Statistics for Business and Economics: Global Edition

Newbold P., Carlson W.L., Thorne B.M.

Chapter 6

Distributions of Sample Statistics - all with Video Answers

Educators

Chapter Questions

Problem 1

A five-a-side soccer club in Singapore buys a set of shirts numbered 1 to 5 .
a. What is the population distribution of shirt numbers?
b. Determine the sampling distribution of the sample mean of the shirt numbers obtained by selecting two shirts.

Khalida Dawar

Khalida Dawar

Numerade Educator

Problem 2

Suppose that you have a fair coin and you label the head side as 1 and the tail side as 0 .
a. Now, you are asked to flip the coin 2 times and write down the numerical value that results from each toss. Without actually flipping the coin, write down the sampling distribution of the sample means.
b. Repeat part (a) with the coin flipped 4 times.
c. Repeat part (a) with the coin flipped 10 times.

Nick Johnson

Numerade Educator

Problem 3

A population contains 6 million $0 \mathrm{~s}$ and 4 million $1 \mathrm{~s}$. What is the approximate sampling distribution of the sample mean in each of the following cases?
a. The sample size is $n=5$
b. The sample size is $n=100$

Nick Johnson

Numerade Educator

Problem 4

Suppose that a mathematician said that it is impossible to obtain a simple random sample from a real-world population. Therefore, the whole basis for applying statistical procedures to real problems is useless. How would you respond?

Nick Johnson

Numerade Educator

Problem 5

Given a population with a mean of $\mu=100$ and a variance of $\sigma^2=81$, the central limit theorem applies when the sample size is $n \geq 25$. A random sample of size $n=25$ is obtained.
a. What are the mean and variance of the sampling distribution for the sample means?
b. What is the probability that $\bar{x}>102$ ?
c. What is the probability that $98 \leq \bar{x} \leq 101$ ?
d. What is the probability that $\bar{x} \leq 101.5$ ?

Nick Johnson

Numerade Educator

Problem 6

Given a population with a mean of $\mu=100$ and a variance of $\sigma^2=900$, the central limit theorem applies when the sample size is $n \geq 25$. A random sample of size $n=30$ is obtained.
a. What are the mean and variance of the sampling distribution for the sample means?
b. What is the probability that $\bar{x}>109$ ?
c. What is the probability that $96 \leq \bar{x} \leq 110$ ?
d. What is the probability that $\bar{x} \leq 107$ ?

Nick Johnson

Numerade Educator

Problem 7

Given a population with a mean of $\mu=200$ and a variance of $\sigma^2=625$, the central limit theorem applies when the sample size $n \geq 25$. A random sample of size $n=25$ is obtained.
a. What are the mean and variance of the sampling distribution for the sample mean?
b. What is the probability that $\bar{x}>209$ ?
c. What is the probability that $198 \leq \bar{x} \leq 211$ ?
d. What is the probability that $\bar{x} \leq 202$ ?

Nick Johnson

Numerade Educator

Problem 8

Given a population with mean $\mu=400$ and variance $\sigma^2=1,600$, the central limit theorem applies when the sample size is $n \geq 25$. A random sample of size $n=35$ is obtained.
a. What are the mean and variance of the sampling distribution for the sample means?
b. What is the probability that $\bar{x}>412$ ?
c. What is the probability that $393 \leq \bar{x} \leq 407$ ?
d. What is the probability that $\bar{x} \leq 389$ ?

Nick Johnson

Numerade Educator

Problem 9

When a production process is operating correctly, the number of units produced per hour has a normal distribution with a mean of 92.0 and a standard deviation of 3.6. A random sample of 4 different hours was taken.
a. Find the mean of the sampling distribution of the sample means.
b. Find the variance of the sampling distribution of the sample mean.
c. Find the standard error of the sampling distribution of the sample mean.
d. What is the probability that the sample mean exceeds 93.0 units?

Nick Johnson

Numerade Educator

Problem 10

The lifetimes of lightbulbs produced by a particular manufacturer have a mean of 1,200 hours and a standard deviation of 400 hours. The population distribution is normal. Suppose that you purchase nine bulbs, which can be regarded as a random sample from the manufacturer's output.
a. What is the mean of the sample mean lifetime?
b. What is the variance of the sample mean?
c. What is the standard error of the sample mean?
d. What is the probability that, on average, those nine lightbulbs have lives of fewer than 1,050 hours?

Nick Johnson

Numerade Educator

Problem 11

The fuel consumption, in miles per gallon, of all cars of a particular model has a mean of 25 and a standard deviation of 2. The population distribution can be assumed to be normal. A random sample of these cars is taken.
a. Find the probability that sample mean fuel consumption will be fewer than 24 miles per gallon if i. a sample of 1 observation is taken.
ii. a sample of 4 observations is taken.
iii. a sample of 16 observations is taken.
b. Explain why the three answers in part (a) differ in the way they do. Draw a graph to illustrate your reasoning.

Nick Johnson

Numerade Educator

Problem 12

The mean selling price of senior condominiums in Green Valley over a year was $$\$ 215,000$$. The population standard deviation was $$\$ 25,000$$. A random sample of 100 new unit sales was obtained.
a. What is the probability that the sample mean selling price was more than $$\$ 210,000$$ ?
b. What is the probability that the sample mean selling price was between $$\$ 213,000$$ and $$\$ 217,000$$ ?
c. What is the probability that the sample mean selling price was between $$\$ 214,000$$ and $$\$ 216,000$$ ?
d. Without doing the calculations, state in which of the following ranges the sample mean selling price is most likely to lie:
$$
\begin{aligned}
& \$ 213,000 \text { to } \$ 215,000 ; \$ 214,000 \text { to } \$ 216,000 \text {; } \\
& \$ 215,000 \text { to } \$ 217,000 ; \$ 216,000 \text { to } \$ 218,000
\end{aligned}
$$
e. Suppose that, after you had done these calculations, a friend asserted that the population distribution of selling prices of senior condominiums in Green Valley was almost certainly not normal. How would you respond?

Nick Johnson

Numerade Educator

Problem 13

Candidates for employment at a city fire department are required to take a written aptitude test. Scores on this test are normally distributed with a mean of 280 and a standard deviation of 60 . A random sample of nine test scores was taken.
a. What is the standard error of the sample mean score?
b. What is the probability that the sample mean score is less than 270 ?
c. What is the probability that the sample mean score is more than 250 ?
d. Suppose that the population standard deviation is, in fact, 40, rather than 60 . Without doing the calculations, state how this would change your answers to parts (a), (b), and (c). Illustrate your conclusions with the appropriate graphs.

Nick Johnson

Numerade Educator

Problem 14

A random sample of 16 junior managers in the offices of corporations in a large city center was taken to estimate average daily commuting time for all such managers. Suppose that the population times have a normal distribution with a mean of 87 minutes and a standard deviation of 22 minutes.
a. What is the standard error of the sample mean commuting time?
b. What is the probability that the sample mean is fewer than 100 minutes?
c. What is the probability that the sample mean is more than 80 minutes?
d. What is the probability that the sample mean is outside the range 85 to 95 minutes?
e. Suppose that a second (independent) random sample of 50 junior managers is taken. Without doing the calculations, state whether the probabilities in parts (b), (c), and (d) would be higher, lower, or the same for the second sample. Sketch graphs to illustrate your answers.

Nick Johnson

Numerade Educator

Problem 15

A company produces breakfast cereal. The true mean weight of the contents of its cereal packages is 20 ounces, and the standard deviation is 0.6 ounce. The population distribution of weights is normal. Suppose that you purchase four packages, which can be regarded as a random sample of all those produced.
a. What is the standard error of the sample mean weight?
b. What is the probability that, on average, the contents of these four packages will weigh fewer than 19.7 ounces?
c. What is the probability that, on average, the contents of these four packages will weigh more than 20.6 ounces?
d. What is the probability that, on average, the contents of these four packages will weigh between 19.5 and 20.5 ounces?
e. Two of the four boxes are chosen at random. What is the probability that the average contents of these two packages will weigh between 19.5 and 20.5 ounces?

Nick Johnson

Numerade Educator

Problem 16

Assume that the standard deviation of monthly rents paid by students in a particular town is $$\$ 40$$. A random sample of 100 students was taken to estimate the mean monthly rent paid by the whole student population.
a. What is the standard error of the sample mean monthly rent?
b. What is the probability that the sample mean exceeds the population mean by more than $$\$ 5$$ ?
c. What is the probability that the sample mean is more than $$\$ 4$$ below the population mean?
d. What is the probability that the sample mean differs from the population mean by more than $$\$ 3$$ ?

Nick Johnson

Numerade Educator

Problem 17

The times spent studying by students in the week before final exams follows a normal distribution with standard deviation 8 hours. A random sample of four students was taken in order to estimate the mean study time for the population of all students.
a. What is the probability that the sample mean exceeds the population mean by more than 2 hours?
b. What is the probability that the sample mean is more than 3 hours below the population mean?
c. What is the probability that the sample mean differs from the population mean by more than 4 hours?
d. Suppose that a second (independent) random sample of 10 students was taken. Without doing the calculations, state whether the probabilities in parts (a), (b), and (c) would be higher, lower, or the same for the second sample.

Nick Johnson

Numerade Educator

Problem 18

An industrial process produces batches of a chemical whose impurity levels follow a normal distribution with standard deviation 1.6 grams per 100 grams of chemical. A random sample of 100 batches is selected in order to estimate the population mean impurity level.
a. The probability is 0.05 that the sample mean impurity level exceeds the population mean by how much?
b. The probability is 0.10 that the sample mean impurity level is below the population mean by how much?
c. The probability is 0.15 that the sample mean impurity level differs from the population mean by how much?

Nick Johnson

Numerade Educator

Problem 19

The price-earnings ratios for all companies whose shares are traded on the New York Stock Exchange follow a normal distribution with a standard deviation of 3.8 . A random sample of these companies is selected in order to estimate the population mean price-earnings ratio.
a. How large a sample is necessary in order to ensure that the probability that the sample mean differs from the population mean by more than 1.0 is less than 0.10 ?
b. Without doing the calculations, state whether a larger or smaller sample size compared to the sample size in part (a) would be required to guarantee that the probability of the sample mean differing from the population mean by more than 1.0 is less than 0.05 .
c. Without doing the calculations, state whether a larger or smaller sample size compared to the sample size in part a would be required to guarantee that the probability of the sample mean differing from the population mean by more than 1.5 hours is less than 0.10 .

Nick Johnson

Numerade Educator

Problem 20

The number of hours spent studying by students on a large campus in the week before final exams follows a normal distribution with a standard deviation of 8.4 hours. A random sample of these students is taken to estimate the population mean number of hours studying.
a. How large a sample is needed to ensure that the probability that the sample mean differs from the population mean by more than 2.0 hours is less than 0.05 ?
b. Without doing the calculations, state whether a larger or smaller sample size compared to the sample size in part (a) would be required to guarantee that the probability of the sample mean differing from the population mean by more than 2.0 hours is less than 0.10 .
c. Without doing the calculations, state whether a larger or smaller sample size compared to the sample size in part (a) would be required to guarantee that the probability of the sample mean differing from the population mean by more than 1.5 hours is less than 0.05 .

Nick Johnson

Numerade Educator

Problem 21

Greenstone Coffee is experiencing financial pressures due to increased competition for its numerous urban coffee shops. Total sales revenue has dropped by $15 \%$ and the company wishes to establish a sales monitoring process to identify shops that are underperforming. Historically, the daily mean sales for a shop have been $$\$ 11,500$$ with a variance of $4,000,000$. Their monitoring plan will take a random sample of 5 days' sales per month and use the sample mean sales to identify shops that are underperforming. Establish the lower limit sales such that only $5 \%$ of the shops would have a sample sales mean below this value.

Nick Johnson

Numerade Educator

Problem 22

In taking a sample of $n$ observations from a population of $N$ members, the variance of the sampling distribution of the sample means is as follows:
$$
\sigma_{\bar{x}}^2=\frac{\sigma_x^2}{n} \cdot \frac{N-n}{N-1}
$$
The quantity $\frac{(N-n)}{(N-1)}$ is called the finite population correction factor.
a. To get some feeling for possible magnitudes of the finite population correction factor, calculate it for samples of $n=20$ observations from populations of members: 20,40,100,1,000,10,000.
b. Explain why the result found in part a, is precisely what one should expect on intuitive grounds.
c. Given the results in part a, discuss the practical significance of using the finite-population correction factor for samples of 20 observations from populations of different sizes.

Nick Johnson

Numerade Educator

Problem 23

A town has 500 real estate agents. The mean value of the properties sold in a year by these agents is $$\$ 800,000$$, and the standard deviation is $$\$ 300,000$$. A random sample of 100 agents is selected, and the value of the properties they sold in a year is recorded.
a. What is the standard error of the sample mean?
b. What is the probability that the sample mean exceeds $$\$ 825,000$$ ?
c. What is the probability that the sample mean exceeds $$\$ 780,000$$ ?
d. What is the probability that the sample mean is between $$\$ 790,000$$ and $$\$ 820,000$$ ?

Nick Johnson

Numerade Educator

Problem 24

An English literature course was taken by 250 students. Each member of a random sample of 50 of these students was asked to estimate the amount of time he or she spent on the previous week's assignment. Suppose that the population standard deviation is 30 minutes.
a. What is the probability that the sample mean exceeds the population mean by more than 2.5 minutes?
b. What is the probability that the sample mean is more than 5 minutes below the population mean?
c. What is the probability that the sample mean differs from the population mean by more than 10 minutes?

Nick Johnson

Numerade Educator

Problem 25

For an audience of 600 people attending a concert, the average time on the journey to the concert was $32 \mathrm{~min}$ utes, and the standard deviation was 10 minutes. A random sample of 150 audience members was taken.
a. What is the probability that the sample mean journey time was more than 31 minutes?
b. What is the probability that the sample mean journey time was less than 33 minutes?
c. Construct a graph to illustrate why the answers to parts (a) and (b) are the same.
d. What is the probability that the sample mean journey time was not between 31 and 33 minutes?

Nick Johnson

Numerade Educator

Problem 26

Suppose that we have a population with proportion $P=0.40$ and a random sample of size $n=100$ drawn from the population.
a. What is the probability that the sample proportion is greater than 0.45 ?
b. What is the probability that the sample proportion is less than 0.29 ?
c. What is the probability that the sample proportion is between 0.35 and 0.51 ?

Nick Johnson

Numerade Educator

Problem 27

Suppose that we have a population with proportion $P=0.25$ and a random sample of size $n=200$ drawn from the population.
a. What is the probability that the sample proportion is greater than 0.31 ?
b. What is the probability that the sample proportion is less than 0.14 ?
c. What is the probability that the sample proportion is between 0.24 and 0.40 ?

Nick Johnson

Numerade Educator

Problem 28

Suppose that we have a population with proportion $P=0.60$ and a random sample of size $n=100$ drawn from the population.
a. What is the probability that the sample proportion is more than 0.66 ?
b. What is the probability that the sample proportion is less than 0.48 ?
c. What is the probability that the sample proportion is between 0.52 and 0.66 ?

Nick Johnson

Numerade Educator

Problem 29

Suppose that we have a population with proportion $P=0.50$ and a random sample of size $n=900$ drawn from the population.
a. What is the probability that the sample proportion is more than 0.52 ?
b. What is the probability that the sample proportion is less than 0.46 ?
c. What is the probability that the sample proportion is between 0.47 and 0.53 ?

Nick Johnson

Numerade Educator

Problem 30

In 1992, Canadians voted in a referendum on a new constitution. In the province of Quebec, $42.4 \%$ of those who voted were in favor of the new constitution. A random sample of 100 voters from the province was taken.
a. What is the mean of the distribution of the sample proportion in favor of a new constitution?
b. What is the variance of the sample proportion?
c. What is the standard error of the sample proportion?
d. What is the probability that the sample proportion is more than 0.5 ?

Nick Johnson

Numerade Educator

Problem 31

According to the Internal Revenue Service, $75 \%$ of all tax returns lead to a refund. A random sample of 100 tax returns is taken.
a. What is the mean of the distribution of the sample proportion of returns leading to refunds?
b. What is the variance of the sample proportion?
c. What is the standard error of the sample proportion?
d. What is the probability that the sample proportion exceeds 0.8 ?

Nick Johnson

Numerade Educator

Problem 32

A record store owner finds that $20 \%$ of customers entering her store make a purchase. One morning 180 people, who can be regarded as a random sample of all customers, enter the store.
a. What is the mean of the distribution of the sample proportion of customers making a purchase?
b. What is the variance of the sample proportion?
c. What is the standard error of the sample proportion?
d. What is the probability that the sample proportion is less than 0.15 ?

Nick Johnson

Numerade Educator

Problem 33

An administrator for a large group of hospitals believes that of all patients $30 \%$ will generate bills that become at least 2 months overdue. A random sample of 200 patients is taken.
a. What is the standard error of the sample proportion that will generate bills that become at least 2 months overdue?
b. What is the probability that the sample proportion is less than 0.25 ?
c. What is the probability that the sample proportion is more than 0.33 ?
d. What is the probability that the sample proportion is between 0.27 and 0.33 ?

Nick Johnson

Numerade Educator

Problem 34

A corporation receives 120 applications for positions from recent college graduates in business. Assuming that these applicants can be viewed as a random sample of all such graduates, what is the probability that between $35 \%$ and $45 \%$ of them are women if $40 \%$ of all recent college graduates in business are women?

Nick Johnson

Numerade Educator

Problem 35

A charity has found that $42 \%$ of all donors from last year will donate again this year. A random sample of 300 donors from last year was taken.
a. What is the standard error of the sample proportion who will donate again this year?
b. What is the probability that more than half of these sample members will donate again this year?
c. What is the probability that the sample proportion is between 0.40 and 0.45 ?
d. Without doing the calculations, state in which of the following ranges the sample proportion is more likely to lie: 0.39 to $0.41,0.41$ to $0.43,0.43$ to 0.45 , or 0.45 to 0.46 .

Nick Johnson

Numerade Educator

Problem 36

A corporation is considering a new issue of convertible bonds. Management believes that the offer terms will be found attractive by $20 \%$ of all its current stockholders. Suppose that this belief is correct. A random sample of 130 current stockholders is taken.
a. What is the standard error of the sample proportion who find this offer attractive?
b. What is the probability that the sample proportion is more than 0.15 ?
c. What is the probability that the sample proportion is between 0.18 and 0.22 ?
d. Suppose that a sample of 500 current stockholders had been taken. Without doing the calculations, state whether the probabilities in parts (b) and (c) would have been higher, lower, or the same as those found.

Nick Johnson

Numerade Educator

Problem 37

A store has determined that $30 \%$ of all lawn mower purchasers will also purchase a service agreement. In 1 month 280 lawn mowers are sold to customers, who can be regarded as a random sample of all purchasers.
a. What is the standard error of the sample proportion of those who will purchase a service agreement?
b. What is the probability that the sample proportion will be less than 0.32 ?
c. Without doing the calculations, state in which of the following ranges the sample proportion is most likely to be: 0.29 to $0.31,0.30$ to $0.32,0.31$ to 0.33 , or 0.32 to 0.34 .

Nick Johnson

Numerade Educator

Problem 38

A random sample of 100 voters is taken to estimate the proportion of a state's electorate in favor of increasing the gasoline tax to provide additional revenue for highway repairs. What is the largest value that the standard error of the sample proportion in favor of this measure can take?

Nick Johnson

Numerade Educator

Problem 39

In the previous exercise, suppose that it is decided that a sample of 100 voters is too small to provide a sufficiently reliable estimate of the population proportion. It is required instead that the probability that the sample proportion differs from the population proportion (whatever its value) by more than 0.03 should not exceed 0.05 . How large a sample is needed to guarantee that this requirement is met?

Nick Johnson

Numerade Educator

Problem 40

A company wants to estimate the proportion of people who are likely to purchase electric shavers from those who watch the nationally telecast baseball playoffs. A random sample obtained information from 120 people who were identified as persons who watch baseball telecasts. Suppose that the proportion of those likely to purchase electric shavers in the population who watch the telecast is 0.25 .
a. The probability is 0.10 that the sample proportion watching the telecast exceeds the population proportion by how much?
b. The probability is 0.05 that the sample proportion is lower than the population proportion by how much?
c. The probability is 0.30 that the sample proportion differs from the population proportion by how much?

Nick Johnson

Numerade Educator

Problem 41

Suppose that $44 \%$ of adult Australians believe that Australia should become a republic. Calculate the probability that more than $50 \%$ of a random sample of 100 adult Australians would believe this.

Nick Johnson

Numerade Educator

Problem 42

Suppose that $50 \%$ of adult Australians believe that Australia should apply to host the next rugby World Cup. Calculate the probability that more than $56 \%$ of a random sample of 150 adult Australians would believe this.

Nick Johnson

Numerade Educator

Problem 43

A journalist wanted to learn the views of the chief executive officers of the 500 largest U.S. corporations on program trading of stocks. In the time available, it was possible to contact only a random sample of 81 of these chief executive officers. If $55 \%$ of all the population members believe that program trading should be banned, what is the probability that less than half the sample members hold this view?

Nick Johnson

Numerade Educator

Problem 44

Forty percent of students at small colleges have brought their own personal computers to campus. A random sample of 120 entering freshmen was taken.
a. What is the standard error of the sample proportion bringing their own personal computers to campus?
b. What is the probability that the sample proportion is less than 0.33 ?
c. What is the probability that the sample proportion is between 0.38 and 0.46 ?

Nick Johnson

Numerade Educator

Problem 45

An employee survey conducted two years ago by Rice Motors, Inc., found that $53 \%$ of its employees were concerned about future health care benefits. A random sample of 80 of these employees were asked if they were now concerned about future health care benefits. Answer the following, assuming that there has been no change in the level of concern about health care benefits compared to the survey two years ago.
a. What is the standard error of the sample proportion who are concerned?
b. What is the probability that the sample proportion is less than 0.5 ?
c. What is the upper limit of the sample proportion such that only $3 \%$ of the time the sample proportion would exceed this value?

Nick Johnson

Numerade Educator

Problem 46

The annual percentage salary increases for the chief executive officers of all midsize corporations are normally distributed with mean $12.2 \%$ and standard deviation $3.6 \%$. A random sample of 81 of these chief executive officers was taken. What is the probability that more than half the sample members had salary increases of less than $10 \%$ ?

Nick Johnson

Numerade Educator

Problem 47

A random sample of size $n=16$ is obtained from a normally distributed population with a population mean of $\mu=100$ and a variance of $\sigma^2=25$.
a. What is the probability that $\bar{x}>101$ ?
b. What is the probability that the sample variance is greater than 45 ?
b. What is the value of the sample variance such that $5 \%$ of the sample variances would be less than this value?
c. What is the value of the sample variance such that $5 \%$ of the sample variances would be greater than this value?

Hoan Nguyen

Numerade Educator

Problem 49

A random sample of size $n=18$ is obtained from a normally distributed population with a population mean of $\mu=46$ and a variance of $\sigma^2=50$.
a. What is the probability that the sample mean is greater than 50 ?
b. What is the value of the sample variance such that $5 \%$ of the sample variances would be less than this value?
c. What is the value of the sample variance such that $5 \%$ of the sample variances would be greater than this value?

Nick Johnson

Numerade Educator

Problem 50

A process produces batches of a chemical whose impurity concentrations follow a normal distribution with a variance of 1.75 . A random sample of 20 of these batches is chosen. Find the probability that the sample variance exceeds 3.10 .

Nick Johnson

Numerade Educator

Problem 51

Monthly rates of return on the shares of a particular common stock are independent of one another and normally distributed with a standard deviation of 1.6. A sample of 12 months is taken.
a. Find the probability that the sample standard deviation is less than 2.5 .
b. Find the probability that the sample standard deviation is more than 1.0.

Nick Johnson

Numerade Educator

Problem 52

It is believed that first-year salaries for newly qualified accountants follow a normal distribution with a standard deviation of $$\$ 2,500$$. A random sample of 16 observations was taken.
a. Find the probability that the sample standard deviation is more than $$\$ 3,000$$.
b. Find the probability that the sample standard deviation is less than $$\$ 1,500$$.

Nick Johnson

Numerade Educator

Problem 53

A mathematics test of 100 multiple-choice questions is to be given to all freshmen entering a large university. Initially, in a pilot study the test was given to a random sample of 20 freshmen. Suppose that, for the population of all entering freshmen, the distribution of the number of correct answers would be normal with a variance of 250 .
a. What is the probability that the sample variance would be less than 100 ?
b. What is the probability that the sample variance would be more than 500 ?

Nick Johnson

Numerade Educator

Problem 54

In a large city it was found that summer electricity bills for single-family homes followed a normal distribution with a standard deviation of $$\$ 100$$. A random sample of 25 bills was taken.
a. Find the probability that the sample standard deviation is less than $$\$ 75$$.
b. Find the probability that the sample standard deviation is more than $$\$ 150$$.

Nick Johnson

Numerade Educator

Problem 55

The number of hours spent watching television by students in the week before final exams has a normal distribution with a standard deviation of 4.5 hours. A random sample of 30 students was taken.
a. Is the probability more than 0.95 that the sample standard deviation exceeds 3.5 hours?
b. Is the probability more than 0.95 that the sample standard deviation is less than 6 hours?

Nick Johnson

Numerade Educator

Problem 56

In Table 6.1 we considered the 15 possible samples of two observations from a population of $N=6$ values of years on the job for employees. The population variance for these six values is as follows:
$$
\sigma=\frac{47}{12}
$$
For each of the 15 possible samples, calculate the sample variance. Find the average of these 15 sample variances, thus confirming that the expected value of the sample variance is not equal to the population variance when the number of sample members is not a small proportion of the number of population members. In fact, as you can verify here,
$$
E\left[s^2\right]=N \sigma^2 /(N-1)
$$

Nick Johnson

Numerade Educator

Problem 57

A production process manufactures electronic components with timing signals whose duration follows a normal distribution. A random sample of six components was taken, and the durations of their timing signals were measured.
a. The probability is 0.05 that the sample variance is greater than what percentage of the population variance?
b. The probability is 0.10 that the sample variance is less than what percentage of the population variance?

Nick Johnson

Numerade Educator

Problem 58

A random sample of 10 stock market mutual funds was taken. Suppose that rates of returns on the population of all stock market mutual funds follow a normal distribution.
a. The probability is 0.10 that sample variance is greater than what percentage of the population variance?
b. Find any pair of numbers, $a$ and $b$, to complete the following sentence: The probability is 0.95 that the sample variance is between $a \%$ and $b \%$ of the population variance.
c. Suppose that a sample of 20 mutual funds had been taken. Without doing the calculations, indicate how this would change your answer to part (b).

Nick Johnson

Numerade Educator

Problem 59

Each member of a random sample of 15 business economists was asked to predict the rate of inflation for the coming year. Assume that the predictions for the whole population of business economists follow a normal distribution with standard deviation $1.8 \%$.
a. The probability is 0.01 that the sample standard deviation is bigger than what number?
b. The probability is 0.025 that the sample standard deviation is less than what number?
c. Find any pair of numbers such that the probability that the sample standard deviation that lies between these numbers is 0.90 .

Nick Johnson

Numerade Educator

Problem 60

A precision instrument is checked by making 12 readings on the same quantity. The population distribution of readings is normal.
a. The probability is 0.95 that the sample variance is more than what percentage of the population variance?
b. The probability is 0.90 that the sample variance is more than what percentage of the population variance?
c. Determine any pair of appropriate numbers, $a$ and $b$, to complete the following sentence: The probability is 0.95 that the sample variance is between $a \%$ and $b \%$ of the population variance.

Nick Johnson

Numerade Educator

Problem 61

A drug company produces pills containing an active ingredient. The company is concerned about the mean weight of this ingredient per pill, but it also requires that the variance (in squared milligrams) be no more than 1.5. A random sample of 20 pills is selected, and the sample variance is found to be 2.05 . How likely is it that a sample variance this high or higher would be found if the population variance is, in fact, 1.5? Assume that the population distribution is normal.

Nick Johnson

Numerade Educator

Problem 62

A manufacturer has been purchasing raw materials from a supplier whose consignments have a variance of 15.4 (in squared pounds) in impurity levels. A rival supplier claims that she can supply consignments of this raw material with the same mean impurity level but with lower variance. For a random sample of 25 consignments from the second supplier, the variance in impurity levels was found to be 12.2 . What is the probability of observing a value this low or lower for the sample variance if, in fact, the true population variance is 15.4? Assume that the population distribution is normal.

Nick Johnson

Numerade Educator

Problem 63

What is meant by the statement that the sample mean has a sampling distribution?

Nick Johnson

Numerade Educator

Problem 64

An investor is considering six different money market funds. The average number of days to maturity for each of these funds is as follows:
$41,39,35,35,33,38$
Two of these funds are to be chosen at random.
a. How many possible samples of two funds are there?
b. List all possible samples.
c. Find the probability function of the sampling distribution of the sample means.
d. Verify directly that the mean of the sampling distribution of the sample means is equal to the population mean.

Nick Johnson

Numerade Educator

Problem 65

Of what relevance is the central limit theorem to the sampling distribution of the sample means?

Nick Johnson

Numerade Educator

Problem 66

The scores of all applicants taking an aptitude test required by a law school have a normal distribution with a mean of 420 and a standard deviation of 100 . A random sample of 25 scores is taken.
a. Find the probability that the sample mean score is higher than 450 .
b. Find the probability that the sample mean score is between 400 and 450 .
c. The probability is 0.10 that the sample mean score is higher than what number?
d. The probability is 0.10 that the sample mean score is lower than what number?
e. The probability is 0.05 that the sample standard deviation of the scores is higher than what number?
f. The probability is 0.05 that the sample standard deviation of the scores is lower than what number?
g. If a sample of 50 test scores had been taken, would the probability of a sample mean score higher than
450 be smaller than, larger than, or the same as the correct answer to part (a)? It is not necessary to do the detailed calculations here. Sketch a graph to illustrate your reasoning.

Nick Johnson

Numerade Educator

Problem 67

A company services home air conditioners. It has been found that times for service calls follow a Normal distribution with a mean of 60 minutes and a standard deviation of 10 minutes. A random sample of four service calls was taken.
a. What is the probability that the sample mean service time is more than 65 minutes?
b. The probability is 0.10 that the sample mean service time is less than how many minutes?
c. The probability is 0.10 that the sample standard deviation of service times is more than how many minutes?
d. The probability is 0.10 that the sample standard deviation of service times is less than how many minutes?
e. What is the probability that more than two of these calls take more than 65 minutes?

Nick Johnson

Numerade Educator

Problem 68

In a particular year, the percentage rates of return of U.S. common stock mutual funds had a normal distribution with a mean of 14.8 and a standard deviation of 6.3. A random sample of nine of these mutual funds was taken.
a. What is the probability that the sample mean percentage rate of return is more than 19.0 ?
b. What is the probability that the sample mean percentage rate of return is between 10.6 and 19.0 ?
c. The probability is 0.25 that the sample mean percentage return is less than what number?
d. The probability is 0.10 that the sample standard deviation of percentage return is more than what number?
e. If a sample of $\mathbf{2 0}$ of these funds was taken, state whether the probability of a sample mean percentage rate of return of more than 19.0 would be smaller than, larger than, or the same as the correct answer to part (a). Sketch a graph to illustrate your reasoning.

Nick Johnson

Numerade Educator

Problem 69

The lifetimes of a certain electronic component are known to be normally distributed with a mean of 1,600 hours and a standard deviation of 400 hours.
a. For a random sample of 16 components, find the probability that the sample mean is more than 1,500 hours.
b. For a random sample of 16 components, the probability is 0.15 that the sample mean lifetime is more than how many hours?
c. For a random sample of 16 components, the probability is 0.10 that the sample standard deviation lifetime is more than how many hours?

Nick Johnson

Numerade Educator

Problem 70

Refer to the chapter appendix in order to derive the mean of the sampling distribution of the sample variances for a sample of $n$ observations from a population of $N$ members when the population variance is $\sigma^2$. By appropriately modifying the argument regarding variances in the chapter appendix, show that
$$
E\left[s^2\right]=N \sigma^2 /(N-1)
$$
Note the intuitive plausibility of this result when $n=N$

Nick Johnson

Numerade Educator

Problem 71

It has been found that times taken by people to complete a particular tax form follow a normal distribution with a mean of 100 minutes and a standard deviation of 30 minutes. A random sample of nine people who have completed this tax form was taken.
a. What is the probability that the sample mean time taken is more than 120 minutes?
b. The probability is 0.20 that the sample mean time taken is less than how many minutes?
c. The probability is 0.05 that the sample standard deviation of time taken is less than how many minutes?

Nick Johnson

Numerade Educator

Problem 72

It was found that $80 \%$ of seniors at a particular college had accepted a job offer before graduation. For those accepting offers, salary distribution was normal with a mean of $$\$ 37,000$$ and a standard deviation of $$\$ 4,000$$.
a. For a random sample of 60 seniors what is the probability that less than $70 \%$ have accepted job offers?
b. For a random sample of 6 seniors, what is the probability that less than $70 \%$ have accepted job offers?
c. For a random sample of 6 seniors who have accepted job offers, what is the probability that the average salary is more than $$\$ 38,000$$ ?
d. A senior is chosen at random. What is the probability that she has accepted a job offer with a salary of more than $$\$ 38,000$$ ?

Nick Johnson

Numerade Educator

Problem 73

Plastic bags used for packaging produce are manufactured so that the breaking strengths of the bags are normally distributed with a standard deviation of 1.8 pounds per square inch. A random sample of $16 \mathrm{bags}$ is selected.
a. The probability is 0.01 that the sample standard deviation of breaking strengths exceeds what number?
b. The probability is 0.15 that the sample mean exceeds the population mean by how much?
c. The probability is 0.05 that the sample mean differs from the population mean by how much?

Nick Johnson

Numerade Educator

Problem 74

A quality-control manager was concerned about variability in the amount of an active ingredient in pills produced by a particular process. A random sample of 21 pills was taken. What is the probability that the sample variance of the amount of an active ingredient was more than twice the population variance?

Nick Johnson

Numerade Educator

Problem 75

A sample of 100 students is to be taken to determine which of two brands of beer is preferred in a blind taste test. Suppose that, in the whole population of students, $50 \%$ would prefer brand $\mathrm{A}$.
a. What is the probability that more than $60 \%$ of the sample members prefer brand A?
b. What is the probability that between $45 \%$ and $55 \%$ of the sample members prefer brand A?
c. Suppose that a sample of only 10 students was available. Indicate how the method of calculation of probabilities would differ, compared with your solutions to parts (a) and (b)?

Nick Johnson

Numerade Educator

Problem 76

Scores on a particular test, taken by a large group of students, follow a normal distribution with a standard deviation of 40 points. A random sample of 16 scores was taken to estimate the population mean score. Let the random variable $\bar{x}$ denote the sample mean. What is the probability that the interval $(\bar{x}-10)$ to $(\bar{x}+10)$ contains the true population mean?

Hoan Nguyen

Numerade Educator

Problem 77

A manufacturer of liquid detergent claims that the mean weight of liquid in containers sold is at least 30 ounces. It is known that the population distribution of weights is normal with a standard deviation of 1.3 ounces. In order to check the manufacturer's claim, a random sample of 16 containers of detergent is examined. The claim will be questioned if the sample mean weight is less than 29.5 ounces. What is the probability that the claim will be questioned if, in fact, the population mean weight is 30 ounces?

Nick Johnson

Numerade Educator

Problem 78

In a particular year $40 \%$ of home sales were partially financed by the seller. A random sample of 250 sales is examined.
a. The probability is 0.8 that the sample proportion is more than what amount?
b. The probability is 0.9 that the sample proportion is less than what amount?
c. The probability is 0.7 that the sample proportion differs from the population proportion by how much?

Nick Johnson

Numerade Educator

Problem 79

A candidate for office intends to campaign in a state if her initial support level exceeds $30 \%$ of the voters. A random sample of 300 voters is taken, and it is decided to campaign if the sample proportion supporting the candidate exceeds 0.28 .
a. What is the probability of a decision not to campaign if, in fact, the initial support level is $20 \%$ ?
b. What is the probability of a decision not to campaign if, in fact, the initial support level is $40 \%$ ?

Nick Johnson

Numerade Educator

Problem 80

It is known that the incomes of subscribers to a particular magazine have a normal distribution with a standard deviation of $$\$ 6,600$$. A random sample of 25 subscribers is taken.
a. What is the probability that the sample standard deviation of their incomes is more than $$\$ 4,000$$ ?
b. What is the probability that the sample standard deviation of their incomes is less than $$\$ 8,000$$ ?

Nick Johnson

Numerade Educator

Problem 81

Batches of chemical are manufactured by a production process. Samples of 20 batches from a production run are selected for testing. If the standard deviation of the percentage of impurity contents in the sample batches exceeds $2.5 \%$, the production process is thoroughly checked. Assume that the population distribution of percentage impurity concentrations is normal. What is the probability that the production process will be thoroughly checked if the population standard deviation of percentage impurity concentrations is $2 \%$ ?

Nick Johnson

Numerade Educator

Problem 82

A consumer product that has flourished in the last few years is bottled natural spring water. Jon Thorne is the CEO of a company that sells natural spring water. He has requested a report of the filling process of the 24-ounce (710-milliliter) bottles to be sure that they are being properly filled. To check if the process needs to be adjusted, Emma Astrom, who monitors the process, randomly samples and weighs five bottles every 15 minutes for a 5 -hour period. The data are contained in the data file Bottles.
a. Compute the sample mean, sample standard deviations for individual bottles, and the standard deviation of the sample mean for each sample.
b. Determine the probability that the sample means are below 685 milliliters if the population mean is 710 .
c. Determine the probability that the sample means are above 720 milliliters.

Nick Johnson

Numerade Educator

Problem 83

Prairie Flower Cereal, Inc., is a small but growing producer of hot and ready-to-eat breakfast cereals. The company was started in 1910 by Gordon Thorson, a successful grain farmer. You have been asked to test the cereal-packing process of 18-ounce (510-gram) boxes of sugar-coated wheat cereal. Two machines are used for the packaging process. Twenty samples of five packages each are randomly sampled and weighed. The data are contained in the file Sugar Coated Wheat.
a. Compute the overall sample mean, sample variance, and variance of the sample means for each machine.
b. Determine the probability that a single sample mean is below 500 if the process is operating properly for each machine.
c. Determine the probability that a single sample mean is above 508 if the process is operating properly for each machine.
d. Using your statistical computer package, obtain 20 random samples of size $n=5$ packages for each machine and compute the sample mean for each sample. Count the number of sample means that are below 500 and the number that are above 508 .

Nick Johnson

Numerade Educator

Problem 84

Another product packaged by Prairie Flower Cereal, Inc., is an apple-cinnamon cereal. To test the packaging process of 40-ounce (1,134-gram) packages of this cereal, 23 samples of six packages each are randomly sampled and weighed. The lower and upper acceptance limits have been set at 1,120 grams and 1,150 grams, respectively. The data are contained in the data file Granola.
a. Compute the overall sample mean, sample variance, and variance of the sample means for each sample.
b. Compute the probability that the sample means will be within the acceptance limits.
c. Using your statistical computer package, obtain 23 random samples of size $n=6$ and compute the sample mean for each sample. Count the number of sample means that are outside the acceptance limits.

Nick Johnson

Numerade Educator