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MATH1-UC 1172, section 002, Spring 2021

L8_HMW_SPRING_2021 (Homework)
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Olivia Mendivil Ramos
New York University

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Assignment Submission & Scoring

Assignment Submission

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.

Assignment Scoring

Your last submission is used for your score.

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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

c

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

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

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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.

x

t = c

E =

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

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

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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

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

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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 c c c c c c c 5. [–/1 Points] BBUNDERSTAT12 7.1.001.DETAILS MY NOTES ASK YOUR TEACHER 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 10/78 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 MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 11/78 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 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 12/78 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 (c) Interpret your results in the context of this problem. 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 MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 13/78 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 c c c c c c 9. [–/1 Points] BBUNDERSTAT12 7.2.009.S.DETAILS MY NOTES ASK YOUR TEACHER 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 14/78 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 (c) Interpret your results in the context of this problem. 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 MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 15/78 (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 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 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 MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 16/78 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 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. (c) Based on your results in part (b), do you think this patient still has a calcium deficiency? Explain. 12. [–/14 Points] BBUNDERSTAT12 7.2.017.MI.SA.DETAILS MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 17/78 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 MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 18/78 (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. 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 19/78 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 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 20/78 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. x 15. [–/4 Points] BBUNDERSTAT12 7.2.013.MI.S.DETAILS MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 21/78 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.) c c 16. [–/1 Points] BBUNDERSTAT12 7.2.004.S.DETAILS MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER 17. [–/1 Points] BBUNDERSTAT12 7.2.003.S.DETAILS MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 22/78 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 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 23/78 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 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 MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 24/78 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 MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 25/78 (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. 4/8/2021 L8_HMW_SPRING_2021 - MATH1-UC 1172, section 002, Spring 2021 | WebAssign https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=26060321&tags=autosave#question4738633_33 26/78 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 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? (c) Interpret your results in the context of this problem. (d) Find the sample size necessary for a 99% confidence …