work. The scholar thinks that workers who change jobs frequently (group A) have lower self-esteem than those who do not (group B). The following data are used to measure the score of self-esteem by sampling each group independently.
Group A: 60 45 42 62 68 54 52 55 44 41
Group B: 70 72 74 74 76 91 71 78 78 83 50 52 66 65 53 52
\(\qquad\)Can this data support the psychologist's idea? Assume that scores of the population are normally distributed and that the population variance is not known but the same. (Significance level = 0.01)
8.8 In a business administration department of a university, a debate arose over claims that men have more knowledge of the stock market than women. To calm the dispute, the instructor sampled each of 15 men and women independently and tested them for knowledge of the stock market. The result is as follows.
Women: 73 96 74 55 91 50 46 82 79 79 50 46 81 83
Men: 57 78 42 44 91 65 63 60 97 85 92 42 86 81 64
\(\qquad\)According to the data, on average, can you say that men have more knowledge of the stock market than women? Use the significance level of 0.05. What assumptions do you need?
8.9 An oil company has developed a gasoline additive that will improve the fuel mileage of gasoline. We used 16 pairs of cars to compare the fuel mileage to see if it actually improved. Each pair of cars has the same details as its structure, model, engine size, and other relationship characteristics. When driving the test course using gasoline, one of the pair selected randomly and added additives, the other of the pair was driving the same course using gasoline without additives. The following table shows the km per liter for each of pairs. Is this data a basis for saying that additives increase fuel mileage? Assume that the fuel mileage is normally distributed. Use 5% significance level.
| pair |
Additive \(X_1\) |
No Additive \(X_2\) |
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16 |
17.1
12.7
11.6
15.8
14.0
17.8
14.7
16.3
10.8
14.9
19.7
11.4
11.4
9.3
19.0
10.1 |
16.3
11.6
11.2
14.9
12.8
17.1
13.4
15.4
10.1
13.7
18.3
11.0
10.5
8.7
17.9
9.4 |
(unit: km / liter)
8.10 A study deals with a survey on whether car accidents in a village can be reduced effectively by increasing the number of street lamps. The following table shows the average number of accidents per night, one year before and one year after putting street lamps on 12 locations. Does this data provide evidence that street lamps have reduced nightly car accidents? Use the 5% significance level.
| Location |
Before |
After |
A
B
C
D
E
F
G
H
I
J
K
L |
8
12
5
4
6
3
4
3
2
6
6
9 |
5
3
2
1
4
2
2
4
3
5
4
3 |
8.11 The survey result of (wife’s age, husband’s age) by sampling 16 couples are as follows .
(28, 28) (29, 30) (18, 21) (29, 33) (22, 22) (18, 21) (40, 35) (24, 29)
(21, 31) (20, 24) (20, 34) (23, 25) (33, 39) (33, 35) (40, 29) (39, 40)
\(\qquad\)Test whether the wife’s age is the same as the husband’s age or not. Use the significance level of 0.05.
8.12 One person is considering the use of a test to compare between two population means. 16 samples are randomly taken from two populations and their sample variances are 28.5 and 9.5. Is this data shows evidence that two population variances are the same? (Significance level = 0.05)
8.13 Certain studies have been planned to compare the two relaxing drugs for office workers
in stressful jobs. A medical team sampled eight workers for each of two drugs and collected data
on the strain. Two sample variances are \(s_1^2\) = 2916 and \(s_2^2\) = 4624. Using the
significance level of 0.05, can this data be said to differ in two population variances of tension?
Explain necessary assumptions.
8.14 Let and be the number of days it takes for a plant to sprout its wide leaves and narrow leaves, respectively. The measured data are as follows.
\(\small n_x = 13, \; {\overline x} = 18.97, \; s_x^2 = 9.88 \)
\(\small n_y = 9, \; {\overline y} = 23.20, \; s_y^2 = 4.08 \)
\(\qquad\)If \(X \sim N(\mu_x , \sigma_x^2 ) \) and \(Y \sim N(\mu_y , \sigma_y^2 )\), test the following hypothesis using the 5% significance level.
$$ \small
H_0 : \frac{\sigma_x^2}{\sigma_y^2} = 1, H_1 : \frac{\sigma_x^2}{\sigma_y^2} > 1
$$
8.15 Both tire products are known to have an average life span of 80,000 km. However, there seems to be a difference in the variance. Sixteen tires from each of two companies were randomly selected and run under similar conditions to measure their life span. Sample variances were 4,500 and 2,200, respectively.
1) Test the null hypothesis that variances of the tire life of two products are the same at the significance levels of both 0.10 and 0.05.
2) Obtain 90% and 95% confidence intervals of the ratio \(\frac{\sigma_1^2}{\sigma_2^2}\).
8.16 A carpet manufacturer is looking for materials that can withstand temperature above 250 degree Fahrenheit.
One of two materials is a natural material and the other is a cheap artificial material, which both have
the same properties except for heat-resistant levels. As a result of a heat-resistant experiment by independently
selecting 250 samples from each of two materials, 36 samples from natural materials and 45 samples
from man-made materials failed at temperatures above 250 degrees Fahrenheit. Is there a difference in
the heat resistance of two materials from this data using the significance level of 0.05?
8.17 A labor union of a company found that 63 percent of 150 salespeople who did not receive a college education wanted to take it back even now. The company did a similar study 10 years ago, when only 58 percent of 160 people wanted it. Test the null hypothesis that the desire for college education is not different from 10 years ago using the significance level of 0.05. Samples were selected independently.
8.18 When we extracted 200 companies of the type A and examined them, we found that 12% of them spent more than 1% of their total sales on advertising. The other 200 companies of the type B independently selected and examined, we found 15% of them spent more than 1% of their total sales on advertising. Test the following hypothesis with the significance level of 0.05.
$$ \small
H_0 : p_B \le p_A , H_1 : p_B \gt p_A
$$
8.19 In a company, a study was conducted on the leisure activities of sales staffs and managing staffs. 400 persons were selected independently from each of sales and managing staffs. 288 sales and 260 managing staffs answered that they usually spend their leisure time on sports activities. From this data, can you say that the percentage of two groups for the leisure time spent on sports activities is the same? Use the significance level of 0.05.
8.20 In September 2013, a research institute surveyed 260 men and 263 women about a political issue and
the response result is as follows. Do you think there are significant differences in their way of thinking
on the political issue?' Specify the null hypothesis and the alternative hypothesis and test at the 5%
significance level.
|
Men |
Women |
| Yes |
57% |
65% |
| No |
43% |
35% |
8.21 In order to see whether the unemployment rate in two cities are different, samples of 500
people were randomly selected from two cities and found unemployed people were 35 and 25
respectively. Can you say that the unemployment rate in two cities is different?
Describe the necessary assumptions and calculate the \(p\)-value.