STAT 5368: Review (Basics)
Fall 2011
Please show all the steps and the final interpretation.
1. The mean life span of an African savanna elephant in the wild used to be 60 years. A
recent sample of 15 deceased elephants in the wild yielded a mean life span of 62 years.
Assume that σ = 6 years and normally distributed population. Test if the population mean
life span of African savanna elephants has increased, using α = 0.05.
2. From a telephone survey of 1,124 adults, it was found 71% support lowering property
taxes. Does this poll substantiate the conjecture that less than 75% are in favor of
lowering property taxes with  = .05?
3. A few years ago, noon-time pedestrian traffic past a store had a mean of  = 225. To see
if any change in traffic has occurred, counts were taken for a sample of 15 weekdays. It
was found that x  240 and s = 20.
(a) Construct an  = .05 test of H0 :   225 against the alternative that some change
has occurred.
(b) Obtain a 95% confidence interval for .
(c) Obtain a 99% confidence interval for .
4. A sample of 20 adult female rats is drawn at random from a large number of rats in an
animal house. Among other measurements the uterine weight (in milligrams) is recorded
at postmortem for each rat. The sample yields a mean uterine weight of 21mg. Assume σ
= 1.6 mg and normally distributed population. Using the critical value, test whether the
population mean uterine weight is less than 21.5 mg, using a level of significance α =
0.10.
5. A marine researcher wanted to compare the lengths of adult humpback whales and gray
whales. A random sample of 14 adult humpback whales had an average length of 42.5
feet with a standard deviation of 4 feet. A random sample of 12 adult gray whales had an
average length of 42 feet with a standard deviation of 2 feet. Assume that the whale
lengths are normally distributed.
6. A random sample of seven 2009 sports cars is taken and their “in the city” miles per
gallon is recorded. The results are as follows: 20 19 20 19 16 18 22
Calculate a 95% confidence interval for , the population mean “in the city” mpg for 2009
sports cars.
7. A consumer group is investigating a producer of diet meals to examine if its prepackages
meal actually contains the advertised 6 ounces of protein in each package. The group
collected the following data.
5.1
4.9
4.9
5.5
6.0
5.6
5.1
5.8
5.7
6.0
5.5
6.1
4.9
6.0
5.8
5.2
4.8
4.7
(a) Check this data set for normality.
(b) Construct a 90% confidence interval for the mean weight.
8. Following is the set of all test scores for a class and you may treat it as the population to
answer following questions.
76
72
73
78
76
73
76
76
78
81
90
70
76
76
78
77
64
78
72
72
76
82
74
86
77
80
75
74
82
71
78
63
77
67
73
78
74
78
84
74
77
85
80
73
73
72
78
74
76
79
(a) Draw a random sample of size 10 from the above population.
(i) Find
∑
∑
̅
̅
(ii) Find
(iii) Construct a 95% confidence interval for µ.
(b) Repeat part(a) 20 times and
(i) Find the mean of .
(ii) Find the mean of
.
(iii) Check the confidence intervals to see if the actual mean is inside the interval.
9. A company was wondering which style of pepperoni pizza was most popular. It set up
an experiment where ten people were each given two types of pizza to eat, Type A and
Type B. Each pizza was carefully weighed at exactly 16 oz. After fifteen minutes, the
remainders of the pizza were weighed, and the amount of each type pizza eaten per
person was calculated. It is assumed that the subject would eat more of the type of pizza
he or she preferred. Here are the data:
Subject
1
Pizza A
12.9
Pizza B
16.0
2
5.7
7.5
3
16
16
4
14.3
15.7
5
2.4
13.2
6
1.6
5.4
7
14.6
15.5
8
10.2
11.3
9
4.3
15.4
10
6.6
Test the claim that.
10.6
10. Suppose a sample of 12 students were given a diagnostic test before studying a particular
module and then again after completing the module. We want to find out if, in general,
our teaching leads to improvements in students’ knowledge/skills (i.e. test scores).
a. Conduct an appropriate test.
b. Find a 99% confidence interval for the mean difference.
Student Pre-module Score Post Module Score
1
18
22
2
21
25
3
16
17
4
22
24
5
19
16
6
24
29
7
17
20
8
21
23
9
23
19
10
18
20
11
14
15
12
16
15
11. The Urban Institute in Washington, DC published an article on child-care expenses in
American families. According to the report, child-care expenses for a single-parent
family are less compared to two-parent families. A random sample of 500 single-parent
households had a yearly average child-care expense of $258 with a sample standard
deviation of $55. Another random sample of 400 two-parent households showed the
average yearly child-care expense to be $297 with a sample standard deviation of $50.
Perform a hypothesis test to test the Urban Institutes claim at level of confidence α =
0.10.
12. In a study designed to compare two new drugs A and B, 150 patients were treated with
drug A and 200 patients with drub B and the following results were obtained.
Drug A Drug B
Cured
85
82
Not cured
65
118
Total
150
200
(a) Do these results demonstrate a significantly higher cure rate with drug A than drug B
Test at  = .01
(b) Construct a 95% confidence interval for the difference in the cure rates of the two drugs.
13. Two work designs are being considered for possible adoption in an assembly plant. A
time study is conducted with 13 workers using design 1 and 10 workers using design 2.
The results of their assembly times (in minutes) are
Design 1
Design 2
x1  74.3
x 2 = 81.6
s1  4.8
s 2  6.5
14. There is an old folk belief that the sex of a baby can be guessed before birth on the basis
of heart rate. Supposedly, male babies have heart rates that are significantly faster than
those of female babies. Fetal heart rates (bpm) were observed for 20 male babies and 23
female babies admitted to a maternity ward with following results.
Female
Male
x1  120.37
x 2 = 135.59
s1  63.0
s 2  62.0
Is the claim that male babies have higher heart rate true?
Is the mean assembly time significantly higher for design 2?
15. Let X , X ,..., X be a random sample from
a. Show that ̅ is an unbiased estimator of .
1
2
n
b.
Show that
c.
Find E{
∑
∑
̅
.
̅
is an unbiased estimator of