Understandable Statistics: Concepts And Methods - 12e Chapter 7, Estimation WebAssign

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MATH1-UC 1172, section 002, Spring 2021

L8_HMW_SPRING_2021 (Homework) INSTRUCTOR

Olivia Mendivil Ramos New York University

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For this assignment, you submit answers by question parts. The number of submissions remaining for each question part only changes if you submit or change the answer.

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Use the Student's t distribution to find t for a 0.95 confidence level when the sample is 21. (Round your answer to three decimal places.)

2.086

1. [1/1 Points] BBUNDERSTAT12 7.2.001.MI.S.DETAILS PREVIOUS ANSWERS

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The method of tree-ring dating gave the following years A.D. for an archaeological excavation site. Assume that the population of x values has an approximately normal distribution.

1,222 1,229 1,236 1,194 1,268 1,316 1,275 1,317 1,275

(a) Use a calculator with mean and standard deviation keys to find the sample mean year x and sample standard deviation s. (Round your answers to four decimal places.)

= A.D.

s = yr

(b) When finding an 90% confidence interval, what is the critical value for confidence level? (Give your answer to three decimal places.)

What is the maximal margin of error when finding a 90% confidence interval for the mean of all tree-ring dates from this archaeological site? (Round your answer to the nearest whole number.)

Find a 90% confidence interval for the mean of all tree-ring dates from this archaeological site. (Round your answers to the nearest whole number.)

lower limit A.D.

upper limit A.D.

t = c

E =

2. [–/6 Points] BBUNDERSTAT12 7.2.013.EP.S.DETAILS

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Jobs and productivity! How do banks rate? One way to answer this question is to examine annual profits per employee. The following is data about annual profits per employee (in units of 1 thousand dollars per employee) for representative companies in financial services. Assume

≈ 10.6 thousand dollars. 27.8 32.0 44.6 37.6 54.4 35.5 39.7 36.9 42.5 33.0 33.6

36.9 27.0 47.1 33.8 28.1 28.5 29.1 36.5 36.1 26.9 27.8

28.8 29.3 31.5 31.7 31.1 38.0 32.0 31.7 32.9 23.1 54.9

43.8 36.9 31.9 25.5 23.2 29.8 22.3 26.5 26.7

(a) Use a calculator or appropriate computer software to find x for the preceding data. (Round your answer to two decimal places.)

thousand dollars

(b) Let us say that the preceding data are representative of the entire sector of (successful) financial services corporations. Find a

75% confidence interval for , the average annual profit per employee for all successful banks. (Round your answers to two decimal places.)

lower limit thousand dollars

upper limit thousand dollars

(c) Let us say that you are the manager of a local bank with a large number of employees. Suppose the annual profits per employee are less than 30 thousand dollars per employee. Do you think this might be somewhat low compared with other successful financial institutions? Explain by referring to the confidence interval you computed in part (b).

Yes. This confidence interval suggests that the bank profits are less than those of other financial institutions.

Yes. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.

No. This confidence interval suggests that the bank profits are less than those of other financial institutions.

No. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.

3. [–/9 Points] BBUNDERSTAT12 7.1.023.DETAILS

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(d) Suppose the annual profits are more than 40 thousand dollars per employee. As manager of the bank, would you feel somewhat better? Explain by referring to the confidence interval you computed in part (b).

No. This confidence interval suggests that the bank profits are higher than those of other financial institutions.

No. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.

Yes. This confidence interval suggests that the bank profits are higher than those of other financial institutions.

Yes. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.

(e) Find a 90% confidence interval for , the average annual profit per employee for all successful banks. (Round your answers to two decimal places.)

lower limit thousand dollars

upper limit thousand dollars

(f) Let us say that you are the manager of a local bank with a large number of employees. Suppose the annual profits per employee are less than 30 thousand dollars per employee. Do you think this might be somewhat low compared with other successful financial institutions? Explain by referring to the confidence interval you computed in part (e).

Yes. This confidence interval suggests that the bank profits are less than those of other financial institutions.

Yes. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.

No. This confidence interval suggests that the bank profits are less than those of other financial institutions.

No. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.

(g) Suppose the annual profits are more than 40 thousand dollars per employee. As manager of the bank, would you feel somewhat better? Explain by referring to the confidence interval you computed in part (e).

No. This confidence interval suggests that the bank profits are higher than those of other financial institutions.

No. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.

Yes. This confidence interval suggests that the bank profits are higher than those of other financial institutions.

Yes. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.

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True or false? If the sample mean x of a random sample from an x distribution is relatively small, when the confidence level c is reduced, the

confidence interval for becomes shorter. True. As the level of confidence decreases, the maximal error of estimate increases.

False. As the level of confidence decreases, the maximal error of estimate decreases.

True. As the level of confidence decreases, the maximal error of estimate decreases.

False. As the level of confidence decreases, the maximal error of estimate increases.

4. [–/1 Points] BBUNDERSTAT12 7.1.008.DETAILS MY NOTES ASK YOUR TEACHER

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True or false? The value z is a value from the standard normal distribution such that P(–z < z < z ) = c.

False. By definition, critical values z are such that 100c% of the area under the standard normal curve falls between –z and z .

False. By definition, critical values z are such that 100c% of the area under the standard normal curve falls in the tails, to the left of

–z and to the right of z .

True. By definition, critical values z are such that 100c% of the area under the standard normal curve falls in the tails, to the left of

–z and to the right of z .

True. By definition, critical values z are such that 100c% of the area under the standard normal curve falls between –z and z .

c c c

c c

c c c

5. [–/1 Points] BBUNDERSTAT12 7.1.001.DETAILS MY NOTES ASK YOUR TEACHER

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This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part.

Tutorial Exercise Thirty-three small communities in Maine (population near 10,000 each) gave an average of x = 124.4 reported cases of larceny per year. Assume that is known to be 40.6 cases per year.

(a) Find a 90% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error?

(b) Find a 95% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error?

(c) Find a 99% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error?

(d) Compare the margins of error for parts (a) through (c). As the confidence levels increase, do the margins of error increase?

(e) Compare the lengths of the confidence intervals for parts (a) through (c). As the confidence levels increase, do the confidence intervals increase in length?

6. [–/20 Points] BBUNDERSTAT12 7.1.019.MI.SA.DETAILS

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True or false? Every random sample of the same size from a given population will produce exactly the same confidence interval for . False. Different random samples may produce different x values, resulting in the same confidence intervals.

False. Different random samples may produce different x values, resulting in different confidence intervals.

True. Different random samples will produce the same x values, resulting in the same confidence intervals.

True. Different random samples may produce different x values, resulting in different confidence intervals.

7. [–/1 Points] BBUNDERSTAT12 7.1.004.DETAILS MY NOTES ASK YOUR TEACHER

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Overproduction of uric acid in the body can be an indication of cell breakdown. This may be an advance indication of illness such as gout, leukemia, or lymphoma. Over a period of months, an adult male patient has taken eleven blood tests for uric acid. The mean concentration was x = 5.45 mg/dl. The distribution of uric acid in healthy adult males can be assumed to be normal, with = 1.91 mg/dl.

(a) Find a 95% confidence interval for the population mean concentration of uric acid in this patient's blood. What is the margin of error? (Round your answers to two decimal places.)

lower limit

upper limit

margin of error

(b) What conditions are necessary for your calculations? (Select all that apply.)

uniform distribution of uric acid

is known

is unknown

n is large

normal distribution of uric acid

We are 95% confident that the true uric acid level for this patient falls within this interval.

The probability that this interval contains the true average uric acid level for this patient is 0.95.

The probability that this interval contains the true average uric acid level for this patient is 0.05.

We are 5% confident that the true uric acid level for this patient falls within this interval.

(d) Find the sample size necessary for a 95% confidence level with maximal margin of error E = 1.02 for the mean concentration of uric acid in this patient's blood. (Round your answer up to the nearest whole number.)

†

8. [–/6 Points] BBUNDERSTAT12 7.1.016.MI.DETAILS

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blood tests

Lorraine computed a confidence interval for based on a sample of size 41. Since she did not know , she used s in her calculations. Lorraine used the normal distribution for the confidence interval instead of a Student's t distribution. Will her interval be longer or shorter than one obtained by using an appropriate Student's t distribution? Explain.

Shorter. For d.f. = 40, z is less than t .

Longer. For d.f. = 40, z is less than t .

Shorter. For d.f. = 40, z is greater than t .

Longer. For d.f. = 40, z is greater than t .

c c

9. [–/1 Points] BBUNDERSTAT12 7.2.009.S.DETAILS MY NOTES ASK YOUR TEACHER

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Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 17 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the

weights of Allen's hummingbirds have a normal distribution, with = 0.36 gram.

(a) Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.)

lower limit

upper limit

margin of error

(b) What conditions are necessary for your calculations? (Select all that apply.)

is unknown

n is large

is known

normal distribution of weights

uniform distribution of weights

We are 20% confident that the true average weight of Allen's hummingbirds falls within this interval.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.

We are 80% confident that the true average weight of Allen's hummingbirds falls within this interval.

10. [–/6 Points] BBUNDERSTAT12 7.1.015.DETAILS

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(d) Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.14 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)

hummingbirds

Tutorial Exercise The method of tree-ring dating gave the following years A.D. for an archaeological excavation site. Assume that the population of x values has an approximately normal distribution.

1,215 1,215 1,187 1,243 1,268 1,316 1,275 1,317 1,275

(a) Use a calculator with mean and standard deviation keys to find the sample mean year x and sample standard deviation s.

(b) Find a 90% confidence interval for the mean of all tree-ring dates from this archaeological site.

11. [–/11 Points] BBUNDERSTAT12 7.2.013.MI.SA.DETAILS

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Tutorial Exercise Over the past several months, an adult patient has been treated for tetany (severe muscle spasms). This condition is associated with an average total calcium level below 6 mg/dl. Recently, the patient's total calcium tests gave the following readings (in mg/dl). Assume that the population of x values has an approximately normal distribution.

9.40 8.30 10.60 8.40 9.40 9.80 10.00 9.90 11.20 12.10

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean reading x and the sample standard deviation s.

(b) Find a 99.9% confidence interval for the population mean of total calcium in this patient's blood.

12. [–/14 Points] BBUNDERSTAT12 7.2.017.MI.SA.DETAILS

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The home run percentage is the number of home runs per 100 times at bat. A random sample of 43 professional baseball players gave the following data for home run percentages.

1.6 2.4 1.2 6.6 2.3 0.0 1.8 2.5 6.5 1.8

2.7 2.0 1.9 1.3 2.7 1.7 1.3 2.1 2.8 1.4

3.8 2.1 3.4 1.3 1.5 2.9 2.6 0.0 4.1 2.9

1.9 2.4 0.0 1.8 3.1 3.8 3.2 1.6 4.2 0.0

1.2 1.8 2.4

(a) Use a calculator with mean and standard deviation keys to find x and s. (Round your answers to four decimal places.)

x = %

s = %

(b) Compute a 90% confidence interval for the population mean of home run percentages for all professional baseball players. Hint: If you use the Student's t distribution table, be sure to use the closest d.f. that is smaller. (Round your answers to two decimal places.)

lower limit %

upper limit %

(c) Compute a 99% confidence interval for the population mean of home run percentages for all professional baseball players. (Round your answers to two decimal places.)

lower limit %

upper limit %

13. [–/8 Points] BBUNDERSTAT12 7.2.022.S.DETAILS

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(d) The home run percentages for three professional players are below.

Player A, 2.5 Player B, 2.3 Player C, 3.8

Examine your confidence intervals and describe how the home run percentages for these players compare to the population average. We can say Player A falls close to the average, Player B is above average, and Player C is below average.

We can say Player A falls close to the average, Player B is below average, and Player C is above average.

We can say Player A and Player B fall close to the average, while Player C is above average.

We can say Player A and Player B fall close to the average, while Player C is below average.

(e) In previous problems, we assumed the x distribution was normal or approximately normal. Do we need to make such an assumption in this problem? Why or why not? Hint: Use the central limit theorem.

Yes. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.

Yes. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal.

No. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.

No. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal.

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Sam computed a 90% confidence interval for from a specific random sample of size n. He claims that at the 90% confidence level, his

confidence interval contains . Is this claim correct? Explain.

No. The probability that this interval contains is either 0 or 1.

No. The proportion of all confidence intervals based on random samples of size n that contain is 0.10.

Yes. The proportion of all confidence intervals based on random samples of size n that contain is 0.90.

Yes. 90% of all confidence intervals will contain .

14. [–/1 Points] BBUNDERSTAT12 7.1.010.DETAILS MY NOTES ASK YOUR TEACHER

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The method of tree-ring dating gave the following years A.D. for an archaeological excavation site. Assume that the population of x values has an approximately normal distribution.

1,222 1,208 1,208 1,180 1,268 1,316 1,275 1,317 1,275

(a) Use a calculator with mean and standard deviation keys to find the sample mean year x and sample standard deviation s. (Round your answers to four decimal places.)

= A.D.

s = yr

(b) Find a 90% confidence interval for the mean of all tree-ring dates from this archaeological site. (Round your answers to the nearest whole number.)

lower limit A.D.

upper limit A.D.

15. [–/4 Points] BBUNDERSTAT12 7.2.013.MI.S.DETAILS

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Use the Student's t distribution to find t for a 0.95 confidence level when the sample is 3. (Round your answer to three decimal places.)

Use the Student's t distribution to find t for a 0.90 confidence level when the sample is 15. (Round your answer to three decimal places.)

16. [–/1 Points] BBUNDERSTAT12 7.2.004.S.DETAILS

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17. [–/1 Points] BBUNDERSTAT12 7.2.003.S.DETAILS

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True or false? If the original x distribution has a relatively small standard deviation, the confidence interval for will be relatively short.

True. As decreases, E decreases, resulting in a shorter confidence interval.

True. As decreases, E increases, resulting in a shorter confidence interval.

False. As decreases, E decreases, resulting in a longer confidence interval.

False. As decreases, E increases, resulting in a longer confidence interval.

18. [–/1 Points] BBUNDERSTAT12 7.1.006.DETAILS MY NOTES ASK YOUR TEACHER

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Tutorial Exercise What price do farmers get for their watermelon crops? In the third week of July, a random sample of 35 farming regions gave a sample

mean of x = $6.88 per 100 pounds of watermelon. Assume that is known to be $1.88 per 100 pounds.

(a) Find a 90% confidence interval for the population mean price (per 100 pounds) that farmers in this region get for their watermelon crop. What is the margin of error? (b) Find the sample size necessary for a 90% confidence level with maximal error of estimate E = 0.41 for the mean price per 100 pounds of watermelon. (c) A farm brings 15 tons of watermelon to market. Find a 90% confidence interval for the population mean cash value of this crop. What is the margin of error? Hint: 1 ton is 2000 pounds.

19. [–/14 Points] BBUNDERSTAT12 7.1.018.MI.SA.DETAILS

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Thirty small communities in Connecticut (population near 10,000 each) gave an average of x = 138.5 reported cases of larceny per year. Assume that is known to be 44.9 cases per year.

(a) Find a 90% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

lower limit

upper limit

margin of error

(b) Find a 95% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

lower limit

upper limit

margin of error

(c) Find a 99% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

lower limit

upper limit

margin of error

(d) Compare the margins of error for parts (a) through (c). As the confidence levels increase, do the margins of error increase?

As the confidence level increases, the margin of error remains the same.

As the confidence level increases, the margin of error decreases.

As the confidence level increases, the margin of error increases.

20. [–/11 Points] BBUNDERSTAT12 7.1.019.MI.DETAILS

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(e) Compare the lengths of the confidence intervals for parts (a) through (c). As the confidence levels increase, do the confidence intervals increase in length?

As the confidence level increases, the confidence interval decreases in length.

As the confidence level increases, the confidence interval remains the same length.

As the confidence level increases, the confidence interval increases in length.

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Tutorial Exercise Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 43 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that = 7.50 ml/kg for the distribution of blood plasma.

(a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error?

(b) What conditions are necessary for your calculations?

(d) Find the sample size necessary for a 99% confidence level with maximal margin of error E = 2.10 for the mean plasma volume in male firefighters.

21. [–/11 Points] BBUNDERSTAT12 7.1.017.MI.SA.DETAILS

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Tutorial Exercise Overproduction of uric acid in the body can be an indication of cell breakdown. This may be an advance indication of illness such as gout, leukemia, or lymphoma. Over a period of months, an adult male patient has taken eight blood tests for uric acid. The mean concentration was x = 6.11 mg/dl. The distribution of uric acid in healthy adult males can be assumed to be normal, with = 1.96 mg/dl.

(a) Find a 95% confidence interval for the population mean concentration of uric acid in this patient's blood. What is the margin of error?

(b) What conditions are necessary for your calculations?

(d) Find the sample size necessary for a 95% confidence level with maximal margin of error E = 1.15 for the mean concentration of uric acid in this patient's blood.

†

22. [–/11 Points] BBUNDERSTAT12 7.1.016.MI.SA.DETAILS

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How hot is the air in the top (crown) of a hot air balloon? Information from Ballooning: The Complete Guide to Riding the Winds, by Wirth and Young (Random House), claims that the air in the crown should be an average of 100°C for a balloon to be in a state of equilibrium. However, the temperature does not need to be exactly 100°C. What is a reasonable and safe range of temperatures? This range may vary with the size and (decorative) shape of the balloon. All balloons have a temperature gauge in the crown. Suppose that 51 readings (for a

balloon in equilibrium) gave a mean temperature of x = 97°C. For this balloon, ≈ 17°C.

(a) Compute a 95% confidence interval for the average temperature at which this balloon will be in a steady-state equilibrium. (Round your answers to one decimal place.)

lower limit °C

upper limit °C

(b) If the average temperature in the crown of the balloon goes above the high end of your confidence interval, do you expect that the balloon will go up or down? Explain.

It will go up because hot air will make the balloon fall.

It will go down because hot air will make the balloon rise.

It will go up because hot air will make the balloon rise.

It will go down because hot air will make the balloon fall.

23. [–/3 Points] BBUNDERSTAT12 7.1.025.DETAILS

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What are the degrees of freedom for Student's t distribution when the sample size is 18?

d.f. =

Use the Student's t distribution to find t for a 0.95 confidence level when the sample is 18. (Round your answer to three decimal places.)

24. [–/2 Points] BBUNDERSTAT12 7.2.001.EP.S.DETAILS

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The price of a share of stock divided by the company's estimated future earnings per share is called the P/E ratio. High P/E ratios usually indicate "growth" stocks, or maybe stocks that are simply overpriced. Low P/E ratios indicate "value" stocks or bargain stocks. A random sample of 51 of the largest companies in the United States gave the following P/E ratios†.

11 35 19 13 15 21 40 18 60 72 9 20

29 53 16 26 21 14 21 27 10 12 47 14

33 14 18 17 20 19 13 25 23 27 5 16

8 49 44 20 27 8 19 12 31 67 51 26

19 18 32

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean x and sample standard deviation s. (Round your answers to four decimal places.)

x =

s =

(b) Find a 90% confidence interval for the P/E population mean of all large U.S. companies. (Round your answers to one decimal place.)

lower limit

upper limit

(c) Find a 99% confidence interval for the P/E population mean of all large U.S. companies. (Round your answers to one decimal place.)

lower limit

25. [–/8 Points] BBUNDERSTAT12 7.2.021.S.DETAILS

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upper limit

(d) Bank One (now merged with J. P. Morgan) had a P/E of 12, AT&T Wireless had a P/E of 72, and Disney had a P/E of 24. Examine the confidence intervals in parts (b) and (c). How would you describe these stocks at the time the sample was taken?

We can say Bank One is above average, AT&T Wireless is below average, and Disney falls close to the average.

We can say Bank One is below average, AT&T Wireless is above average, and Disney falls close to the average.

We can say Bank One is below average, AT&T Wireless is above average, and Disney is above average.

We can say Bank One is below average, AT&T Wireless is above average, and Disney is below average.

(e) In previous problems, we assumed the x distribution was normal or approximately normal. Do we need to make such an assumption in this problem? Why or why not? Hint: Use the central limit theorem.

Yes. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal.

No. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.

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Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 42 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight).

Assume that = 7.30 ml/kg for the distribution of blood plasma.

(a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.)

lower limit

upper limit

margin of error

(b) What conditions are necessary for your calculations? (Select all that apply.)

n is large

the distribution of volumes is uniform

is unknown

the distribution of volumes is normal

is known

We are 99% confident that the true average blood plasma volume in male firefighters falls within this interval.

The probability that this interval contains the true average blood plasma volume in male firefighters is 0.01.

We are 1% confident that the true average blood plasma volume in male firefighters falls within this interval.

The probability that this interval contains the true average blood plasma volume in male firefighters is 0.99.

26. [–/6 Points] BBUNDERSTAT12 7.1.017.DETAILS

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(d) Find the sample size necessary for a 99% confidence level with maximal margin of error E = 3.00 for the mean plasma volume in male firefighters. (Round up to the nearest whole number.)

male firefighters

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Tutorial Exercise How much does a sleeping bag cost? Let's say you want a sleeping bag that should keep you warm in temperatures from 20°F to 45°F. A random sample of prices ($) for sleeping bags in this temperature range is given below. Assume that the population of x values has an approximately normal distribution.

105 101 77 73 111 69 30 23 100 110

105 95 105 60 110 120 95 90 60 70

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean price x and sample standard deviation s.

(b) Using the given data as representative of the population of prices of all summer sleeping bags, find a 90% confidence interval for the mean price of all summer sleeping bags.

27. [–/12 Points] BBUNDERSTAT12 7.2.014.MI.SA.DETAILS

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Over the past several months, an adult patient has been treated for tetany (severe muscle spasms). This condition is associated with an average total calcium level below 6 mg/dl. Recently, the patient's total calcium tests gave the following readings (in mg/dl). Assume that the population of x values has an approximately normal distribution.

9.90 9.40 10.90 9.50 9.40 9.80 10.00 9.90 11.20 12.10

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean reading x and the sample standard deviation s. (Round your answers to four decimal places.)

x = mg/dl

s = mg/dl

(b) Find a 99.9% confidence interval for the population mean of total calcium in this patient's blood. (Round your answer to two decimal places.)

lower limit mg/dl

upper limit mg/dl

Yes. This confidence interval suggests that the patient may still have a calcium deficiency.

Yes. This confidence interval suggests that the patient no longer has a calcium deficiency.

No. This confidence interval suggests that the patient may still have a calcium deficiency.

No. This confidence interval suggests that the patient no longer has a calcium deficiency.

28. [–/5 Points] BBUNDERSTAT12 7.2.017.MI.S.DETAILS

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True or false? A larger sample size produces a longer confidence interval for . False. As the sample size increases, the maximal error decreases, resulting in a shorter confidence interval.

True. As the sample size increases, the maximal error decreases, resulting in a longer confidence interval.

True. As the sample size increases, the maximal error increases, resulting in a longer confidence interval.

False. As the sample size increases, the maximal error increases, resulting in a shorter confidence interval.

29. [–/1 Points] BBUNDERSTAT12 7.1.005.DETAILS MY NOTES ASK YOUR TEACHER

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Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther. Suppose a small group of 11 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with = 0.28 gram.

(a) Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.)

lower limit

upper limit

margin of error

(b) What conditions are necessary for your calculations? (Select all that apply.)

is known

uniform distribution of weights

normal distribution of weights

n is large

is unknown

We are 20% confident that the true average weight of Allen's hummingbirds falls within this interval.

We are 80% confident that the true average weight of Allen's hummingbirds falls within this interval.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.

(d) Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.13 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)

†

30. [–/6 Points] BBUNDERSTAT12 7.1.015.FB.DETAILS

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hummingbirds

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The following data represent crime rates per 1000 population for a random sample of 46 Denver neighborhoods.†

63.2 36.3 26.2 53.2 65.3 32.0 65.0

66.3 68.9 35.2 25.1 32.5 54.0 42.4

77.5 123.2 66.3 92.7 56.9 77.1 27.5

69.2 73.8 71.5 58.5 67.2 78.6 33.2

74.9 45.1 132.1 104.7 63.2 59.6 75.7

39.2 69.9 87.5 56.0 154.2 85.5 77.5

84.7 24.2 37.5 41.1

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean x and sample standard deviation s. (Round your answers to four decimal places.)

x = crimes per 1000 people

s = crimes per 1000 people

(b) Let us say the preceding data are representative of the population crime rates in Denver neighborhoods. Compute an 80%

confidence interval for , the population mean crime rate for all Denver neighborhoods. (Round your answers to one decimal place.)

lower limit crimes per 1000 people

upper limit crimes per 1000 people

(c) Suppose you are advising the police department about police patrol assignments. One neighborhood has a crime rate of 56 crimes per 1000 population. Do you think that this rate is below the average population crime rate and that fewer patrols could safely be assigned to this neighborhood? Use the confidence interval to justify your answer.

Yes. The confidence interval indicates that this crime rate is below the average population crime rate.

31. [–/11 Points] BBUNDERSTAT12 7.2.020.S.DETAILS

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Yes. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

No. The confidence interval indicates that this crime rate is below the average population crime rate.

No. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

(d) Another neighborhood has a crime rate of 73 crimes per 1000 population. Does this crime rate seem to be higher than the population average? Would you recommend assigning more patrols to this neighborhood? Use the confidence interval to justify your answer.

Yes. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

Yes. The confidence interval indicates that this crime rate is higher than the average population crime rate.

No. The confidence interval indicates that this crime rate is higher than the average population crime rate.

No. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

(e) Compute a 95% confidence interval for , the population mean crime rate for all Denver neighborhoods. (Round your answers to one decimal place.)

lower limit crimes per 1000 people

upper limit crimes per 1000 people

(f) Suppose you are advising the police department about police patrol assignments. One neighborhood has a crime rate of 56 crimes per 1000 population. Do you think that this rate is below the average population crime rate and that fewer patrols could safely be assigned to this neighborhood? Use the confidence interval to justify your answer.

Yes. The confidence interval indicates that this crime rate is below the average population crime rate.

Yes. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

No. The confidence interval indicates that this crime rate is below the average population crime rate.

No. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

(g) Another neighborhood has a crime rate of 73 crimes per 1000 population. Does this crime rate seem to be higher than the population average? Would you recommend assigning more patrols to this neighborhood? Use the confidence interval to justify your answer.

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Yes. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

Yes. The confidence interval indicates that this crime rate is higher than the average population crime rate.

No. The confidence interval indicates that this crime rate is higher than the average population crime rate.

No. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

(h) In previous problems, we assumed the x distribution was normal or approximately normal. Do we need to make such an assumption in this problem? Why or why not? Hint: Use the central limit theorem.

Yes. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.

Yes. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal.

No. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.

No. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal.

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At wind speeds above 1000 centimeters per second (cm/sec), significant sand-moving events begin to occur. Wind speeds below 1000 cm/sec deposit sand and wind speeds above 1000 cm/sec move sand to new locations. The cyclic nature of wind and moving sand determines the shape and location of large dunes. At a test site, the prevailing direction of the wind did not change noticeably. However, the velocity did change. Sixty-one wind speed readings gave an average velocity of x = 1075 cm/sec. Based on long-term experience, can be assumed to be 230 cm/sec.

(a) Find a 95% confidence interval for the population mean wind speed at this site. (Round your answers to the nearest whole number.)

lower limit cm/sec

upper limit cm/sec

(b) Does the confidence interval indicate that the population mean wind speed is such that the sand is always moving at this site? Explain.

No. This interval indicates that the population mean wind speed is such that the sand may not always be moving at this site.

Yes. This interval indicates that the population mean wind speed is such that the sand may not always be moving at this site.

Yes. This interval indicates that the population mean wind speed is such that the sand is always moving at this site.

No. This interval indicates that the population mean wind speed is such that the sand is always moving at this site.

32. [–/3 Points] BBUNDERSTAT12 7.1.022.MI.DETAILS

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The method of tree-ring dating gave the following years A.D. for an archaeological excavation site. Assume that the population of x values has an approximately normal distribution.

1,194 1,180 1,236 1,313 1,268 1,316 1,275 1,317 1,275

(a) Use a calculator with mean and standard deviation keys to find the sample mean year x and sample standard deviation s. (Round your answers to four decimal places.)

= A.D.

s = yr

(b) Find a 90% confidence interval for the mean of all tree-ring dates from this archaeological site. (Round your answers to the nearest whole number.)

lower limit A.D.

upper limit A.D.

33. [–/4 Points] BBUNDERSTAT12 7.2.013.FB.S.DETAILS

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When is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for .

Method 1: Use the Student's t distribution with d.f. = n − 1. This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method.

Method 2: When n ≥ 30, use the sample standard deviation s as an estimate for , and then use the standard normal distribution.

This method is based on the fact that for large samples, s is a fairly good approximation for . Also, for large n, the critical values for the Student's t distribution approach those of the standard normal distribution.

Consider a random sample of size n = 36, with sample mean x = 45.5 and sample standard deviation s = 5.9.

(a) Compute 90%, 95%, and 99% confidence intervals for using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

90% 95% 99%

lower limit

upper limit

(b) Compute 90%, 95%, and 99% confidence intervals for using Method 2 with the standard normal distribution. Use s as an

estimate for . Round endpoints to two digits after the decimal.

90% 95% 99%

lower limit

upper limit

(c) Compare intervals for the two methods. Would you say that confidence intervals using a Student's t distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?

No. The respective intervals based on the t distribution are longer.

34. [–/27 Points] BBUNDERSTAT12 7.2.023.DETAILS

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Yes. The respective intervals based on the t distribution are shorter.

Yes. The respective intervals based on the t distribution are longer.

No. The respective intervals based on the t distribution are shorter.

(d) Now consider a sample size of 81. Compute 90%, 95%, and 99% confidence intervals for using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

90% 95% 99%

lower limit

upper limit

(e) Compute 90%, 95%, and 99% confidence intervals for using Method 2 with the standard normal distribution. Use s as an

estimate for . Round endpoints to two digits after the decimal.

90% 95% 99%

lower limit

upper limit

(f) Compare intervals for the two methods. Would you say that confidence intervals using a Student's t distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?

No. The respective intervals based on the t distribution are longer.

Yes. The respective intervals based on the t distribution are shorter.

No. The respective intervals based on the t distribution are shorter.

Yes. The respective intervals based on the t distribution are longer.

With increased sample size, do the two methods give respective confidence intervals that are more similar?

As the sample size increases, the difference between the two methods remains constant.

As the sample size increases, the difference between the two methods becomes greater.

As the sample size increases, the difference between the two methods is less pronounced.

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Do you want to own your own candy store? Wow! With some interest in running your own business and a decent credit rating, you can probably get a bank loan on startup costs for franchises such as Candy Express, The Fudge Company, Karmel Corn, and Rocky Mountain Chocolate Factory. Startup costs (in thousands of dollars) for a random sample of candy stores are given below. Assume that the population of x values has an approximately normal distribution.

99 178 130 92 75 94 116 100 85

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean startup cost x and sample standard deviation s. (Round your answers to four decimal places.)

x = thousand dollars

s = thousand dollars

(b) Find a 90% confidence interval for the population average startup costs for candy store franchises. (Round your answers to one decimal place.)

lower limit thousand dollars

upper limit thousand dollars

35. [–/4 Points] BBUNDERSTAT12 7.2.016.S.DETAILS

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Lorraine was in a hurry when she computed a confidence interval for . Because was not known, she used a Student's t distribution. However, she accidentally used degrees of freedom n instead of n − 1. Will her confidence interval be longer or shorter than one found using the correct degrees of freedom n − 1? Explain.

Longer. As the degrees of freedom increase, the value for t decreases.

Shorter. As the degrees of freedom increase, the value for t increases.

Longer. As the degrees of freedom increase, the value for t increases.

Shorter. As the degrees of freedom increase, the value for t decreases.

36. [–/1 Points] BBUNDERSTAT12 7.2.010.S.DETAILS MY NOTES ASK YOUR TEACHER

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Sam computed a 95% confidence interval for from a specific random sample. His confidence interval was 10.1 < < 12.2. He claims that

the probability that is in this interval is 0.95. What is wrong with his claim?

A probability can not be assigned to the event of falling in this interval.

Either is in the interval or it is not. Therefore, the probability that is in this interval is 0.95 or 0.05.

The probability that is in this interval is 0.05.

Either is in the interval or it is not. Therefore, the probability that is in this interval is 0 or 1.

As the degrees of freedom increase, what distribution does the Student's t distribution become more like? uniform

chi-square

binomial

standard normal

37. [–/1 Points] BBUNDERSTAT12 7.1.009.DETAILS MY NOTES ASK YOUR TEACHER

38. [–/1 Points] BBUNDERSTAT12 7.2.006.DETAILS MY NOTES ASK YOUR TEACHER

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Consider a 90% confidence interval for . Assume is not known. For which sample size, n = 10 or n = 20, is the critical value t larger?

neither

39. [–/1 Points] BBUNDERSTAT12 7.2.007.S.DETAILS MY NOTES ASK YOUR TEACHER

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Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther. Suppose a small group of 18 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with = 0.34 gram.

When finding an 80% confidence interval, what is the critical value for confidence level? (Give your answer to two decimal places.)

(a) Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.)

lower limit

upper limit

margin of error

(b) What conditions are necessary for your calculations? (Select all that apply.)

uniform distribution of weights

is known

is unknown

normal distribution of weights

n is large

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.

We are 20% confident that the true average weight of Allen's hummingbirds falls within this interval.

We are 80% confident that the true average weight of Allen's hummingbirds falls within this interval.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.

†

z = c

40. [–/8 Points] BBUNDERSTAT12 7.1.015.EP.DETAILS

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(d) Which equation is used to find the sample size n for estimating when is known?

Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.16 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)

hummingbirds

n = z E 2

n = z E

n = z

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Tutorial Exercise Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther. Suppose a small group of 20 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with = 0.35 gram.

(a) Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error?

(b) What conditions are necessary for your calculations?

(d) Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.15 for the mean weights of the hummingbirds.

†

41. [–/11 Points] BBUNDERSTAT12 7.1.015.MI.SA.DETAILS

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Student's t distributions are symmetric about a value of t. What is that t value?

42. [–/1 Points] BBUNDERSTAT12 7.2.005.DETAILS MY NOTES ASK YOUR TEACHER

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How much do wild mountain lions weigh? Adult wild mountain lions (18 months or older) captured and released for the first time in the San Andres Mountains gave the following weights (pounds):

65 102 127 121 60 64

Assume that the population of x values has an approximately normal distribution.

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean weight x and sample standard deviation s. (Round your answers to four decimal places.)

x = lb

s = lb

(b) Find a 75% confidence interval for the population average weight of all adult mountain lions in the specified region. (Round your answers to one decimal place.)

lower limit lb

upper limit lb

43. [–/4 Points] BBUNDERSTAT12 7.2.015.S.DETAILS

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Tutorial Exercise Use the Student's t distribution to find t for a 0.99 confidence level when the sample is 8.

True or false? If the sample mean of a random sample from an x distribution is relatively small, then the confidence interval for will be relatively short.

True. The maximal error of estimate controls the length of the confidence interval regardless of the value of x.

False. The maximal error of estimate controls the length of the confidence interval dependent on the value of x.

True. The maximal error of estimate controls the length of the confidence interval dependent on the value of x.

False. The maximal error of estimate controls the length of the confidence interval regardless of the value of x.

44. [–/4 Points] BBUNDERSTAT12 7.2.002.MI.SA.DETAILS

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45. [–/1 Points] BBUNDERSTAT12 7.1.007.DETAILS MY NOTES ASK YOUR TEACHER

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The method of tree ring dating gave the following years A.D. for an archaeological excavation site. Assume that the population of x values has an approximately normal distribution.

1,306 1,264 1,292 1,257 1,268 1,316 1,275 1,317 1,275

(a) Use a calculator with mean and standard deviation keys to find the sample mean year x and sample standard deviation s. (Round your answers to four decimal places.)

x = A.D.

s = yr

(b) Find a 90% confidence interval for the mean of all tree ring dates from this archaeological site. (Round your answers to the nearest whole number.)

lower limit A.D.

upper limit A.D.

46. [–/4 Points] BBUNDERSTAT12 7.2.013.S.DETAILS

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True or false? Consider a random sample of size n from an x distribution. For such a sample, the margin of error for estimating is the

magnitude of the difference between x and .

True. By definition, the margin of error is the magnitude of the difference between x and .

False. By definition, the margin of error is the magnitude of the difference between x and .

True. By definition, the margin of error is the magnitude of the difference between x and .

False. By definition, the margin of error is the magnitude of the difference between x and .

47. [–/1 Points] BBUNDERSTAT12 7.1.003.DETAILS MY NOTES ASK YOUR TEACHER

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What percentage of hospitals provide at least some charity care? Based on a random sample of hospital reports from eastern states, the following information is obtained (units in percentage of hospitals providing at least some charity care):

57.5 56.5 53.3 66.1 59.0 64.7 70.1 64.7 53.5 78.2

Assume that the population of x values has an approximately normal distribution.

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean percentage x and the sample standard deviation s. (Round your answers to four decimal places.)

x = %

s = %

(b) Find a 90% confidence interval for the population average of the percentage of hospitals providing at least some charity care. (Round your answers to one decimal place.)

lower limit %

upper limit %

48. [–/4 Points] BBUNDERSTAT12 7.2.018.MI.S.DETAILS

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Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 45 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that = 7.90 ml/kg for the distribution of blood plasma.

(a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.)

lower limit

upper limit

margin of error

(b) What conditions are necessary for your calculations? (Select all that apply.)

is known

n is large

the distribution of volumes is normal

is unknown

the distribution of volumes is uniform

We are 99% confident that the true average blood plasma volume in male firefighters falls within this interval.

We are 1% confident that the true average blood plasma volume in male firefighters falls within this interval.

The probability that this interval contains the true average blood plasma volume in male firefighters is 0.99.

The probability that this interval contains the true average blood plasma volume in male firefighters is 0.01.

(d) Find the sample size necessary for a 99% confidence level with maximal margin of error E = 2.60 for the mean plasma volume in male

49. [–/6 Points] BBUNDERSTAT12 7.1.017.FB.DETAILS

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firefighters. (Round up to the nearest whole number.)

male firefighters

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Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 41 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that = 7.90 ml/kg for the distribution of blood plasma.

When finding an 99% confidence interval, what is the critical value for confidence level? (Give your answer to two decimal places.)

(a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.)

lower limit

upper limit

margin of error

(b) What conditions are necessary for your calculations? (Select all that apply.)

is known

is unknown

the distribution of volumes is normal

the distribution of volumes is uniform

n is large

The probability that this interval contains the true average blood plasma volume in male firefighters is 0.99.

We are 99% confident that the true average blood plasma volume in male firefighters falls within this interval.

We are 1% confident that the true average blood plasma volume in male firefighters falls within this interval.

z = c

50. [–/8 Points] BBUNDERSTAT12 7.1.017.EP.DETAILS

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The probability that this interval contains the true average blood plasma volume in male firefighters is 0.01.

(d) Which equation is used to find the sample size n for estimating when is known?

Find the sample size necessary for a 99% confidence level with maximal margin of error E = 2.60 for the mean plasma volume in male firefighters. (Round up to the nearest whole number.)

male firefighters

n = z

n = z E

n = z E 2

n = z

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Tutorial Exercise What percentage of hospitals provide at least some charity care? Based on a random sample of hospital reports from western states, the following information is obtained (units in percentage of hospitals providing at least some charity care):

57 56.6 52.6 66.4 59.0 64.7 70.1 64.7 53.5 78.2

Assume that the population of x values has an approximately normal distribution.

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean percentage x and the sample standard deviation s.

(b) Find a 90% confidence interval for the population average of the percentage of hospitals providing at least some charity care.

51. [–/12 Points] BBUNDERSTAT12 7.2.018.MI.SA.DETAILS

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True or false? The point estimate for the population mean of an x distribution is x, computed from a random sample of the x distribution. False. The mean of the x distribution does not equal the mean of the x distribution and the standard error of the x distribution increases as n increases.

True. The mean of the x distribution equals the mean of the x distribution and the standard error of the x distribution increases as n increases.

True. The mean of the x distribution equals the mean of the x distribution and the standard error of the x distribution decreases as n increases.

False. The mean of the x distribution does not equal the mean of the x distribution and the standard error of the x distribution decreases as n increases.

52. [–/1 Points] BBUNDERSTAT12 7.1.002.DETAILS MY NOTES ASK YOUR TEACHER

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The distribution of heights of 18-year-old men in the United States is approximately normal, with mean 68 inches and standard deviation 3 inches (U.S. Census Bureau). In Minitab, we can simulate the drawing of random samples of size 20 from this population (� Calc � Random Data � Normal, with 20 rows from a distribution with mean 68 and standard deviation 3). Then we can have Minitab compute a 95% confidence interval and draw a boxplot of the data (� Stat � Basic Statistics � 1 -- Sample t, with boxplot selected in the graphs). The boxplots and confidence intervals for four different samples are shown in the accompanying figures. The four confidence intervals are as follows.

VARIABLE N MEAN STDEV SEMEAN 95.0 % CI

Sample 1 20 68.050 2.901 0.649 (66.692 , 69.407)

Sample 2 20 67.958 3.137 0.702 (66.490 , 69.426)

Sample 3 20 67.976 2.639 0.590 (66.741 , 69.211)

Sample 4 20 66.908 2.440 0.546 (65.766 , 68.050)

(a) Examine the figure [parts (a) to (d)]. How do the boxplots for the four samples differ? Why should you expect the boxplots to differ?

They differ in interquartile length, median location, and whisker length. The boxplots differ because they come from the same sample.

They differ in interquartile length and whisker length. The boxplots differ because they come from different samples.

They differ in interquartile length, median location, and whisker length. The boxplots differ because they come from different samples.

They differ in interquartile length and median location. The boxplots differ because they come from different samples.

53. [–/3 Points] BBUNDERSTAT12 7.2.019.DETAILS MY NOTES ASK YOUR TEACHER

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(b) Examine the 95% confidence intervals for the four samples shown in the printout. Do the intervals differ in length? Do the intervals all contain the expected population mean of 68 inches? ---Select---

If we draw more samples, do you expect all of the resulting 95% confidence intervals to contain = 68? Why or why not? We expect all of the samples to generate intervals that contain the mean of the population.

We expect none of the samples to generate intervals that contain the mean of the population.

We expect 5% of the samples to generate intervals that contain the mean of the population.

We expect 95% of the samples to generate intervals that contain the mean of the population.

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How much does a sleeping bag cost? Let's say you want a sleeping bag that should keep you warm in temperatures from 20°F to 45°F. A random sample of prices ($) for sleeping bags in this temperature range is given below. Assume that the population of x values has an approximately normal distribution.

55 75 105 100 85 55 30 23 100 110

105 95 105 60 110 120 95 90 60 70

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean price x and sample standard deviation s. (Round your answers to four decimal places.)

x = $

s = $

(b) Using the given data as representative of the population of prices of all summer sleeping bags, find a 90% confidence interval for the mean price of all summer sleeping bags. (Round your answers to two decimal places.)

lower limit $

upper limit $

54. [–/4 Points] BBUNDERSTAT12 7.2.014.MI.S.DETAILS

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What price do farmers get for their watermelon crops? In the third week of July, a random sample of 45 farming regions gave a sample

mean of x = $6.88 per 100 pounds of watermelon. Assume that is known to be $1.92 per 100 pounds.

(a) Find a 90% confidence interval for the population mean price (per 100 pounds) that farmers in this region get for their watermelon crop. What is the margin of error? (Round your answers to two decimal places.)

lower limit $

upper limit $

margin of error $

(b) Find the sample size necessary for a 90% confidence level with maximal error of estimate E = 0.37 for the mean price per 100 pounds of watermelon. (Round up to the nearest whole number.)

farming regions

(c) A farm brings 15 tons of watermelon to market. Find a 90% confidence interval for the population mean cash value of this crop. What is the margin of error? Hint: 1 ton is 2000 pounds. (Round your answers to two decimal places.)

lower limit $

upper limit $

margin of error $

55. [–/7 Points] BBUNDERSTAT12 7.1.018.MI.DETAILS

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Use the Student's t distribution to find t for a 0.99 confidence level when the sample is 11.

56. [–/1 Points] BBUNDERSTAT12 7.2.002.MI.S.DETAILS

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(a) Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.)

lower limit

upper limit

margin of error

(b) What conditions are necessary for your calculations? (Select all that apply.)

uniform distribution of weights

is known

is unknown

normal distribution of weights

n is large

We are 80% confident that the true average weight of Allen's hummingbirds falls within this interval.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.

We are 20% confident that the true average weight of Allen's hummingbirds falls within this interval.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.

(d) Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.06 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)

†

57. [–/6 Points] BBUNDERSTAT12 7.1.015.MI.DETAILS

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hummingbirds

Use the Student's t distribution to find t for a 0.95 confidence level when the sample is 25. (Round your answer to three decimal places.)

58. [–/1 Points] BBUNDERSTAT12 7.2.001.FB.S.DETAILS

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Consider a 90% confidence interval for . Assume is not known. For which sample size, n = 10 or n = 20, is the confidence interval longer?

neither

59. [–/1 Points] BBUNDERSTAT12 7.2.008.S.DETAILS MY NOTES ASK YOUR TEACHER

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Jobs and productivity! How do retail stores rate? One way to answer this question is to examine annual profits per employee. The following

data give annual profits per employee (in units of 1 thousand dollars per employee) for companies in retail sales. Assume ≈ 3.7 thousand dollars.

(a) Use a calculator or appropriate computer software to find x for the preceding data. (Round your answer to two decimal places.)

thousand dollars per employee

(b) Let us say that the preceding data are representative of the entire sector of retail sales companies. Find an 80% confidence

interval for , the average annual profit per employee for retail sales. (Round your answers to two decimal places.)

lower limit thousand dollars

upper limit thousand dollars

(c) Let us say that you are the manager of a retail store with a large number of employees. Suppose the annual profits are less than 3 thousand dollars per employee. Do you think this might be low compared with other retail stores? Explain by referring to the confidence interval you computed in part (b).

Yes. This confidence interval suggests that the profits per employee are less than those of other retail stores.

No. This confidence interval suggests that the profits per employee do not differ from those of other retail stores.

(d) Suppose the annual profits are more than 6.5 thousand dollars per employee. As store manager, would you feel somewhat better? Explain by referring to the confidence interval you computed in part (b).

Yes. This confidence interval suggests that the profits per employee are greater than those of other retail stores.

No. This confidence interval suggests that the profits per employee do not differ from those of other retail stores.

4.1 6.9 4.4 8.3 8.3 5.5 8.9 6.3 2.6 2.9 8.1 −1.9

11.9 8.2 6.4 4.7 5.5 4.8 3.0 4.3 −6.0 1.5 2.9 4.8

−1.7 9.4 5.5 5.8 4.7 6.2 15.0 4.1 3.7 5.1 4.2

60. [–/9 Points] BBUNDERSTAT12 7.1.024.DETAILS

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(e) Find an 95% confidence interval for , the average annual profit per employee for retail sales. (Round your answers to two decimal places.)

lower limit thousand dollars

upper limit thousand dollars

Let us say that you are the manager of a retail store with a large number of employees. Suppose the annual profits are less than 3 thousand dollars per employee. Do you think this might be low compared with other retail stores? Explain by referring to the confidence interval you computed in part (b).

Yes. This confidence interval suggests that the profits per employee are less than those of other retail stores.

No. This confidence interval suggests that the profits per employee do not differ from those of other retail stores.

Suppose the annual profits are more than 6.5 thousand dollars per employee. As store manager, would you feel somewhat better? Explain by referring to the confidence interval you computed in part (b).

Yes. This confidence interval suggests that the profits per employee are greater than those of other retail stores.

No. This confidence interval suggests that the profits per employee do not differ from those of other retail stores.

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(a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.)

lower limit

upper limit

margin of error

(b) What conditions are necessary for your calculations? (Select all that apply.)

n is large

is known

the distribution of volumes is uniform

the distribution of volumes is normal

is unknown

The probability that this interval contains the true average blood plasma volume in male firefighters is 0.99.

We are 1% confident that the true average blood plasma volume in male firefighters falls within this interval.

The probability that this interval contains the true average blood plasma volume in male firefighters is 0.01.

We are 99% confident that the true average blood plasma volume in male firefighters falls within this interval.

(d) Find the sample size necessary for a 99% confidence level with maximal margin of error E = 2.60 for the mean plasma volume in male

61. [–/6 Points] BBUNDERSTAT12 7.1.017.MI.DETAILS

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firefighters. (Round up to the nearest whole number.)

male firefighters

Tutorial Exercise Use the Student's t distribution to find t for a 0.95 confidence level when the sample is 26.

62. [–/5 Points] BBUNDERSTAT12 7.2.001.MI.SA.DETAILS

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Tutorial Exercise At wind speeds above 1000 centimeters per second (cm/sec), significant sand-moving events begin to occur. Wind speeds below 1000 cm/sec deposit sand and wind speeds above 1000 cm/sec move sand to new locations. The cyclic nature of wind and moving sand determines the shape and location of large dunes. At a test site, the prevailing direction of the wind did not change noticeably. However, the velocity did change. Sixty-eight wind speed readings gave an average velocity of x = 1,117 cm/sec. Based on long-term experience, can be assumed to be 246 cm/sec.

(a) Find a 95% confidence interval for the population mean wind speed at this site.

(b) Does the confidence interval indicate that the population mean wind speed is such that the sand is always moving at this site? Explain.

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63. [–/8 Points] BBUNDERSTAT12 7.1.022.MI.SA.DETAILS

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