Chapter 9 Examples
1. A random sample of 36 GE light bulbs lasted on average 418 hours and had a standard
deviation of 24 hours. A random sample of 49 Sylvania light bulbs lasted on average 402
hours and had a standard deviation of 21 hours. Construct a 95% confidence interval for
the difference between the mean lifetimes of the two kinds of light bulbs. Interpret the
interval. Can you conclude that one brand is usually better than the other?
2. A study has been made to compare the nicotine contents of two brands of cigarettes. 11
Virginia Slims cigarettes had an average nicotine content of 3.1 milligrams with a
standard deviation of 0.4 milligrams, while 9 Jacks cigarettes had an average nicotine
content of 8.1 milligrams with a standard deviation of 0.5 milligrams. Assuming that the
two sets of data are independent random samples from normal populations with equal
variances, construct a 90% confidence interval for the difference between the mean
nicotine contents of the two brands of cigarettes. Interpret the interval. Can you
conclude that one brand contains more nicotine content than the other?
3. In the comparison of two kinds of paint, a consumer testing service finds that 4 1-gallon
cans of Benjamin Moore paint cover on the average 540 square feet with a standard
deviation of 30 feet, whereas 4 1-gallon cans of Pittsburgh paint cover on the average 490
square feet with a standard deviation of 25 feet. Assuming that the two populations
sampled are normal and have equal variances, test to see whether or not the Benjamin
Moore paints cover a larger area on average, at the 1% significance level, and compute
the observed significance level of the test.
4. To find out whether the inhabitants of two South Pacific islands may be regarded as
having the same racial ancestry, an anthropologist determines the cephalic indices of 64
adult males from each island. The measurements from Tonga yielded an average of 77
cm with a standard deviation of 16 cm, while the measurements from Kiribati yielded an
average of 72 cm with a standard deviation of 8 cm. Test whether or not the two islands
have the same racial ancestry, at the 5% significance level, and compute the p-value.
5. To determine the effectiveness of a new traffic-control system, the number of accidents
that occurred at 12 dangerous intersections during four weeks before and four weeks after
the installation of the new system were observed, and the following data were observed:
Before
3
5
2
3
3
3
After
1
2
0
2
2
0
Difference
2
3
2
1
1
3
Before
0
4
1
6
4
1
After
2
3
3
4
1
0
Difference
-2
1
-2
2
3
1
Chapter 9 Examples
The mean difference from this data is 1.25 with a standard deviation of 1.71.
a. Construct an 80% confidence interval for the true mean difference. Interpret the
interval.
b. Test at the 2% significance level whether or not the new traffic-control system has
any effect on the number of accidents that occur and compute the p-value.
6. A pharmaceutical company wants to test whether a new medication reduces Medicare
beneficiaries’ systolic heart rate. 64 senior citizens have their systolic blood pressure
measured and then after taking the new medication have their systolic blood pressure
measured again, after the medication has a suitable amount of time to take effect. The
mean difference of the reduction in systolic blood pressure in the sample is found to be
11.2 beats per minute with a standard deviation of 4 beats per minute.
a. Construct a 95% confidence interval for the true mean difference of reduction in
systolic blood pressure.
b. Test at the 5% significance level whether or not the true mean difference of
reduction in systolic blood pressure is greater than 10 beats per minute and
compute the p-value.
7. If 132 of 200 male voters and 90 of 150 female voters favor a certain candidate running
for governor of Illinois, do the following.
a. Check to see whether the normal approximation of the sampling distribution is
appropriate.
b. Construct a 99% confidence interval for the difference between the true
proportions of male and female voters who favor the candidate. Interpret the
interval. Can you conclude that one sex favors the candidate more?
c. Test at the 10% significance level whether the true proportion of male voters is
less than the true proportion of female voters favoring the candidate, and compute
the p-value.
8. A high school counselor wants to estimate the difference in mean income per day
between high school graduates who have a college education and those who have not
gone on to college. Suppose it is decided to compare the daily incomes of 30-year-olds,
and the range of daily incomes for both groups is approximately $200 per day. How
many people from each group should be sampled in order to estimate the true difference
between mean daily incomes correct to within $10 per day with 90% confidence?
9. Suppose you want to estimate the difference in proportions of men and women who say
they have used or expect to use televised home shopping using a 95% confidence interval
with a width of .06 or less. Approximately how many people should be included in your
samples?