Section 7–1 Confidence Intervals for the Mean When σ Is Known
379
3. What type of data collection technique would you use?
4. Assume you found out that from your sample of 85 people, an average of about 57 tissues
are used throughout the duration of a cold, with a population standard deviation of 15. Use a
confidence interval to help you decide how many tissues will go in the boxes.
5. Explain how you decided how many tissues will go in the boxes.
See page 411 for the answers.
Exercises 7–1
1. What is the difference between a point estimate and an
interval estimate of a parameter? Which is better? Why?
2. What information is necessary to calculate a confidence
interval?
3. What is the margin of error?
4. What is meant by the 95% confidence interval of the mean?
5. What are three properties of a good estimator?
6. What statistic best estimates 𝜇?
7. Find each.
a. zα⧸2 for the 99% confidence interval
b. zα⧸2 for the 98% confidence interval
c. zα⧸2 for the 95% confidence interval
d. zα⧸2 for the 90% confidence interval
e. zα⧸2 for the 94% confidence interval
8. What is necessary to determine the sample size?
9. Fuel Efficiency of Cars and Trucks Since 1975 the average fuel efficiency of U.S. cars and light trucks (SUVs)
has increased from 13.5 to 25.8 mpg, an increase of over
90%! A random sample of 40 cars from a large community got a mean mileage of 28.1 mpg per vehicle. The population standard deviation is 4.7 mpg. Estimate the true
mean gas mileage with 95% confidence.
Source: World Almanac 2012.
10. Fast-Food Bills for Drive-Thru Customers A random
sample of 50 cars in the drive-thru of a popular fast food
restaurant revealed an average bill of $18.21 per car.
The population standard deviation is $5.92. Estimate the
mean bill for all cars from the drive-thru with 98% confidence.
11. Overweight Men For a random sample of 60 overweight men, the mean of the number of pounds that they
were overweight was 30. The standard deviation of the
population is 4.2 pounds.
a. Find the best point estimate of the average number of
excess pounds that they weighed.
b. Find the 95% confidence interval of the mean of
these pounds.
c. Find the 99% confidence interval of these pounds.
d. Which interval is larger? Why?
12. Number of Jobs A sociologist found that in a random
sample of 50 retired men, the average number of jobs
they had during their lifetimes was 7.2. The population
standard deviation is 2.1.
a. Find the best point estimate of the population
mean.
b. Find the 95% confidence interval of the mean number of jobs.
c. Find the 99% confidence interval of the mean number of jobs.
d. Which is smaller? Explain why.
13. Number of Faculty The numbers of faculty at
32 randomly selected state-controlled colleges and
universities with enrollment under 12,000 students
are shown below. Use these data to estimate the mean
number of faculty at all state-controlled colleges and
universities with enrollment under 12,000 with 92%
confidence. Assume σ = 165.1.
211
621
318
471
384
367
836
638
396
408
203
425
211
515
374
159
224
280
224
324
337
289
121
395
180
412
121
431
134
356
176
539
Source: World Almanac.
14. Freshmen GPAs First-semester GPAs for a random
selection of freshmen at a large university are shown.
Estimate the true mean GPA of the freshman class with
99% confidence. Assume σ = 0.62.
1.9
2.8
2.5
3.1
2.0
2.1
3.2
3.0
2.7
2.7
2.8
2.4
2.0
3.8
2.8
3.5
1.9
3.0
2.9
2.7
3.2
3.8
4.0
3.4
2.7
2.0
3.0
3.9
2.2
2.9
3.3
1.9
3.8
2.7
2.8
2.1
15. Carbohydrate Grams in Commercial Subs The number of grams of carbohydrates in various commercially
prepared 7-inch subs is recorded below. The population
7–11
380
Chapter 7 Confidence Intervals and Sample Size
standard deviation is 6.46. Estimate the mean number of carbs in all similarly sized subs with 95%
confidence.
63
55
70
53
67
60
65
65
61
55
49
68
64
57
51
63
51
60
61
48
42
60
54
54
56
66
50
56
70
55
55
57
ing a sample mean of 190.7 days and the population
standard deviation of 54.2 days. Estimate for all U.S.
cities the true mean of the growing season with 95%
confidence.
61
58
56
Source: The Old Farmer’s Almanac.
16. Number of Farms A random sample of the number of
farms (in thousands) in various states follows. Estimate
the mean number of farms per state with 90% confidence. Assume σ = 31.0.
47
8
68
29
95
90
7
54
3
15
33
49
21
64
4
4 44
52
6
8
79
78
57
80
109
9
48
40
80
16
50
Source: New York Times Almanac.
17. Gasoline Use A random sample of 36 drivers used on
average 749 gallons of gasoline per year. If the standard deviation of the population is 32 gallons, find the
95% confidence interval of the mean for all drivers. If a
driver said that he used 803 gallons per year, would you
believe that?
18. Day Care Tuition A random sample of 50 four-yearolds attending day care centers provided a yearly tuition
average of $3987 and the population standard deviation
of $630. Find the 90% confidence interval of the true
mean. If a day care center were starting up and wanted
to keep tuition low, what would be a reasonable amount
to charge?
19. Hospital Noise Levels Noise levels at various area
urban hospitals were measured in decibels. The mean
of the noise levels in 84 randomly selected corridors
was 61.2 decibels, and the standard deviation of the
population was 7.9. Find the 95% confidence interval
of the true mean.
Source: M. Bayo, A. Garcia, and A. Garcia, “Noise Levels in an Urban
Hospital and Workers’ Subjective Responses,” Archives of Environmental
Health 50(3): 247-51 (May-June 1995).
20. Length of Growing Seasons The growing seasons for
a random sample of 35 U.S. cities were recorded, yield-
Technology
TI-84 Plus
Step by Step
21. Christmas Presents How large a sample is needed to
estimate the population mean for the amount of money a
person spends on Christmas presents within $2 and be 95%
confident? The standard deviation of the population
is $7.50.
22. Hospital Noise Levels In the hospital study cited in
Exercise 19, the mean noise level in 171 randomly selected ward areas was 58.0 decibels, and the population
standard deviation was 4.8. Find the 90% confidence
interval of the true mean.
Source: M. Bayo, A. Garcia, and A. Garcia, “Noise Levels in an Urban Hospital and Workers’ Subjective Responses,” Archives of Environmental Health
50(3): 247-51 (May-June 1995).
23. Internet Viewing A researcher wishes to estimate
the average number of minutes per day a person
spends on the Internet. How large a sample must
she select if she wishes to be 90% confident that the
population mean is within 10 minutes of the sample
mean? Assume the population standard deviation is
42 minutes.
24. Cost of Pizzas A pizza shop owner wishes to find the
95% confidence interval of the true mean cost of a large
cheese pizza. How large should the sample be if she
wishes to be accurate to within $0.15? A previous study
showed that the standard deviation of the price was
$0.26.
25. Water Temperature If the variance of the water
temperature in a lake is 28°, how many days should
the researcher select to measure the temperature to
estimate the true mean within 3° with 99% confidence?
26. Undergraduate GPAs It is desired to estimate the
mean GPA of each undergraduate class at a large university. How large a sample is necessary to estimate
the GPA within 0.25 at the 99% confidence level? The
population standard deviation is 1.2.
Step by Step
Finding a z Confidence Interval for the Mean (Data)
1. Enter the data into L1.
2. Press STAT and move the cursor to TESTS.
3. Press 7 for ZInterval.
4. Move the cursor to Data and press ENTER.
5. Type in the appropriate values.
6. Move the cursor to Calculate and press ENTER.
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