Exercise Chapter 8

254 CHAPTER 8 Confidence Interval Estimation

Example 8.2 illustrates the effect of using a 99% confidence interval.

EXAMPLE 8.2Estimating the Mean Paper Length with 99% Confidence

Construct a 99% confidence interval estimate for the population mean paper length.

SOLUTION Using Equation (8.1) on page 252, with Za>2 = 2.58 for 99% confidence,

X { Za>2 s1n= 10.998 { 12.582 0.021100

= 10.998 { 0.00516 10.9928 … m … 11.0032

Once again, because 11 is included within this wider interval, you have no reason to believe that anything is wrong with the production process.

As discussed in Section 7.2, the sampling distribution of the sample mean, X, is normally distributed if the population for your characteristic of interest, X, follows a normal distribution. And if the population of X does not follow a normal distribution, the Central Limit Theorem almost always ensures that X is approximately normally distributed when n is large. However, when dealing with a small sample size and a population that does not follow a normal distribu-tion, the sampling distribution of X is not normally distributed, and therefore the confidence interval discussed in this section is inappropriate. In practice, however, as long as the sample size is large enough and the population is not very skewed, you can use the confidence interval defined in Equation (8.1) to estimate the population mean when s is known. To assess the as-sumption of normality, you can evaluate the shape of the sample data by constructing a histo-gram, stem-and-leaf display, boxplot, or normal probability plot.

Can You Ever Know the Population Standard Deviation?To solve Equation (8.1), you must know the value for s, the population standard deviation. To know s implies that you know all the values in the entire population. (How else would you know the value of this population parameter?) If you knew all the values in the entire popu-lation, you could directly compute the population mean. There would be no need to use the inductive reasoning of inferential statistics to estimate the population mean. In other words, if you know s, you really do not have a need to use Equation (8.1) to construct a confidence interval estimate of the mean (s known).

More significantly, in virtually all real-world business situations, you would never know the standard deviation of the population. In business situations, populations are often too large to examine all the values. So why study the confidence interval estimate of the mean (s known) at all? This method serves as an important introduction to the concept of a con-fidence interval because it uses the normal distribution, which has already been thoroughly discussed in Chapters 6 and 7. In the next section, you will see that constructing a confidence interval estimate when s is not known requires another distribution (the t distribution) not pre-viously mentioned in this book.

Student TipBecause understanding the confidence interval concept is very impor-tant when reading the rest of this book, review this section carefully to understand the underly-ing concept—even if you never have a practical reason to use the confidence interval estimate of the mean (s known) method.

Problems for Section 8.1LEARNING THE BASICS8.1 If X = 85, s = 8, and n = 64, construct a 95% confidence interval estimate for the population mean, m.

8.2 If X = 125, s = 24, and n = 36, construct a 99% confi-dence interval estimate for the population mean, m.

8.3 Why is it not possible in Example 8.1 on page 253 to have 100% confidence? Explain.

8.4 Is it true in Example 8.1 on page 253 that you do not know for sure whether the population mean is between 10.9941 and 11.0019 inches? Explain.

254 CHAPTER 8 Confidence Interval Estimation

Example 8.2 illustrates the effect of using a 99% confidence interval.

EXAMPLE 8.2Estimating the Mean Paper Length with 99% Confidence

Construct a 99% confidence interval estimate for the population mean paper length.

SOLUTION Using Equation (8.1) on page 252, with Za>2 = 2.58 for 99% confidence,

X { Za>2 s1n= 10.998 { 12.582 0.021100

= 10.998 { 0.00516 10.9928 … m … 11.0032

Once again, because 11 is included within this wider interval, you have no reason to believe that anything is wrong with the production process.

As discussed in Section 7.2, the sampling distribution of the sample mean, X, is normally distributed if the population for your characteristic of interest, X, follows a normal distribution. And if the population of X does not follow a normal distribution, the Central Limit Theorem almost always ensures that X is approximately normally distributed when n is large. However, when dealing with a small sample size and a population that does not follow a normal distribu-tion, the sampling distribution of X is not normally distributed, and therefore the confidence interval discussed in this section is inappropriate. In practice, however, as long as the sample size is large enough and the population is not very skewed, you can use the confidence interval defined in Equation (8.1) to estimate the population mean when s is known. To assess the as-sumption of normality, you can evaluate the shape of the sample data by constructing a histo-gram, stem-and-leaf display, boxplot, or normal probability plot.

Can You Ever Know the Population Standard Deviation?To solve Equation (8.1), you must know the value for s, the population standard deviation. To know s implies that you know all the values in the entire population. (How else would you know the value of this population parameter?) If you knew all the values in the entire popu-lation, you could directly compute the population mean. There would be no need to use the inductive reasoning of inferential statistics to estimate the population mean. In other words, if you know s, you really do not have a need to use Equation (8.1) to construct a confidence interval estimate of the mean (s known).

More significantly, in virtually all real-world business situations, you would never know the standard deviation of the population. In business situations, populations are often too large to examine all the values. So why study the confidence interval estimate of the mean (s known) at all? This method serves as an important introduction to the concept of a con-fidence interval because it uses the normal distribution, which has already been thoroughly discussed in Chapters 6 and 7. In the next section, you will see that constructing a confidence interval estimate when s is not known requires another distribution (the t distribution) not pre-viously mentioned in this book.

Student TipBecause understanding the confidence interval concept is very impor-tant when reading the rest of this book, review this section carefully to understand the underly-ing concept—even if you never have a practical reason to use the confidence interval estimate of the mean (s known) method.

Problems for Section 8.1LEARNING THE BASICS8.1 If X = 85, s = 8, and n = 64, construct a 95% confidence interval estimate for the population mean, m.

8.2 If X = 125, s = 24, and n = 36, construct a 99% confi-dence interval estimate for the population mean, m.

8.3 Why is it not possible in Example 8.1 on page 253 to have 100% confidence? Explain.

8.4 Is it true in Example 8.1 on page 253 that you do not know for sure whether the population mean is between 10.9941 and 11.0019 inches? Explain.

8.2 Confidence Interval Estimate for the Mean (s Unknown) 261

In Figure 8.11, observe that the standard deviation of the samples varies from 6.25 (sample 17) to 14.83 (sample 3). Thus, the width of the confidence interval developed varies from 8.94 in sample 17 to 21.22 in sample 3. Because you know that the population mean or-der time m = 69.637 minutes, you can see that the interval for sample 8 169.68 - 85.482 and the interval for sample 10 156.41 - 68.692 do not correctly estimate the population mean. All the other intervals correctly estimate the population mean. Once again, remember that in prac-tice you select only one sample, and you are unable to know for sure whether your one sample provides a confidence interval that includes the population mean.

F I G U R E 8 . 1 1Confidence interval estimates of the mean for 20 samples of n = 10 randomly selected from the population of N = 200 orders with s unknown

Problems for Section 8.2LEARNING THE BASICS8.11 If X = 75, S = 24, and n = 36, and assuming that the population is normally distributed, construct a 95% confidence interval estimate for the population mean, m.

8.12 Determine the critical value of t in each of the following circumstances:a. 1 - a = 0.95, n = 10b. 1 - a = 0.99, n = 10c. 1 - a = 0.95, n = 32d. 1 - a = 0.95, n = 65e. 1 - a = 0.90, n = 16

8.13 Assuming that the population is normally distributed, con-struct a 95% confidence interval estimate for the population mean for each of the following samples:

Sample A: 1 1 1 1 8 8 8 8

Sample B: 1 2 3 4 5 6 7 8

Explain why these two samples produce different confidence intervals even though they have the same mean and range.

8.14 Assuming that the population is normally distributed, con-struct a 95% confidence interval for the population mean, based on the following sample of size n = 7:

1 2 3 4 5 6 20

Change the value of 20 to 7 and recalculate the confidence inter-val. Using these results, describe the effect of an outlier (i.e., an extreme value) on the confidence interval.

APPLYING THE CONCEPTS8.15 A marketing researcher wants to estimate the mean savings ($) realized by shoppers who showroom. Showrooming is the practice of inspecting products in retail stores and then purchas-ing the products online at a lower price. A random sample of 100 shoppers who recently purchased a consumer electronics item online after making a visit to a retail store yielded a mean savings of $58 and a standard deviation of $55.a. Construct a 95% confidence interval estimate for the mean sav-

ings for all showroomers who purchased a consumer electron-ics item.

b. Suppose the owners of a consumer electronics retailer wants to estimate the total value of lost sales attributed to the next 1,000 showroomers that enter their retail store. How are the results in (a) useful in assisting the consumer electronics retailer in their estimation?

8.16 A survey of nonprofit organizations showed that online fundraising has increased in the past year. Based

on a random sample of 55 nonprofits, the mean one-time gift do-nation in the past year was $75, with a standard deviation of $9.a. Construct a 95% confidence interval estimate for the popula-

tion mean one-time gift donation.b. Interpret the interval constructed in (a).

8.17 The U.S. Department of Transportation requires tire manu-facturers to provide tire performance information on the sidewall of a tire to better inform prospective customers as they make purchasing decisions. One very important measure of tire perfor-mance is the tread wear index, which indicates the tire’s resistance to tread wear compared with a tire graded with a base of 100.

SELF Test

8.2 Confidence Interval Estimate for the Mean (s Unknown) 261

In Figure 8.11, observe that the standard deviation of the samples varies from 6.25 (sample 17) to 14.83 (sample 3). Thus, the width of the confidence interval developed varies from 8.94 in sample 17 to 21.22 in sample 3. Because you know that the population mean or-der time m = 69.637 minutes, you can see that the interval for sample 8 169.68 - 85.482 and the interval for sample 10 156.41 - 68.692 do not correctly estimate the population mean. All the other intervals correctly estimate the population mean. Once again, remember that in prac-tice you select only one sample, and you are unable to know for sure whether your one sample provides a confidence interval that includes the population mean.

F I G U R E 8 . 1 1Confidence interval estimates of the mean for 20 samples of n = 10 randomly selected from the population of N = 200 orders with s unknown

Problems for Section 8.2LEARNING THE BASICS8.11 If X = 75, S = 24, and n = 36, and assuming that the population is normally distributed, construct a 95% confidence interval estimate for the population mean, m.

8.12 Determine the critical value of t in each of the following circumstances:a. 1 - a = 0.95, n = 10b. 1 - a = 0.99, n = 10c. 1 - a = 0.95, n = 32d. 1 - a = 0.95, n = 65e. 1 - a = 0.90, n = 16

8.13 Assuming that the population is normally distributed, con-struct a 95% confidence interval estimate for the population mean for each of the following samples:

Sample A: 1 1 1 1 8 8 8 8

Sample B: 1 2 3 4 5 6 7 8

Explain why these two samples produce different confidence intervals even though they have the same mean and range.

8.14 Assuming that the population is normally distributed, con-struct a 95% confidence interval for the population mean, based on the following sample of size n = 7:

1 2 3 4 5 6 20

Change the value of 20 to 7 and recalculate the confidence inter-val. Using these results, describe the effect of an outlier (i.e., an extreme value) on the confidence interval.

APPLYING THE CONCEPTS8.15 A marketing researcher wants to estimate the mean savings ($) realized by shoppers who showroom. Showrooming is the practice of inspecting products in retail stores and then purchas-ing the products online at a lower price. A random sample of 100 shoppers who recently purchased a consumer electronics item online after making a visit to a retail store yielded a mean savings of $58 and a standard deviation of $55.a. Construct a 95% confidence interval estimate for the mean sav-

ings for all showroomers who purchased a consumer electron-ics item.

b. Suppose the owners of a consumer electronics retailer wants to estimate the total value of lost sales attributed to the next 1,000 showroomers that enter their retail store. How are the results in (a) useful in assisting the consumer electronics retailer in their estimation?

8.16 A survey of nonprofit organizations showed that online fundraising has increased in the past year. Based

on a random sample of 55 nonprofits, the mean one-time gift do-nation in the past year was $75, with a standard deviation of $9.a. Construct a 95% confidence interval estimate for the popula-

tion mean one-time gift donation.b. Interpret the interval constructed in (a).

8.17 The U.S. Department of Transportation requires tire manu-facturers to provide tire performance information on the sidewall of a tire to better inform prospective customers as they make purchasing decisions. One very important measure of tire perfor-mance is the tread wear index, which indicates the tire’s resistance to tread wear compared with a tire graded with a base of 100.

SELF Test

262 CHAPTER 8 Confidence Interval Estimation

A tire with a grade of 200 should last twice as long, on average, as a tire graded with a base of 100. A consumer organization wants to estimate the actual tread wear index of a brand name of tires that claims “graded 200” on the sidewall of the tire. A random sample of n = 18 indicates a sample mean tread wear index of 195.3 and a sample standard deviation of 21.4.a. Assuming that the population of tread wear indexes is normally

distributed, construct a 95% confidence interval estimate for the population mean tread wear index for tires produced by this manufacturer under this brand name.

b. Do you think that the consumer organization should accuse the manufacturer of producing tires that do not meet the performance information provided on the sidewall of the tire? Explain.

c. Explain why an observed tread wear index of 210 for a particu-lar tire is not unusual, even though it is outside the confidence interval developed in (a).

8.18 The file FastFood contains the amount that a sample of 15 customers spent for lunch ($) at a fast-food restaurant:

7.42 6.29 5.83 6.50 8.34 9.51 7.10 6.80 5.904.89 6.50 5.52 7.90 8.30 9.60

a. Construct a 95% confidence interval estimate for the popula-tion mean amount spent for lunch ($) at a fast-food restaurant, assuming a normal distribution.

b. Interpret the interval constructed in (a).

8.19 The file Sedans contains the overall miles per gallon (MPG) of 2014 midsized sedans:

38 26 30 26 25 27 24 22 27 32 3926 24 24 23 24 25 31 26 37 22 33

Source: Data extracted from “Which Car Is Right for You,” Consumer Reports, April 2014, pp. 40–41.

a. Construct a 95% confidence interval estimate for the popula-tion mean MPG of 2014 midsized sedans, assuming a normal distribution.

b. Interpret the interval constructed in (a).c. Compare the results in (a) to those in Problem 8.20(a).

8.20 The file SUV contains the overall MPG of 2014 small SUVs:

26 22 23 21 25 24 22 26 25 2221 21 22 22 23 24 23 22 21 22

Source: Data extracted from “Which Car Is Right for You,” Consumer Reports, April 2014, pp. 60–61.

a. Construct a 95% confidence interval estimate for the population mean MPG of 2014 small SUVs, assuming a nor-mal distribution.

b. Interpret the interval constructed in (a).c. Compare the results in (a) to those in Problem 8.19(a).

8.21 Is there a difference in the yields of different types of in-vestments? The file CDRate contains the yields for a one-year certificate of deposit (CD) and a five-year CD for 22 banks in the United States as of March 12, 2014. (Data extracted from www .Bankrate.com, March 12, 2014.)a. Construct a 95% confidence interval estimate for the mean

yield of one-year CDs.

b. Construct a 95% confidence interval estimate for the mean yield of five-year CDs.

c. Compare the results of (a) and (b).

8.22 One of the major measures of the quality of service pro-vided by any organization is the speed with which the organization responds to customer complaints. A large family-held department store selling furniture and flooring, including carpet, had under-gone a major expansion in the past several years. In particular, the flooring department had expanded from 2 installation crews to an installation supervisor, a measurer, and 15 installation crews. The store had the business objective of improving its response to com-plaints. The variable of interest was defined as the number of days between when the complaint was made and when it was resolved. Data were collected from 50 complaints that were made in the past year. The data, stored in Furniture , are as follows:

54 5 35 137 31 27 152 2 123 81 74 27

11 19 126 110 110 29 61 35 94 31 26 5

12 4 165 32 29 28 29 26 25 1 14 13

13 10 5 27 4 52 30 22 36 26 20 23

33 68

a. Construct a 95% confidence interval estimate for the popula-tion mean number of days between the receipt of a complaint and the resolution of the complaint.

b. What assumption must you make about the population distribution in order to construct the confidence interval estimate in (a)?

c. Do you think that the assumption needed in order to construct the confidence interval estimate in (a) is valid? Explain.

d. What effect might your conclusion in (c) have on the validity of the results in (a)?

8.23 A manufacturing company produces electric insulators. You define the variable of interest as the strength of the insulators. If the insulators break when in use, a short circuit is likely. To test the strength of the insulators, you carry out destructive testing to deter-mine how much force is required to break the insulators. You measure force by observing how many pounds are applied to the insulator be-fore it breaks. You collect the force data for 30 insulators selected for the experiment and organize and store these data in Force :

1,870 1,728 1,656 1,610 1,634 1,784 1,522 1,696

1,592 1,662 1,866 1,764 1,734 1,662 1,734 1,774

1,550 1,756 1,762 1,866 1,820 1,744 1,788 1,688

1,810 1,752 1,680 1,810 1,652 1,736

a. Construct a 95% confidence interval estimate for the popula-tion mean force.

b. What assumption must you make about the population dis-tribution in order to construct the confidence interval esti-mate in (a)?

c. Do you think that the assumption needed in order to construct the confidence interval estimate in (a) is valid? Explain.

8.24 The file Market Penetration contains Facebook penetration values (the percentage of the country population that are Facebook users) for 22 of the world’s largest economies:

56 57 43 55 42 35 7 25 42 17 43

6 31 28 59 20 27 36 45 80 57 56Source: Data extracted from slidesha.re/ODv6vG.

8.3 Confidence Interval Estimate for the Proportion 265

Equation (8.3) contains a Z statistic because you can use the normal distribution to approximate the binomial distribution when the sample size is sufficiently large. In Example 8.4, the confidence interval using Z provides an excellent approximation for the population propor-tion because both X and n - X are greater than 5. However, if you do not have a sufficiently large sample size, you should use the binomial distribution rather than Equation (8.3) (see refer-ences 1, 3, and 9). The exact confidence intervals for various sample sizes and proportions of items of interest have been tabulated by Fisher and Yates (reference 3) and can also be computed using Minitab.

Problems for Section 8.3LEARNING THE BASICS8.26 If n = 200 and X = 50, construct a 95% confidence inter-val estimate for the population proportion.

8.27 If n = 400 and X = 25, construct a 99% confidence inter-val estimate for the population proportion.

APPLYING THE CONCEPTS8.28 A cellphone provider has the business objective of wanting to estimate the proportion of subscribers

who would upgrade to a new cellphone with improved features if it were made available at a substantially reduced cost. Data are collected from a random sample of 500 subscribers. The results indicate that 135 of the subscribers would upgrade to a new cell-phone at a reduced cost.a. Construct a 99% confidence interval estimate for the popula-

tion proportion of subscribers that would upgrade to a new cell-phone at a reduced cost.

b. How would the manager in charge of promotional programs use the results in (a)?

8.29 In a survey of 529 travelers, 386 said that location was very important and 323 said that room quality was very important in choosing a hotel. (Source: C. Jones, “Top Reason for Picking a Hotel? It’s Where It’s At,” USA Today, April 28, 2014, p. 3B.)a. Construct a 95% confidence interval estimate for the popula-

tion proportion of travelers who said that location was very im-portant for choosing a hotel.

b. Construct a 95% confidence interval estimate for the popula-tion proportion of travelers who said that room quality was very important in choosing a hotel.

c. Write a short summary of the information derived from (a) and (b).

8.30 Are you likely to purchase an item promoted by a celebrity on a social media site? According to a Brand Republic survey, 26% of social media users have made such a purchase (Source: “Celebrity endorsement on social media,” bit.ly/1o4yNHa.)a. Suppose that the survey had a sample size of n = 1,000. Con-

struct a 95% confidence interval estimate for the population proportion of social media users that have purchased an item promoted by a celebrity on a social media site.

b. Based on (a), can you claim that more than a quarter of all so-cial media users have purchased an item promoted by a celeb-rity on a social media site?

c. Repeat parts (a) and (b), assuming that the survey had a sample size of n = 10,000.

d. Discuss the effect of sample size on confidence interval estimation.

8.31 In a survey of 239 organizations, 75 responded that “the need for collaboration among increasing number of locations” is a business driver that led them to implement cloud solutions. Cloud solutions enable more effective employee communication and higher decision maker visibility into real-time data. (Source: The Benefits of Cloud ERP: It’s About Transforming Your Business, Aberdeen Group, available at bit.ly/1meEC3D.)

Construct a 95% confidence interval estimate for the popula-tion proportion of organizations that indicated “the need for collabo-ration among increasing number of locations” as a business driver for cloud solution implementation.

8.32 In a Pew Research Center survey of 960 Facebook users, 452 cited “seeing photos or videos” as a major reason why they use Facebook, while 298 cited “keeping up with news and current events” as a major reason why they use Facebook. (Source: “6 new facts about Facebook,” bit.ly/1lAmkv5.)a. Construct a 95% confidence interval estimate for the popula-

tion proportion of Facebook users who cite “seeing photos or videos” as a major reason for why they use Facebook.

b. Construct a 95% confidence interval estimate for the popula-tion proportion of Facebook users who cite “keeping up with news and current events” as a major reason why they use Facebook.

c. Compare the results of (a) and (b).

8.33 What are the global trends that technology CEOs believe will transform their business? According to a PwC white paper, 105 of 117 technology CEOs from around the world responded that technological advances will transform their business and 42 responded that resource scarcity and climate change will trans-form their business. (Source: Fit for the Future: 17th Annual Global CEO Survey, available at pwc.to/PRQZYr.)a. Construct a 95% confidence interval estimate for the popula-

tion proportion of tech CEOs who indicate technological ad-vances as one of the global trends that will transform their business.

b. Construct a 95% confidence interval estimate for the popula-tion proportion of tech CEOs who indicate resource scarcity and climate change as one of the global trends that will trans-form their business.

c. Interpret the intervals in (a) and (b).

SELF Test

8.3 Confidence Interval Estimate for the Proportion 265

Equation (8.3) contains a Z statistic because you can use the normal distribution to approximate the binomial distribution when the sample size is sufficiently large. In Example 8.4, the confidence interval using Z provides an excellent approximation for the population propor-tion because both X and n - X are greater than 5. However, if you do not have a sufficiently large sample size, you should use the binomial distribution rather than Equation (8.3) (see refer-ences 1, 3, and 9). The exact confidence intervals for various sample sizes and proportions of items of interest have been tabulated by Fisher and Yates (reference 3) and can also be computed using Minitab.

Problems for Section 8.3LEARNING THE BASICS8.26 If n = 200 and X = 50, construct a 95% confidence inter-val estimate for the population proportion.

8.27 If n = 400 and X = 25, construct a 99% confidence inter-val estimate for the population proportion.

APPLYING THE CONCEPTS8.28 A cellphone provider has the business objective of wanting to estimate the proportion of subscribers

who would upgrade to a new cellphone with improved features if it were made available at a substantially reduced cost. Data are collected from a random sample of 500 subscribers. The results indicate that 135 of the subscribers would upgrade to a new cell-phone at a reduced cost.a. Construct a 99% confidence interval estimate for the popula-

tion proportion of subscribers that would upgrade to a new cell-phone at a reduced cost.

b. How would the manager in charge of promotional programs use the results in (a)?

8.29 In a survey of 529 travelers, 386 said that location was very important and 323 said that room quality was very important in choosing a hotel. (Source: C. Jones, “Top Reason for Picking a Hotel? It’s Where It’s At,” USA Today, April 28, 2014, p. 3B.)a. Construct a 95% confidence interval estimate for the popula-

tion proportion of travelers who said that location was very im-portant for choosing a hotel.

b. Construct a 95% confidence interval estimate for the popula-tion proportion of travelers who said that room quality was very important in choosing a hotel.

c. Write a short summary of the information derived from (a) and (b).

8.30 Are you likely to purchase an item promoted by a celebrity on a social media site? According to a Brand Republic survey, 26% of social media users have made such a purchase (Source: “Celebrity endorsement on social media,” bit.ly/1o4yNHa.)a. Suppose that the survey had a sample size of n = 1,000. Con-

struct a 95% confidence interval estimate for the population proportion of social media users that have purchased an item promoted by a celebrity on a social media site.

b. Based on (a), can you claim that more than a quarter of all so-cial media users have purchased an item promoted by a celeb-rity on a social media site?

c. Repeat parts (a) and (b), assuming that the survey had a sample size of n = 10,000.

d. Discuss the effect of sample size on confidence interval estimation.

8.31 In a survey of 239 organizations, 75 responded that “the need for collaboration among increasing number of locations” is a business driver that led them to implement cloud solutions. Cloud solutions enable more effective employee communication and higher decision maker visibility into real-time data. (Source: The Benefits of Cloud ERP: It’s About Transforming Your Business, Aberdeen Group, available at bit.ly/1meEC3D.)

Construct a 95% confidence interval estimate for the popula-tion proportion of organizations that indicated “the need for collabo-ration among increasing number of locations” as a business driver for cloud solution implementation.

8.32 In a Pew Research Center survey of 960 Facebook users, 452 cited “seeing photos or videos” as a major reason why they use Facebook, while 298 cited “keeping up with news and current events” as a major reason why they use Facebook. (Source: “6 new facts about Facebook,” bit.ly/1lAmkv5.)a. Construct a 95% confidence interval estimate for the popula-

tion proportion of Facebook users who cite “seeing photos or videos” as a major reason for why they use Facebook.

b. Construct a 95% confidence interval estimate for the popula-tion proportion of Facebook users who cite “keeping up with news and current events” as a major reason why they use Facebook.

c. Compare the results of (a) and (b).

8.33 What are the global trends that technology CEOs believe will transform their business? According to a PwC white paper, 105 of 117 technology CEOs from around the world responded that technological advances will transform their business and 42 responded that resource scarcity and climate change will trans-form their business. (Source: Fit for the Future: 17th Annual Global CEO Survey, available at pwc.to/PRQZYr.)a. Construct a 95% confidence interval estimate for the popula-

tion proportion of tech CEOs who indicate technological ad-vances as one of the global trends that will transform their business.

b. Construct a 95% confidence interval estimate for the popula-tion proportion of tech CEOs who indicate resource scarcity and climate change as one of the global trends that will trans-form their business.

c. Interpret the intervals in (a) and (b).

SELF Test

8.3 Confidence Interval Estimate for the Proportion 265

Equation (8.3) contains a Z statistic because you can use the normal distribution to approximate the binomial distribution when the sample size is sufficiently large. In Example 8.4, the confidence interval using Z provides an excellent approximation for the population propor-tion because both X and n - X are greater than 5. However, if you do not have a sufficiently large sample size, you should use the binomial distribution rather than Equation (8.3) (see refer-ences 1, 3, and 9). The exact confidence intervals for various sample sizes and proportions of items of interest have been tabulated by Fisher and Yates (reference 3) and can also be computed using Minitab.

Problems for Section 8.3LEARNING THE BASICS8.26 If n = 200 and X = 50, construct a 95% confidence inter-val estimate for the population proportion.

8.27 If n = 400 and X = 25, construct a 99% confidence inter-val estimate for the population proportion.

APPLYING THE CONCEPTS8.28 A cellphone provider has the business objective of wanting to estimate the proportion of subscribers

who would upgrade to a new cellphone with improved features if it were made available at a substantially reduced cost. Data are collected from a random sample of 500 subscribers. The results indicate that 135 of the subscribers would upgrade to a new cell-phone at a reduced cost.a. Construct a 99% confidence interval estimate for the popula-

tion proportion of subscribers that would upgrade to a new cell-phone at a reduced cost.

b. How would the manager in charge of promotional programs use the results in (a)?

8.29 In a survey of 529 travelers, 386 said that location was very important and 323 said that room quality was very important in choosing a hotel. (Source: C. Jones, “Top Reason for Picking a Hotel? It’s Where It’s At,” USA Today, April 28, 2014, p. 3B.)a. Construct a 95% confidence interval estimate for the popula-

tion proportion of travelers who said that location was very im-portant for choosing a hotel.

b. Construct a 95% confidence interval estimate for the popula-tion proportion of travelers who said that room quality was very important in choosing a hotel.

c. Write a short summary of the information derived from (a) and (b).

8.30 Are you likely to purchase an item promoted by a celebrity on a social media site? According to a Brand Republic survey, 26% of social media users have made such a purchase (Source: “Celebrity endorsement on social media,” bit.ly/1o4yNHa.)a. Suppose that the survey had a sample size of n = 1,000. Con-

struct a 95% confidence interval estimate for the population proportion of social media users that have purchased an item promoted by a celebrity on a social media site.

b. Based on (a), can you claim that more than a quarter of all so-cial media users have purchased an item promoted by a celeb-rity on a social media site?

c. Repeat parts (a) and (b), assuming that the survey had a sample size of n = 10,000.

d. Discuss the effect of sample size on confidence interval estimation.

8.31 In a survey of 239 organizations, 75 responded that “the need for collaboration among increasing number of locations” is a business driver that led them to implement cloud solutions. Cloud solutions enable more effective employee communication and higher decision maker visibility into real-time data. (Source: The Benefits of Cloud ERP: It’s About Transforming Your Business, Aberdeen Group, available at bit.ly/1meEC3D.)

Construct a 95% confidence interval estimate for the popula-tion proportion of organizations that indicated “the need for collabo-ration among increasing number of locations” as a business driver for cloud solution implementation.

8.32 In a Pew Research Center survey of 960 Facebook users, 452 cited “seeing photos or videos” as a major reason why they use Facebook, while 298 cited “keeping up with news and current events” as a major reason why they use Facebook. (Source: “6 new facts about Facebook,” bit.ly/1lAmkv5.)a. Construct a 95% confidence interval estimate for the popula-

tion proportion of Facebook users who cite “seeing photos or videos” as a major reason for why they use Facebook.

b. Construct a 95% confidence interval estimate for the popula-tion proportion of Facebook users who cite “keeping up with news and current events” as a major reason why they use Facebook.

c. Compare the results of (a) and (b).

8.33 What are the global trends that technology CEOs believe will transform their business? According to a PwC white paper, 105 of 117 technology CEOs from around the world responded that technological advances will transform their business and 42 responded that resource scarcity and climate change will trans-form their business. (Source: Fit for the Future: 17th Annual Global CEO Survey, available at pwc.to/PRQZYr.)a. Construct a 95% confidence interval estimate for the popula-

tion proportion of tech CEOs who indicate technological ad-vances as one of the global trends that will transform their business.

b. Construct a 95% confidence interval estimate for the popula-tion proportion of tech CEOs who indicate resource scarcity and climate change as one of the global trends that will trans-form their business.

c. Interpret the intervals in (a) and (b).

SELF Test