Selection of Appropriate Statistical Techniques:
18 Problems
For each of the following problems, choose the appropriate statistical
techinique to use. Identify the technique by one of the 9 numbers
indicated below.
Choices of statistical methods/techniques
1. Test of an hypothesis about a population mean µ, assumed to
be known or n large, using a z-statistic.
2. Test of an hypothesis about a population mean µ, not known,
n small, using a t-statistic.
3. Test about two population means µ1 and µ2:
large samples, using a z-statistic.
4. Test about two population means µ1 and µ2:
small samples, using a t-statistic.
5. Test about the difference of two means in a paired comparison study
using a t-statistic.
6. Test about a population proportion p using a z-statistic.
7. Test of the equality of two population proportions p1
and p2 using a z-statistic.
8. Description of categorical variables: marginal and conditional distributions
(or chi-square test)
9. Finding the best fitting line to a set of data with a quantitative
explanatory variable x and a quantitative response variable y and examining
the slope of the regression line (or testing whether the slope of a line
is zero).
1. Is there a relationship between the number of votes received
by candidates for public office and the amount spent on their campaign?
The following gives sample information on 4 candidates in a recent election:
|
Candidate
|
Amount spent ($)
|
Votes received
|
|
Weber
|
30,000
|
14000
|
|
Taite
|
40,000
|
7000
|
|
Spencer
|
20,000
|
5000
|
|
Lopez
|
50,000
|
12000
|
Test to see if there is a significant linear relationship between amount
spent and the number of votes received.
Statistical test to use:_________
2. The following data are the weight gains, measured in pounds, of
babies from birth to age 6 months. 13 babies were randomly split
into two groups: 7 breast-fed babies who were breast-fed and 6 babies who
were formula-fed. The results:
| Breast-fed babies |
7
|
8
|
6
|
10
|
9
|
8
|
9
|
| Formula-fed babies |
9
|
10
|
8
|
6
|
7
|
8
|
|
Is there evidence that the weight gains are different among the two
groups?
Statistical test to use:_________
3. Are the proportions of males and females who like women's basketball
the same or not? 400 randomly selected males and 400 randomly selected
females were asked about this. It was found that 42% of the males said
they liked women’s basketball, compared to 47% of the females. Test the
hypothesis that the two proportions are the same against the alternative
that they are not equal.
Statistical test to use:_________
4. A manufacturer of women's apparel is interested in determining
if age is a factor in whether women believe they would buy a particular
garment. Accordingly, the firm surveyed 140 women in each of three age
groups and asked each person to rate the garment as either excellent, average,
or poor. The results follow. Test the hypothesis at the .05 level that
the ratings are the same in each age group.
| |
Rating
|
|
|
Age
|
Excellent |
Average |
Poor |
Total |
|
15-24
|
44
|
53
|
43
|
140
|
|
25-39
|
47
|
74
|
19
|
140
|
|
40-55
|
46
|
57
|
37
|
140
|
|
Total
|
137
|
184
|
99
|
420
|
Statistical test to use:_________
5. To determine if a new gas additive improves the mileage performance
of gasoline, seven test runs were conducted with the additive and six runs
were made without it. The test results appear below. Is there sufficient
evidence to conclude that the additive increases gasoline mileage?
| With additive |
32.6 |
30.1 |
29.8 |
34.6 |
33.5 |
29.6 |
33.8 |
| Without additive |
31.3 |
29.7 |
29.1 |
30.3 |
30.9 |
29.9 |
|
Statistical test to use:_________
6. Six junior executives were sent to a class to improve their verbal
skills. In order to test the quality of the program, the executives were
tested before and after taking the class, with the following results:
|
Name of executive
|
| |
Levin |
Baker |
Craft |
Denny |
Lonny |
Larry |
| Before |
18 |
30 |
8 |
10 |
12 |
12 |
| After |
30 |
70 |
20 |
4 |
10 |
20 |
Do these results indicate a significant improvement in verbal skills?
Statistical test to use:_________
7. College financial aid offices expect students to use summer earnings
to help pay for college. But how large are these earnings? One college
studied this question by asking a sample of 675 males and 621 females how
much they earned in the summer. For the sample of males, the mean and standard
deviation were $1884.52 and $1386.37, respectively. For the females the
mean and standard deviation were $1360.39 and $1037.46. Are the means the
same for males and females?
Statistical test to use:_________
8. A drug company tests the effectiveness of a new drug for relieving
anxiety among young adults, compared with a placebo. The new drug is administered
to 200 young adults and 120 experienced relief. A placebo is administered
to 100 young adults and 40 said their anxiety was relieved. Is there sufficient
evidence to conclude that the new drug is better than the placebo?
Statistical test to use:_________
9. The average height of all 18-24 females in the U.S. is 65.5" with
a standard deviation of 2.5". A researcher, studying Native Americans (Indians)
suspects that the average height of this population of people is less than
that of all females in the U.S. She obtains a simple random sample of 25
Native Americans and finds that the average in the sample is 62.8". Is
the difference statistically significant?
Statistical test to use:_________
10. A bank compares two proposals to increase the amount that its
credit card customers charge on their cards. Proposal A offers to eliminate
the annual fee for customers who charge $2400 or more during the year.
Proposal B offers a small percentage of the total amount charged as a cash
rebate at the end of the year. The bank offers each proposal to a sample
of 150 of its existing credit card customers. At the end of the year, the
total amount charged by each customer is recorded. Here are the results:
|
Group
|
n
|
Mean
|
Std Dev
|
|
A
|
150
|
$1987
|
$392
|
|
B
|
150
|
$2056
|
$413
|
Is there convincing evidence that one of the two proposals is better
than the other?
Statistical test to use:_________
11. The average IQ on the Stanford-Binet IQ test is 100 with a standard
deviation of 16. An epidemiologist wishes to see if the average IQ of Cuban
immigrants over the past 5 years is the same or whether it is lower. Test
the hypothesis that the immigrants average IQ is 100 versus the alternative
that it is lower. The results of a simple random sample of size 64 yielded
an average IQ of 98.5.
Statistical test to use:_________
12. After a frost, the owner of 4 orange groves randomly sampled
100 trees from each grove to assess the proportion of trees in each grove
that had been damaged. The results were as follows:
|
Grove Number
|
1
|
2
|
3
|
4
|
|
# Trees Damaged
|
38
|
45
|
32
|
25
|
Is there sufficient evidence to reject the hypothesis that the proportion
of trees damaged in the four groves is the same?
Statistical test to use:_________
13. In order to determine whether a pro-choice leader's claim that
70% of all women support a woman's right to have an abortion is true or
not, a simple random sample of 1200 adult women was obtained. It was found
that 64% of these women believe that a woman does have this right. Is there
sufficient evidence to reject the pro-choice leader's claim?
Statistical test to use:_________
14. The university library is interested in determining whether the
average number of books checked out per visit has increased. In the past,
the average was 3 books per student visit. A random sample of 10 students
revealed an average of 4.1 books with a standard deviation of 2.07 books.
Does this provide sufficient evidence to show that the average has increased?
Statistical test to use:_________
15. A stock market advisory service bases its weekly recommendations
on the percentage of financial analysts who are bullish on stocks. For
its next newsletter, the advisory service surveys 150 randomly chosen analysts
and finds that 60 (or 40%) say yes, they are bullish. Do these results
provide sufficient evidence to conclude that the percentage of bullish
analysts has changed from the previous week's value of 35 percent?
Statistical test to use:_________
16. A home builder claims that the addition of a heat pump will reduce
electric bills in all-electric homes. To support this claim he obtains
the electricity bills for 7 customers for the month of January for two
consecutive years, one before the heat pump was installed and one after.
Is there sufficient evidence to show that heating bills were reduced?
| |
CUSTOMER
|
| |
1
|
2
|
3
|
4
|
5
|
6
|
7
|
| |
Garcia
|
Huffman
|
Johnson
|
Palmer
|
Kerby
|
Beard
|
Sawyer
|
|
Before
|
180
|
156
|
188
|
132
|
208
|
196
|
190
|
|
After
|
160
|
164
|
172
|
130
|
200
|
190
|
184
|
Statistical test to use:_________
17. Is there a relationship between the length of stay of surgical
patients in a hospital and age of the patient? The following data were
obtained in a recent study:
|
Age of Patient
|
40
|
36
|
30
|
27
|
24
|
22
|
20
|
|
Days in Hospital
|
11
|
9
|
10
|
5
|
12
|
4
|
7
|
Test the hypothesis that there is no relationship between age and days
in hospital.
Statistical test to use:_________
18. Your new car has an EPA rating of 26.0 miles per gallon. You
wonder if this is right for your car. You record the mpg for 5 "fill-ups"
and obtain the following results: 21.4, 25.0, 26.8, 23.6, and 24.0. For
this sample, the average is 24.16 mpg and the standard deviation is 1.98
mpg. Test the hypothesis that the average for your car is 26.0 versus the
alternative that it is less than 26.0.
Statistical test to use:_________
This exercise was submitted by William L. Harkness: wlh@stat.psu.edu