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9-1 Copyright © 2017 Pearson Education, Inc. Chapter 9: Sampling Distributions and Confidence Intervals for Proportions – Quiz A Name ________________________________________ 9.2.6 Find and interpret margins of error. 1. In a metal fabrication process, metal rods are produced to a specified target length of 15 feet. Suppose that the lengths are normally distributed. A quality control specialist collects a random sample of 16 rods and finds the sample mean length to be 14.8 feet and a standard deviation of 0.65 feet. a. Describe the sampling distribution for the sample mean. b. What is the standard error? c. For 95% confidence, what is the margin of error? d. Based on the sample results, create the 95% confidence interval and interpret. 9.2.7 Find and interpret confidence intervals. 2. A manufacturer of cheese filled ravioli supplies a pizza restaurant chain. Based on data collected from its automatic filling process, the amount of cheese inserted into the ravioli is normally distributed. To make sure that the automatic filling process is on target, quality control inspectors take a sample of 25 ravioli and measure the weight of cheese filling. They find a sample mean weight of 15 grams with a standard deviation of 1.5 grams. a. Describe the sampling distribution for the sample mean. b. What is the standard error? c. What is the margin of error for 99% confidence? d. What is the margin of error for 90% confidence? e. Based on the sample results, find the 99% confidence interval and interpret. f. Based on the sample results, find the 90% confidence interval and interpret. g. For a more accurate determination of the mean weight, the quality control inspectors wish to estimate it within .25 grams with 99% confidence. How many ravioli should they sample?
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Chapter 9: Sampling Distributions and Confidence Intervals for Proportions – Quiz A Name ________________________________________ 9.2.6 Find and interpret margins of error. 1. In a metal fabrication process, metal rods are produced to a specified target length of 15 feet. Suppose that the lengths are normally distributed. A quality control specialist collects a random sample of 16 rods and finds the sample mean length to be 14.8 feet and a standard deviation of 0.65 feet. a. Describe the sampling distribution for the sample mean. b. What is the standard error? c. For 95% confidence, what is the margin of error? d. Based on the sample results, create the 95% confidence interval and interpret. 9.2.7 Find and interpret confidence intervals. 2. A manufacturer of cheese filled ravioli supplies a pizza restaurant chain. Based on data collected from its automatic filling process, the amount of cheese inserted into the ravioli is normally distributed. To make sure that the automatic filling process is on target, quality control inspectors take a sample of 25 ravioli and measure the weight of cheese filling. They find a sample mean weight of 15 grams with a standard deviation of 1.5 grams. a. Describe the sampling distribution for the sample mean. b. What is the standard error? c. What is the margin of error for 99% confidence? d. What is the margin of error for 90% confidence? e. Based on the sample results, find the 99% confidence interval and interpret. f. Based on the sample results, find the 90% confidence interval and interpret. g. For a more accurate determination of the mean weight, the quality control inspectors wish to estimate it within .25 grams with 99% confidence. How many ravioli should they sample?
9-2 Chapter 9 Sampling Distributions and Confidence Intervals for Proportions
9.2.6 Find and interpret confidence intervals. 3. Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. Last year the average life expectancy of all policyholders was 77 years. ABI Insurance wants to determine if their clients now have a longer life expectancy, on average, so they randomly sample some of their recently paid policies. The ages of the clients in the sample are shown below.
86 75 83 84 81 77 78 79 79 81 76 85 70 76 79 81 73 74 72 83
a. Based on the sample results, find the 90% confidence interval and interpret. b. For more accurate cost determination, ABI Insurance wants to estimate the average life expectancy to within one year with 95% confidence. How many randomly selected recently paid policies would they need to sample? c. Suppose ABI samples 100 recently paid policies. This sample yields a mean of 77.7 years and a standard deviation of 3.6 years. Find a 90% confidence interval and interpret. 9.1.2 Find the mean and standard deviation of proportions. 4. Suppose that the store chain of Electronics Plus sells extended warranties to 20% of customers who purchase electronic devices. A local Seattle store samples 300 customers from their data base. a. What proportion of customers would they expect to have purchased extended warranties? b. What is the standard deviation of the sample proportion? c. What shape would you expect the sampling distribution of the proportion to have? d. Would you be surprised to find out that in a sample of 300, 40 of the customers purchased extended warranties? Explain. What might account for this high percentage?
Quiz A 9-3
Chapter 9: Sampling Distributions and Confidence Intervals for Proportions – Quiz A – Key 1. In a metal fabrication process, metal rods are produced to a specified target length of 15 feet. Suppose that the lengths are normally distributed. A quality control specialist collects a random sample of 16 rods and finds the sample mean length to be 14.8 feet and a standard deviation of 0.65 feet. a. Describe the sampling distribution for the sample mean. The sampling distribution for the sample mean can be modeled using the t-distribution with 15 degrees of freedom. b. What is the standard error?
ft n sySE 1625.0
c. For 95% confidence, what is the margin of error?
ftftySEtn 346.01625.0131.2)(* 1 =×=×−
d. Based on the sample results, create the 95% confidence interval and interpret.
346.08.14)(* 1 ±=×± − ySEty n
The 95% confidence interval for true mean length is from 14.454 to 15.146 ft. We are 95% confident that the average length of metal rods from this process is between 14.454 and 15.146 ft. 2. A manufacturer of cheese filled ravioli supplies a pizza restaurant chain. Based on data collected from its automatic filling process, the amount of cheese inserted into the ravioli is normally distributed. To make sure that the automatic filling process is on target, quality control inspectors take a sample of 25 ravioli and measure the weight of cheese filling. They find a sample mean weight of 15 grams with a standard deviation of 1.5 grams. a. Describe the sampling distribution for the sample mean. The sampling distribution for the sample mean can be modeled using the t-distribution with 24 degrees of freedom.
9-4 Chapter 9 Sampling Distributions and Confidence Intervals for Proportions
b. What is the standard error?
grams n sySE 3.0
c. What is the margin of error for 99% confidence?
gramsgramsySEtn 8391.03.0797.2)(* 1 =×=×−
d. What is the margin of error for 90% confidence?
gramsgramsySEtn 5133.03.0711.1)(* 1 =×=×−
e. Based on the sample results, find the 99% confidence interval and interpret.
8391.015)(* 1 ±=×± − ySEty n
The 99% confidence interval for true mean weight is 14.16 to 15.84 grams. We are 99% confident that the mean weight of cheese in the ravioli made by this process is between 14.16 and 15.84 grams. f. Based on the sample results, find the 90% confidence interval and interpret.
5133.015)(* 1 ±=×± − ySEty n
The 90% confidence interval for true mean weight is 14.49 to 15.51 grams. We are 90% confident that the mean weight of cheese in the ravioli made by this process is between 14.49 and 15.51 grams. g. For a more accurate determination of the mean weight, the quality control inspectors wish to estimate it within .25 grams with 99% confidence. How many ravioli should they sample?
2
22
25.0
Quiz A 9-5
3. Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. Last year the average life expectancy of all policyholders was 77 years. ABI Insurance wants to determine if their clients now have a longer life expectancy, on average, so they randomly sample some of their recently paid policies. The ages of the clients in the sample are shown below.
86 75 83 84 81 77 78 79 79 81 76 85 70 76 79 81 73 74 72 83
a. Based on the sample results, find the 90% confidence interval and interpret. The sample has a mean of 78.6 years and a standard deviation of 4.48 years.
732.16.78 20
48.4 729.16.78)(*
1 ±=±=×± − ySEty n
The 90% confidence interval for true mean age is 76.87 to 80.33 years. We are 90% confident that the average age of clients with recently paid policies is between 76.87 and 80.33 years. b. For more accurate cost determination, ABI Insurance wants to estimate the average life expectancy to within one year with 95% confidence. How many randomly selected recently paid policies would they need to sample?
2
22
1
)48.4()96.1(=n
n = 77.1 or 78 policies c. Suppose ABI samples 100 recently paid policies. This sample yields a mean of 77.7 years and a standard deviation of 3.6 years. Find a 90% confidence interval and interpret.
100
1 ±=×± − ySEty n
The 90% confidence interval for true mean age is 77.1 to 78.3 years. We are 90% confident that the average age of clients with recently paid policies is between 77.1 years to 78.3 years.
9-6 Chapter 9 Sampling Distributions and Confidence Intervals for Proportions
4. Suppose that the store chain of Electronics Plus sells extended warranties to 20% of customers who purchase electronic devices. A local Seattle store samples 300 customers from their data base to compare to the nationwide average. a. What proportion of customers would they expect to have purchased extended warranties?
Because 20% of customers nationwide purchased extended warranties, we would expect the same for the sample proportion. b. What is the standard deviation of the sample proportion?
(0 2)(0 8) ( ) 0 023
300
= = =
c. What shape would you expect the sampling distribution of the proportion to have? Normal d. Would you be surprised to find out that in a sample of 300, 40 of the customers purchased extended warranties? Explain. What might account for this high percentage? 40 customers results in a sample proportion of 13.3% well below the national average. The mean is 0.20 with a standard deviation of 0.023. This sample proportion is less than
the national average by about 3 standard deviations (0 133 0 20)
0 023
− = –2.91. It would be
very unusual to find such a low proportion in a random sample. Either it is a very unusual sample, or the proportion in this regions is not the same as the national average. Perhaps it would be warranted to increase the campaign to sell extended warranties.
Quiz B 9-7
Chapter 9: Sampling Distributions and Confidence Intervals for Proportions – Quiz B Name ________________________________________ 9.2.6 Find and interpret confidence intervals. 1. A small business ships specialty homemade candies to anywhere in the world. Past records indicate that the weight of orders is normally distributed. Suppose a random sample of 16 orders is selected and each is weighed. The sample mean was found to be 110 grams with a standard deviation of 14 grams. a. Describe the sampling distribution for the sample mean. b. What is the standard error? c. For 90% confidence, what is the margin of error? d. Based on the sample results, create the 90% confidence interval and interpret. 9.2.6 Find and interpret confidence intervals. 2. Grandma Gertrude’s Chocolates, a family owned business, has an opportunity to supply its product for distribution through a large coffee house chain. However, the coffee house chain has certain specifications regarding cacao content as it wishes to advertise the health benefits (antioxidants) of the chocolate products it sells. In order to determine the mean % cacao in its dark chocolate products, quality inspectors sample 36 pieces. They find a sample mean of 55% with a standard deviation of 4%. a. Describe the sampling distribution for the sample mean. b. What is the standard error? c. What is the margin of error for 90% confidence? d. What is the margin of error for 95% confidence? e. Based on the sample results, find the 90% confidence interval and interpret. f. Based on the sample results, find the 95% confidence interval and interpret. g. For a more accurate determination of the mean weight, the quality control inspectors wish to estimate it within 1% with 95% confidence. How many pieces of dark chocolate should they sample?
9-8 Chapter 9 Confidence Intervals and Hypothesis Tests for Means
9.2.6 Find and interpret confidence intervals. 3. A large software development firm recently relocated its facilities. Top management is interested in fostering good relations with their new local community and has encouraged their professional employees to engage in local service activities. They wish to determine the average number of hours the firm’s professionals volunteer per month. A random sample of 24 professionals reported the following number of hours:
12 13 14 14 15 15 15 16 16 16 16 16 17 17 17 18 18 18 18 19 19 19 20 21
a. Based on the sample results, find the 95% confidence interval and interpret. b. For a more accurate determination, top management wants to estimate the average number of hours volunteered per month by their professional staff to within one hour with 99% confidence. How many randomly selected professional employees would they need to sample? c. Suppose 40 professional employees are randomly selected. This sample yields a mean of 15.2 hours and a standard deviation of 1.8 hours. Find a 95% confidence interval and interpret. 9.4.8 Find sample sizes. 4. A clothing store is about to send out an electronic mailing to test the market for a specific credit card for their store. From that sample, the want to estimate the true proportion of people who will sign up for the card nationwide. To be within a tenth of a percentage point, or 0.001 of the true rate with 90% confidence, how big does the mailing have to be? Similar mailings in the past lead them to expect about 0.5% of the people receiving the offer will accept it. 9.2.6 Find and interpret confidence intervals. 5. EU (European Union) countries report that 46% of their labor force is female. The United Nations wants to determine if the percentage of females in the U.S. labor force is the same. Representatives from the United States Department of Labor plan to check a random sample of over 10,000 employment records on file to estimate the percentage of females in the U.S. labor force. a. The Department of Labor wants to estimate the percentage of females in the U.S. labor force to within ±5%, with 90% confidence. How many employment records should be sampled? b. They actually select a random sample of 525 employment records, and find that 229 of the people are females. Construct the 90% confidence interval. c. Interpret the confidence interval in this context. d. Explain what 90% confidence means in this context.
Quiz B 9-9
e. Should representatives from the Department of Labor conclude that the percentage of females in the U.S. labor force is lower than Europe’s rate of 46%? Explain. f. Are the assumptions and conditions for constructing a confidence interval met? Explain.
9-10 Chapter 9 Confidence Intervals and Hypothesis Tests for Means
Chapter 9: Sampling Distributions and Confidence Intervals for Proportions – Quiz B – Key 1. A small business ships specialty homemade candies to anywhere in the world. Past records indicate that the weight of orders is normally distributed. Suppose a random sample of 16 orders is selected and each is weighed. The sample mean was found to be 110 grams with a standard deviation of 14 grams. a. Describe the sampling distribution for the sample mean. The sampling distribution for the sample mean can be modeled using the t-distribution with 15 degrees of freedom. b. What is the standard error?
grams n sySE 5.3
c. For 90% confidence, what is the margin of error?
gramsySEtn 135.65.3753.1)(* 1 =×=×−
d. Based on the sample results, create the 90% confidence interval and interpret.
135.6110)(* 1 ±=×± − ySEty n
The 90% confidence interval for true mean weight of orders is from 103.87 to 116.14 grams. We are 90% confident that the average weight of candy orders is between 103.87 and 116.14 grams. 2. Grandma Gertrude’s Chocolates, a family owned business, has an opportunity to supply its product for distribution through a large coffee house chain. However, the coffee house chain has certain specifications regarding cacao content as it wishes to advertise the health benefits (antioxidants) of the chocolate products it sells. In order to determine the mean % cacao in its dark chocolate products, quality inspectors sample 36 pieces. They find a sample mean of 55% with a standard deviation of 4%. a. Describe the sampling distribution for the sample mean. The sampling distribution for the sample mean can be modeled using the t-distribution with 35 degrees of freedom. b. What is the standard error?
Quiz B 9-11
%67.0 36
c. What is the margin of error for 90% confidence?
%13.167.0690.1)(* 1 =×=×− ySEtn
d. What is the margin of error for 95% confidence?
%36.167.0030.2)(* 1 =×=×− ySEtn
e. Based on the sample results, find the 90% confidence interval and interpret.
13.155)(* 1 ±=×± − ySEty n
The 90% confidence interval for true % cacao is 53.87% to 56.13%. We are 90% confident that the mean percentage of cacao in Grandma Gertrude’s dark chocolate is between 53.87% and 56.13%. f. Based on the sample results, find the 95% confidence interval and interpret.
36.155)(* 1 ±=×± − ySEty n
The 95% confidence interval for true % cacao is 53.64% to 56.36%. We are 95% confident that the mean percentage of cacao in Grandma Gertrude’s dark chocolate is between 53.64% and 56.36%. g. For a more accurate determination of the mean weight, the quality control inspectors wish to estimate it within 1% with 95% confidence. How many pieces of dark chocolate should they sample?
2
22
1
9-12 Chapter 9 Confidence Intervals and Hypothesis Tests for Means
3. A large software development firm recently relocated its facilities. Top management is interested in fostering good relations with their new local community and has encouraged their professional employees to engage in local service activities. They wish to determine the average number of hours the firm’s professionals volunteer per month. A random sample of 24 professionals reported the following number of hours:
12 13 14 14 15 15 15 16 16 16 16 16 17 17 17 18 18 18 18 19 19 19 20 21
a. Based on the sample results, find the 95% confidence interval and interpret. The sample has a mean of 16.6 hours and a standard deviation of 2.22 hours.
938.06.16 24
22.2 069.26.16)(*
1 ±=±=×± − ySEty n
The 95% confidence interval for true mean number of volunteer hours is 15.66 to 17.54. We are 95% confident that the average number of hours volunteered by professionals employed with this firm is between 15.66 and 17.54. b. For a more accurate determination, top management wants to estimate the average number of hours volunteered per month by their professional staff to within one hour with 99% confidence. How many randomly selected professional employees would they need to sample?
2
22
1
)22.2()575.2(=n
n = 32.67 or 33 professionals c. Suppose 40 professional employees are randomly selected. This sample yields a mean of 15.2 hours and a standard deviation of 1.8 hours. Find a 95% confidence interval and interpret.
576.02.15 40
8.1 023.22.15)(*
1 ±=±=×± − ySEty n
14.62 to 15.78 hours The 95% confidence interval for true mean number of volunteer hours is 14.62 to 15.78. We are 95% confident that the average number of hours volunteered by professionals employed with this firm is between 14.62 and 15.78.
Quiz B 9-13
4. A clothing store is about to send out an electronic mailing to test the market for a specific credit card for their store. From that sample, the want to estimate the true proportion of people who will sign up for the card nationwide. To be within a tenth of a percentage point, or 0.001 of the true rate with 90% confidence, how big does the mailing have to be?
(0 005)(0 995) 0 001 1 645* pq . .ME . z .
n n = = =
(0 001)
= 13,462.47 or 13,463
5. EU (European Union) countries report that 46% of their labor force is female. The United Nations wants to determine if the percentage of females in the U.S. labor force is the same. Representatives from the United States Department of Labor plan to check a random sample of over 10,000 employment records on file to estimate the percentage of females in the U.S. labor force. a. The Department of Labor wants to estimate the percentage of females in the U.S. labor force to within ±5%, with 90% confidence. How many employment records should be sampled? n = (1.645)2 (.46) (.54) / (.05)2 = 268.87 = 269 records b. They actually select a random sample of 525 employment records, and find that 229 of the people are females. Construct the 90% confidence interval. Confidence interval: (0.3998, 0.4722) c. Interpret the confidence interval in this context. We are 90% confident that between 39.98% and 47.22% of the employment records from the United States labor force are female. d. Explain what 90% confidence means in this context. If many random samples were taken, 90% of the confidence intervals produced would contain the actual percentage of all female employment records in the United States labor force. e. Should representatives from the Department of Labor conclude that the percentage of females in the U.S. labor force is lower than Europe’s rate of 46%? Explain.
9-14 Chapter 9 Confidence Intervals and Hypothesis Tests for Means
No. Since 46% lies in the confidence interval, (0.3998, 0.4722), it is possible that the percentage of females in the labor force matches Europe’s rate of 46% female in the labor force. f. Are the assumptions and conditions for constructing a confidence interval met? Explain. We have a random sample of less than 10% of the employment records, with 229 successes (females) and 296 failures (males), so the normal model applies.
Quiz C 9-15
Chapter 9: Sampling Distributions and Confidence Intervals for Proportions – Quiz C Multiple Choice Name ________________________________________ 9.2.6 Find and interpret confidence intervals. 1. EU (European Union) countries report that 46% of their labor force is female. The United Nations wants to determine if the percentage of females in the U.S. labor force is the same. Representatives from the United States Department of Labor select a random sample of 525 from over 10,000 employment records on file and find that 229 are female. The 90% confidence interval for the proportion of females in the U.S. labor force is A. 0.3998 to 0.4722 B. 0.2747 to 0.5973 C. 0.1776 to 0.6944 D. 0.4235 to 0.5679 E. 0.1243 to 0.7100 9.2.6 Find and interpret confidence intervals. 2. EU (European Union) countries report that 46% of their labor force is female. The United Nations wants to determine if the percentage of females in the U.S. labor force is the same. Based on a sample of 500 employment records, representatives from the United States Department of Labor find that the 95% confidence interval for the proportion of females in the U.S. labor force is .357 to .443. Which of the following is the correct interpretation? A. The percentage of females in the U.S. labor force is between 35.7% and 44.3%. B. We are 95% confident that between 35.7% and 44.3% of the persons in the U.S. labor force is female. C. The margin of error for the true percentage of females in the U.S. labor force is between 35.7% and 44.3%. D. All samples of size 500 will yield a percentage of females in the U.S. labor force that falls within 35.7% and 44.3%. E. None of the above. 9.2.6 Find and interpret margins of error. 3. EU (European Union) countries report that 46% of their labor force is female. The United Nations wants to determine if the percentage of females in the U.S. labor force is the same. Based on a sample of 500 employment records, representatives from the United States Department of Labor find that 240 are female. What is the margin of error for the 95% confidence interval of the proportion of females in the U.S. labor force? A. 0.022 B. 0.044 C. 0.036 D. 0.056 E. 0.089
9-16 Chapter 9 Sampling Distributions and Confidence Intervals for Proportions
9.2.7 Find and interpret confidence intervals. 4. EU (European Union) countries report that 46% of their labor force is female. The United Nations wants to determine if the percentage of females in the U.S. labor force is the same. Based on a sample of 500 employment records, representatives from the United States Department of Labor find that the 95% confidence interval for the proportion of females in the U.S. labor force is .357 to .443. If the Department of Labor wishes to tighten its interval, they should A. increase the confidence level. B. decrease the sample size. C. increase the sample size. D. Both A and B E. Both A and C 9.2.6 Check that inference conditions are satisfied. 5. All else being equal, increasing the level of confidence desired will A. tighten the confidence interval. B. decrease the margin of error. C. increase precision. D. increase the margin of error. E. Both A and D. 9.2.6 Check that inference conditions are satisfied. 6. Which of the following is not an assumption and/or condition required for constructing a confidence interval for the proportion? A. Randomization condition B. Linearity condition C. Success/Failure condition D. 10% condition E. None of the above 9.2.6 Find and interpret margins of error. 7. Automobile mechanics conduct diagnostic tests on 150 new cars of particular make and model to determine the extent to which they are affected by a recent recall due to faulty catalytic converters. They find that 42 of the new cars tested do have faulty catalytic converters. What is the margin of error for a 99% confidence interval based on these sample results? A. 0.0366 B. 0.0719 C. 0.0944 D. 0.1140 E. 0.2876
Quiz C 9-17
9.2.7 Find and interpret confidence intervals. 8. Automobile mechanics conduct diagnostic tests on 150 new cars of particular make and model to determine the extent to which they are affected by a recent recall due to faulty catalytic converters. They find that 42 of the new cars tested do have faulty catalytic converters. The 99% confidence interval for the true proportion of new cars with faulty catalytic converters is A. 0.1856 to 0.3744 B. 0.2434 to 0.3166 C. 0.2081 to 0.3519 D. 0.1660 to 0.3940 E. 0.1243 to 0.4123 9.4.8 Find sample sizes. 9. The U.S. Department of Labor wants to estimate the percentage of females in the labor force to within ±5% with 90% confidence. As a planning value they use 46%, the reported percentage of females in the labor force of EU (European Union) countries. How many employment records should be sampled? A. 121 B. 269 C. 451 D. 382 E. 1000 9.4.8 Find sample sizes. 10. The U.S. Department of Labor wants to estimate the percentage of females in the labor force to within ±2% with 90% confidence. As a planning value they use 46%, the reported percentage of females in the labor force of EU (European Union) countries. How many employment records should be sampled? A. 121 B. 269 C. 451 D. 382 E. 1681
9-18 Chapter 9 Sampling Distributions and Confidence Intervals for Proportions
Chapter 9: Confidence Intervals and Hypothesis Tests for Means – Quiz C – Key
1. A 2. B 3. B 4. C 5. D 6. B 7. C 8. A 9. B 10. E
Quiz D 9-19