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.
6.9 What is wrong?
Explain what is wrong in each of the following statements.
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 .
6.10 What is wrong?
Explain what is wrong in each of the following statements.
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.
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.
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.
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.
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.
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 .
6.15
(a) Larger, to decrease the standard deviation. (b) 0.085. (c) .
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.
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.
With your SRS calculate the sample mean . This statistic is an estimate of .
6.17
(a) (b) Software, answers will vary. (c) Software, answers will vary.
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.
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.
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.
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.
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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.
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.