Download Homework 3 due Monday, Feb. 13

STAT412
Homework 3
Due Monday, Feb. 13
1. Listed below are the lengths (in minutes) of randomly selected CDs of country, rock, and movie soundtracks. Construct
a 97% confidence interval for the mean length of all such CDs. Construct a 97% confidence interval for the mean length
of all such CDs. Use MINITAB to find mean and standard deviation.
62.28
50.16
75.12
50.97
56.13
52.15
32.04
52.07
54.03
53.15
32.57
43.16
20.37
47.42
60.41
38.48
56.13
49.19
32.07
73.23
54.08
51.15
37.06
36.26
29.01
49.63
62.06
39.57
58.02
52.00
39.29
69.71
74.14
61.57
33.15
49.22
54.08
57.06
51.37
48.19
2. A 99% confidence interval (in inches) for the mean height of a population is 65.89 < < 67.51. This result is based on a
sample of size 144. If the confidence interval 66.11< < 67.29 is obtained from the same sample data, what is the degree
of confidence?
Scores on a certain test are normally distributed and a researcher wishes to estimate the mean score achieved by all
students on the test. He was told that he would need a sample size of 112 to assure a 98 percent confidence interval with
width=4. What is the sample size required if he is interested in constructing a 95% confidence interval with width=4?
Problems 4-6 are to be done using traditional and p-value approach to hypothesis testing. Show all of the steps as
illustrated in class.
4. Humerus bones from the same species of animal tend to have approximately the same length-to-width ratios. When
fossils of humerus bones are discovered, archaeologists can often determine the species of animal by examining the
length-to-width ratios of the bones. It is known that Species A exhibits a mean ratio of 8.5. Suppose 41 fossils of humerus
bones were unearthed at an archaeological site in East Africa, where species A is believed to have lived. (Assume that the
unearthed bones were all from the same unknown species.) The length-to-width ratios of the bones were calculated and
listed, as shown in the table.
10.73
8.48
8.52
8.91
8.93
9.38
8.89
8.71
8.87
11.77
8.80
9.07
9.57
6.23
10.48
10.02
9.20
9.29
9.41
10.39
8.38
10.33
9.94
6.66
9.39
11.67
9.98
8.07
9.35
9.17
8.30
9.84
8.37
8.86
9.89
9.17
9.59
6.85
9.93
8.17
12.00
Test whether the population mean ratio of all bones of this particular species differs from 8.5. Use =.01.
5. The length of stay (in days) from 100 randomly selected hospital patients are presented in the table below. Suppose we
want to test the hypothesis that the population mean length of stay at the hospital is less than 5 days. Conduct hypothesis
using =.05.
2
3
8
6
4
4
6
4
2
5
8
10
4
4
4
2
1
3
2
10
1
3
2
3
4
3
5
2
4
1
2
9
1
7
17
9
9
9
4
4
1
1
1
3
1
6
3
3
2
5
1
3
3
14
2
3
9
6
6
3
5
1
4
6
11
22
1
9
6
5
2
2
5
4
3
6
1
5
1
6
17
1
2
4
5
4
4
3
2
3
3
5
2
3
3
2
10
2
4
2
6. A random sample of 35 standard metropolitan statistical areas (SMSAs) was selected and the ratio (per 1,000) of
registered voters to the total number of persons 18 years and over was recorded in each area. Use the sample data given to
test the research hypothesis that the population average ratio is greater than 675, last year’s average ratio. Use =.025.
802
751
807
641
694
497
730
747
848
854
653
635
728
672
674
600
605
561
740
683
729
760
696
818
695
812
681
710
725
803
743
811
735
646
632
7. You have been asked by a researcher to help design an experiment to test the hypothesis H o:  = 200 versus
Ha:  < 200. The population standard deviation, , is known to be 16. The researcher wants to use =.025. The
researcher plans to use a sample size of n = 100. The researcher wants to calculate the probability of committing a type-2
error if = 195.
a. Fill in the blanks below to describe the sampling distribution of x assuming Ho is true.
Mean (  x ) =
Standard deviation ( x ) =
Shape:
Sketch the sampling distribution of x assuming Ho is true.
b.
Specify the rejection region when x is used as the test statistic. Locate the rejection region on your graph from
part a.
c. Describe the sampling distribution of x if = 195.
Mean (  x ) =
Standard deviation ( x ) =
Shape:
On your graph of part a, sketch the sampling distribution of x if =195.
d. Find the probability of failing to reject Ho if  is actually equal to 195. That is, find the probability of making a
Type II error if = 195. Shade the area corresponding to this probability on your graph.