For Exercises 6.23 to 6.25, see page 304; for 6.26 and 6.27, see pages 305–306; for 6.28 to 6.30, see page 309; for 6.31 and 6.32, see page 311; and for 6.33, see page 313.
6.34 Margin of error and the confidence interval.
A study based on a sample of size 30 reported a mean of 82 with a margin of error of 7 for 95% confidence.
6.35 Change the sample size.
Consider the setting of the previous exercise. Suppose that the sample mean is again 82 and the population standard deviation is 7. Make a diagram similar to Figure 6.10 (page 309) that illustrates the effect of sample size on the width of a 95% interval. Use the following sample sizes: 10, 20, 40, and 80. Summarize what the diagram shows.
6.35
As the sample size increases, the width of the interval decreases.
6.36 Change the confidence.
Consider the setting of the previous two exercises. Suppose that the sample mean is still 82, the sample size is 30, and the population standard deviation is 7. Make a diagram similar to Figure 6.11 (page 310) that illustrates the effect of the confidence level on the width of the interval. Use 80%, 90%, 95%, and 99%. Summarize what the diagram shows.
6.37 Populations sampled and margins of error.
Consider the following two scenarios. (A) Take a simple random sample of 100 sophomore students at your college or university. (B) Take a simple random sample of 100 sophomore students in your major at your college or university. For each of these samples you will record the amount spent on textbooks used for classes during the fall semester. Which sample should have the smaller margin of error for 95% confidence? Explain your answer.
6.37
They will have the same margin of error because the sample sizes are the same, .
6.38 Reporting margins of error.
A U.S. News & World Report article of July 17, 2014, reported Commerce Department estimates of changes in the construction industry:
Construction fell 9.3 percent last month to a seasonally adjusted annual rate of 893,000 homes, the Commerce Department said Thursday.
If we turn to the original Commerce Department report (released on July 17, 2014), we read:
Privately-owned housing starts in June were at a seasonally adjusted annual rate of 893,000. This is 9.3 percent (10.3%) below the revised May estimate of 985,000.
6.39 Confidence interval mistakes and misunderstandings.
Suppose that 500 randomly selected alumni of the University of Okoboji were asked to rate the university’s academic advising services on a 1 to 10 scale. The sample mean was found to be 8.6. Assume that the population standard deviation is known to be .
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6.39
(a) She forgot to divide the standard deviation by . (b) Inference is about the population mean, not the sample mean. (c) Confidence does not mean probability; furthermore, making probability statements about doesn’t make sense because it’s fixed, not random. (d) The central limit theorem guarantees that the sample mean will be Normally distributed, not the original values. “… the sample mean of alumni ratings will be approximately Normal.”
6.40 More confidence interval mistakes and misunderstandings.
Suppose that 100 randomly selected members of the Karaoke Channel were asked how much time they typically spend on the site during the week.11 The sample mean was found to be 3.8 hours. Assume that the population standard deviation is known to be .
6.41 In the extremes.
As suggested in our discussions, 90%, 95%, and 99% are probably the most common confidence levels chosen in practice.
6.41
(a) . This is useless because it gives us no information about what is. (b) . The chance that is exactly is has 0 probability, so our confidence is 0%.
6.42 Average starting salary.
The University of Texas at Austin McCombs School of Business performs and reports an annual survey of starting salaries for recent bachelor’s in business administration graduates.12 For 2013, there were a total of 430 respondents.
6.43 Survey response and margin of error.
Suppose that a business conducts a marketing survey. As is often done, the survey is conducted by telephone. As it turns out, the business was only able to illicit responses from less than 10% of the randomly chosen customers. The low response rate is attributable to many factors, including caller ID screening. Undaunted, the marketing manager was pleased with the sample results because the margin of error was quite small, and thus the manager felt that the business had a good sense of the customers’ perceptions on various issues. Do you think the small margin of error is a good measure of the accuracy of the survey’s results? Explain.
6.43
Because there is nonresponse, the accuracy is in question regardless of the small margin of error. There is no guarantee the respondents are similar to the nonrespondents.
6.44 Fuel efficiency.
Computers in some vehicles calculate various quantities related to performance. One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon (mpg). For one vehicle equipped in this way, the car was set to 60 miles per hour by cruise control, and the mpg were recorded at random times.13 Here are the mpg values from the experiment:
mileage
| 37.2 | 21.0 | 17.4 | 24.9 | 27.0 | 36.9 | 38.8 | 35.3 | 32.3 | 23.9 |
| 19.0 | 26.1 | 25.8 | 41.4 | 34.4 | 32.5 | 25.3 | 26.5 | 28.2 | 22.1 |
Suppose that the standard deviation of the population of mpg readings of this vehicle is known to be .
6.45 Fuel efficiency in metric units.
In the previous exercise, you found an estimate with a margin of error for the average miles per gallon. Convert your estimate and margin of error to the metric units kilometers per liter (kpl). To change mpg to kpl, use the fact that and .
6.45
; the margin of error is 1.21.
6.46 Confidence intervals for average annual income.
Based on a 2012 survey, the National Statistics Office of the Republic of the Philippines released a report on various estimates related to family income and expenditures in Philippine pesos. With respect to annual family income, we would find the following reported:14
| Estimate | Standard error |
Lower | Upper | |
|---|---|---|---|---|
| Average annual income |
234,615 | 3,235 | ? | 240,958 |
The “Lower” and “Upper” headers signify lower and upper confidence interval limits. As will be noted in Chapter 7, the “standard error” for estimating the mean is . But because the sample sizes of the national survey are large, is approximately equal to the population standard deviation.
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6.47 What is the cost?
In Exercise 6.44, you found an estimate with a margin of error for the fuel efficiency expressed in miles per gallon. Suppose that fuel costs $3.80 per gallon. Find the estimate and margin of error for fuel efficiency in terms of miles per dollar. To convert miles per gallon to miles per dollar, divide miles per gallon by the cost in dollars per gallon.
6.47
; the margin of error is 0.75.
6.48 More than one confidence interval.
As we prepare to take a sample and compute a 95% confidence interval, we know that the probability that the interval we compute will cover the parameter is 0.95. That’s the meaning of 95% confidence. If we plan to use several such intervals, however, our confidence that all of them will give correct results is less than 95%. Suppose that we plan to take independent samples each month for five months and report a 95% confidence interval for each set of data.
6.49 Satisfied with your job?
The Gallup-Healthways Well-Being Index is a single metric on a 0 to 100 percentage scale based on six domains of well-being, including life evaluation, emotional health, work environment, physical health, healthy behaviors, and basic access. In 2013, the estimate for the index on the national level is 66.2. Material provided with the results of the poll noted:
Interviews are conducted with respondents on landline telephones and cellular phones, with interviews conducted in Spanish for respondents who are primarily Spanish-speaking.
In 2013, for results based on 178,072 respondents, one can say with 95% confidence that the margin of sampling error for those results is ±0.3 percentage points.15
The poll uses a complex multistage sample design, but the sample percent has approximately a Normal sampling distribution.
What is the standard deviation of the estimated percent?
6.49
(a) No, we are only 95% confident that the interval covers the true index value for the population. (b) We believe the actual index for the population is in this interval with 95% confidence. (c) 0.153%. (d) No, the interval only accounts for error due to random sampling.
6.50 Sample size determination.
Refer to Example 6.3 (page 293) to find the standard deviation of the delay departures for Delta Airlines is given by .