SECTION 6.1 Exercises

For Exercise 6.1, see page 289; for 6.2 and 6.3, see page 293; for 6.4, see page 294; for 6.5, see page 295; for 6.6 and 6.7, see page 297; and for 6.8, see page 298.

Question 6.9

6.9 What is wrong?

Explain what is wrong in each of the following statements.

  1. If the population standard deviation is 20, then the standard deviation of for an SRS of 10 observations will be .
  2. When taking SRSs from a large population, larger sample sizes will result in larger standard deviations of .
  3. For an SRS from a large population, both the mean and the standard deviation of depend on the sample size .

6.9

(a) The standard deviation will be . (b) A larger sample will result in a smaller standard deviation. (c) Only the standard deviation depends on the sample size .

Question 6.10

6.10 What is wrong?

Explain what is wrong in each of the following statements.

  1. The central limit theorem states that for large , the population mean is approximately Normal.
  2. For large , the distribution of observed values will be approximately Normal.
  3. For sufficiently large , the 68–95–99.7 rule says that should be within about 95% of the time.

Question 6.11

6.11 Business employees.

There are more than 7 million businesses in the United States with paid employees. The mean number of employees in these businesses is about 16. A university selects a random sample of 100 businesses in Colorado and finds that they average about 11 employees. Is each of the bold numbers a parameter or a statistic?

6.11

Parameter, Statistic.

Question 6.12

6.12 Number of apps on a smartphone.

At a recent Appnation conference, Nielsen reported an average of 41 apps per smartphone among U.S. smartphone subscribers.4 State the population for this survey, the statistic, and some likely values from the population distribution.

Question 6.13

6.13 Why the difference?

Refer to the previous exercise. In Exercise 6.1 (page 289), a survey by AppsFire reported a median of 108 apps per device. This is very different from the average reported in the previous exercise.

  1. Do you think that the two populations are comparable? Explain your answer.
  2. The AppsFire report provides a footnote stating that its data exclude users who do not use any apps at all. Explain how this might contribute to the difference in the two reported statistics.

6.13

(a) In 6.1, the population was all AppsFire users, while in 6.12, the population is all U.S. smartphone subscribers. It is likely that one group has more apps. (b) Excluding those with no apps will increase the mean/median and could account for the difference.

Question 6.14

6.14 Total sleep time of college students.

In Example 6.1 (page 289), the total sleep time per night among college students was approximately Normally distributed with mean hours and standard deviation hours. You plan to take an SRS of size and compute the average total sleep time.

  1. What is the standard deviation for the average time?
  2. Use the 95 part of the 68–95–99.7 rule to describe the variability of this sample mean.
  3. What is the probability that your average will be below 6.9 hours?

Question 6.15

6.15 Determining sample size.

Refer to the previous exercise. Now you want to use a sample size such that about 95% of the averages fall within ±10 minutes (0.17 hour) of the true mean .

  1. Based on your answer to part (b) in Exercise 6.14, should the sample size be larger or smaller than 150? Explain.
  2. What standard deviation of do you need such that 95% of all samples will have a mean within 10 minutes of ?
  3. Using the standard deviation you calculated in part (b), determine the number of students you need to sample.

6.15

(a) Larger, to decrease the standard deviation. (b) 0.085. (c) .

Question 6.16

6.16 Number of friends on Facebook.

Facebook recently examined all active Facebook users (more than 10% of the global population) and determined that the average user has 190 friends. This distribution takes only integer values, so it is certainly not Normal. It is also highly skewed to the right, with a median of 100 friends.5 Suppose that and you take an SRS of 70 Facebook users.

  1. For your sample, what are the mean and standard deviation of , the mean number of friends per user?
  2. Use the central limit theorem to find the probability that the average number of friends for 70 Facebook users is greater than 250.
  3. What are the mean and standard deviation of the total number of friends in your sample? (Hint: For parts (c) and (d), use rules for means and variances for a sum of independent random variables found in Section 4.5, pages 226 and 231.)
  4. What is the probability that the total number of friends among your sample of 70 Facebook users is greater than 17,500?

Question 6.17

6.17 Generating a sampling distribution.

Let’s illustrate the idea of a sampling distribution in the case of a very small sample from a very small population. The population is the sizes of 10 medium-sized businesses, where size is measured in terms of the number of employees. For convenience, the 10 companies have been labeled with the integers 1 to 10.

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Company 1 2 3 4 5 6 7 8 9 10
Size 82 62 80 58 72 73 65 66 74 62

The parameter of interest is the mean size in this population. The sample is an SRS of size drawn from the population. Software can be used to generate an SRS.

  1. Find the mean of the 10 sizes in the population. This is the population mean.
  2. Use now software to make an SRS of size 3.
    • Excel users: A simple way to draw a random sample is to enter “=RANDBETWEEN(1,10)” in any cell. Take note of the number that represents company and record in another column the corresponding size. Hit the F9 key to change the random entry. If you get a repeat, hit the F9 again. Do this until you get three distinct values.
    • JMP users: Enter the size values in a data table. Do the following pull-down sequence: Table → Subset. In the drag-down dialog box named Initialize Data, pick Random option. Choose the bullet option of Random - sample size and enter “3” in its dialog box and then click OK. You will find an SRS of three company sizes in a new data table.
    • Minitab users: Enter the size values in column one (c1) a data table. Do the following pull-down sequence: Calc → Random Data → Sample from Samples. Enter “3” in the Number of rows to sample, type “c1” in the From columns box, and type “c2” in the Store samples in box, and then click OK. You will find an SRS of three company sizes in c2.

    With your SRS calculate the sample mean . This statistic is an estimate of .

  3. Repeat this process nine more times. Make a histogram of the 10 values of You are constructing the sampling distribution of Is the center of your histogram close to ?

6.17

(a) (b) Software, answers will vary. (c) Software, answers will vary.

Question 6.18

6.18 ACT scores of high school seniors.

The scores of your state’s high school seniors on the ACT college entrance examination in a recent year had mean and standard deviation . The distribution of scores is only roughly Normal.

  1. What is the approximate probability that a single student randomly chosen from all those taking the test scores 27 or higher?
  2. Now consider an SRS of 16 students who took the test. What are the mean and standard deviation of the sample mean score of these 16 students?
  3. What is the approximate probability that the mean score of these 16 students is 27 or higher?
  4. Which of your two Normal probability calculations in parts (a) and (c) is more accurate? Why?

Question 6.19

6.19 Safe flying weight.

In response to the increasing weight of airline passengers, the Federal Aviation Administration told airlines to assume that passengers average 190 pounds in the summer, including clothing and carry-on baggage. But passengers vary: the FAA gave a mean but not a standard deviation. A reasonable standard deviation is 35 pounds. Weights are not Normally distributed, especially when the population includes both men and women, but they are not very non-Normal. A commuter plane carries 19 passengers. What is the approximate probability that the total weight of the passengers exceeds 4000 pounds?

(Hint: To apply the central limit theorem, restate the problem in terms of the mean weight.)

6.19

0.0052.

Question 6.20

6.20 Grades in a math course.

Indiana University posts the grade distributions for its courses online.6 Students in one section of Math 118 in the fall 2012 semester received 33% A’s, 33% B’s, 20% C’s, 12% D’s, and 2% F’s.

  1. Using the common scale , , , , , take to be the grade of a randomly chosen Math 118 student. Use the definitions of the mean (page 220) and standard deviation (page 229) for discrete random variables to find the mean and the standard deviation of grades in this course.
  2. Math 118 is a large enough course that we can take the grades of an SRS of 25 students to be independent of each other. If is the average of these 25 grades, what are the mean and standard deviation of ?
  3. What is the probability that a randomly chosen Math 118 student gets a B or better, ?
  4. What is the approximate probability that the grade point average for 25 randomly chosen Math 118 students is a B or better?

Question 6.21

6.21 Increasing sample size.

Heights of adults are well approximated by the Normal distribution. Suppose that the population of adult U.S. males has mean of 69 inches and standard deviation of 2.8 inches.

  1. What is the probability that a randomly chosen male adult is taller than 6 feet?
  2. What is the probability that the sample mean of two randomly chosen male adults is greater than 6 feet?

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  3. What is the probability that the sample mean of five randomly chosen make adults is greater than 6 feet?
  4. Provide an intuitive argument as to why the probability of the sample mean being greater than 6 feet decreases as gets larger.

6.21

(a) 0.1423. (b) 0.0643. (c) 0.0082. (d) It becomes less likely to have a sample mean greater than 6 ft. because as the sample size increases we are more likely to get a few short males in our sample, which will pull the mean closer to the population mean, which is 69 in.

Question 6.22

6.22 Supplier delivery times.

Supplier on-time delivery performance is critical to enabling the buyer’s organization to meet its customer service commitments. Therefore, monitoring supplier delivery times is critical. Based on a great deal of historical data, a manufacturer of personal computers finds for one of its just-in-time suppliers that the delivery times are random and well approximated by the Normal distribution with mean 51.7 minutes and standard deviation 9.5 minutes.

  1. What is the probability that a particular delivery will exceed one hour?
  2. Based on part (a), what is the probability that a particular delivery arrives in less than one hour?
  3. What is the probability that the mean time of five deliveries will exceed one hour?