Question 1: It is known that the average height of a man residing in the Canada is 174 cm and the standard deviation is 8 cm. Assume the heights are normally distributed. To test the hypothesis that men in your city are "average," a sample of 20 men have been chosen. The heights of the men in the sample follow:
|
Man |
Height in Centimetres |
Man |
| 1 |
180.1 |
175.9 |
11 |
| 2 |
169.8 |
190.6 |
12 |
| 3 |
172.7 |
181.4 |
13 |
| 4 |
182.2 |
185.4 |
14 |
| 5 |
178 |
179.6 |
15 |
| 6 |
180.2 |
183.7 |
16 |
| 7 |
176.8 |
199.6 |
17 |
| 8 |
185.3 |
186.4 |
18 |
| 9 |
164.4 |
201 |
19 |
| 10 |
175.8 |
203.1 |
20 |
a) Test the hypothesis at the significance level of α1 = 0.05
b) Test the hypothesis at the significance level of α2 = 0.01.
c) Find the p-value of the test, and use the p-value to verify your answers.
Question 2: Twenty years ago, entering male high school students of Central High could do an average of 24 pushups in 60 seconds. To see whether this remains true today, a random sample of 36 freshmen was chosen. If their sample mean was 22.5 with a sample standard deviation of 3.1, can we conclude that the mean is no longer equal to 24? Use the 5 percent level of significance.
Question 3: Below are the points scored by the winning team in Week 10 of NFL professional football games this Fall, 2011.
24 17 44 13 21 24 30 20 37 26 17 22 27 37 37 45
a) Use these values to test the hypothesis that the average number of points scored by winning teams is less than or equal to 28. Use the 10 percent level of significance.
b) Find the p-value of the test, and use the p-value to verify your answer.
c) Repeat a) and b) with the points scored by all the winning teams in Week 12 of NFL games.
You can get the necessary data by going to https://www.nfl.com/scores/2011/REG12 after the last game on Monday, Nov 28, 2011.
Question 4: The manufacturer of a type of PVC pipe claims that the impact value for the pipe has a standard deviation of 0.1. An Izod impact test was performed on 20 specimens of the PVC pipe. The sample mean of the measured impact value is 1.25 and the sample standard deviation is 0.0649.
a) Give the appropriate choice of the null hypothesis. Mention any assumptions that made about the underlying distribution of the data.
b) Test the null hypothesis. Use the 5 percent level of significance.
c) Find the p-value of the test and comment on the appropriate choice of a significance level for this problem.
Question 5: An article in the Journal of Environmental Engineering (Vol. 115, No. 3,1989, pp. 608-619) reported the results of a study on the occurrence of sodium and chloride in surface streams in central Rhode Island. The following data are chloride concentration y (in milligrams per liter) and roadway area in the watershed x (in percentage).
|
xi
|
yi
|
xi
|
yi
|
xi
|
yi
|
|
.19
|
4.4
|
.47
|
11.8
|
.69
|
19.2
|
|
.15
|
6.6
|
.70
|
12.1
|
1.30
|
23.1
|
|
.57
|
9.7
|
.60
|
14.3
|
1.05
|
27.4
|
|
.70
|
10.6
|
.78
|
14.7
|
1.06
|
27.7
|
|
.67
|
10.8
|
.81
|
15.0
|
1.74
|
31.8
|
|
.63
|
10.9
|
.78
|
17.3
|
1.62
|
39.5
|
a) Draw a scatter diagram of the data. Does a simple linear regression model seem appropriate here?
b) Fit the simple linear regression model using the method of least squares.
c) Find an (unbiased) estimate of σ2.
d) Estimate the mean chloride concentration for a watershed that has 1% roadway area.
Question 6: The following data give the average price of all books reviewed in the journal Science in the years from 1990 to 1996.
|
Year
|
1990
|
1991
|
1992
|
1993
|
1994
|
1995
|
1996
|
|
Price (Dollars)
|
54.43
|
54.08
|
57.58
|
51.21
|
59.96
|
60.52
|
62.13
|
Give an interval that, with 95 percent confidence, will contain the average price of all books reviewed in Science in 1998. Hint: use a prediction interval.
Question 7: The following data relating cigarette smoking and death rates for lung cancers in 14 states in the USA. The data are based in part on records concerning 1960 cigarette tax receipts.
|
Cigarettes per Person
|
Lung Cancer Deaths per Year per 100K People
|
|
2618
|
20.3
|
|
2,860
|
22.07
|
|
2,010
|
13.58
|
|
2,791
|
22.8
|
|
2212
|
16.59
|
|
2184
|
16.84
|
|
2344
|
17.71
|
|
2692
|
22.04
|
|
2206
|
14.2
|
|
2914
|
25.02
|
|
3034
|
25.88
|
|
4240
|
23.03
|
|
1400
|
12.01
|
|
2257
|
20.74
|
a) Draw a scatter diagram relating cigarette use and death rates from lung cancer.
b) Estimate the regression parameters α and β.
c) Test at the .05 level of significance the hypothesis that cigarette consumption does not affect the death rate from lung cancer.
d) What is the p-value of the test in part c)?