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1Define Target Parameter
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem.
1)Which statement best describes a parameter?
A)A parameter is a numerical measure of a population that is almost always unknown and must be estimated.
B)A parameter is a level of confidence associated with an interval about a sample mean or proportion.
C)A parameter is a sample size that guarantees the error in estimation is within acceptable limits.
D)A parameter is an unbiased estimate of a statistic found by experimentation or polling.
2)A study was conducted to determine what proportion of all college students considered themselves as full - time students. A random sample of 300 college students was selected and 210 of the students responded that they considered themselves full- time students. Which of the following would represent the target parameter of interest?
3)Parking at a large university can be extremely difficult at times. One particular university is trying to determine the location of a new parking garage. As part of their research, officials are interested in estimating the average parking time of students from within the various colleges on campus. Which of the following would represent the target parameter of interest?
Answer the question True or False.
4)For data with two outcomes (success or failure), the binomial proportion of successes is likely to be the parameter of interest.
5)For quantitative data, the target parameter is most likely to be the mode of the data.
1Understand Confidence Level, Confidence Interval, α
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem.
1)What is z α /2 when α = 0.05?
A)1.96B)2.575C)2.33D)1.645
2)What is the confidence level of the following confidence interval for μ?
Page 166
3)The registrar's office at State University would like to determine a 95% confidence interval for the mean commute time of its students. A member of the staff randomly chooses a parking lot and surveys the first 125 students who park in the chosen lot on a given day. The confidence interval is
A)not meaningful because of the lack of random sampling.
B)meaningful because the sample size exceeds 30 and the Central Limit Theorem ensures normality of the sampling distribution of the sample mean.
C)not meaningful because the sampling distribution of the sample mean is not normal.
D)meaningful because the sample is representative of the population.
4)A 90% confidence interval for the mean percentage of airline reservations being canceled on the day of the flight is (1.3%, 5%). What is the point estimator of the mean percentage of reservations that are canceled on the day of the flight?
A)3.15%B)1.85%C)2.50%D)3.7%
5)A 90% confidence interval for the average salary of all CEOs in the electronics industry was constructed using the results of a random survey of 45 CEOs. The interval was ($130,852, $147,990). To make more useful inferences from the data, it is desired to reduce the width of the confidence interval. Which of the following will result in a reduced interval width?
A)Increase the sample size and decrease the confidence level.
B)Decrease the sample size and decrease the confidence level.
C)Increase the sample size and increase the confidence level.
D)Decrease the sample size and increase the confidence level.
6)Suppose a large labor union wishes to estimate the mean number of hours per month a union member is absent from work. The union decides to sample 343 of its members at random and monitor the working time of each of them for 1 month. At the end of the month, the total number of hours absent from work is recorded for each employee. Which of the following should be used to estimate the parameter of interest for this problem?
A)A large sample confidence interval for μ.B)A small sample confidence interval for μ
C)A large sample confidence interval for p .D)A small sample confidence interval for p
7)Explain what the phrase 95% confident means when we interpret a 95% confidence interval for μ
A)In repeated sampling, 95% of similarly constructed intervals contain the value of the population mean.
B)95% of similarly constructed intervals would contain the value of the sampled mean.
C)95% of the observations in the population fall within the bounds of the calculated interval.
D)The probability that the sample mean falls in the calculated interval is 0.95.
8)Parking at a large university can be extremely difficult at times. One particular university is trying to determine the location of a new parking garage. As part of their research, officials are interested in estimating the average parking time of students from within the various colleges on campus. A survey of 338 College of Business (COBA) students yields the following descriptive information regarding the length of time (in minutes) it took them to find a parking spot. Note that the "Lo 95%" and "Up 95%" refer to the endpoints of the desired confidence interval.
University officials have determined that the confidence interval would be more useful if the interval were narrower. Which of the following changes in the confidence level would result in a narrower interval?
A)The university could increase their confidence level.
B)The university could decrease their confidence level. Page 167
9)A retired statistician was interested in determining the average cost of a $200,000.00 term life insurance policy for a 60 - year- old male non - smoker. He randomly sampled 65 subjects (60 - year- old male non - smokers) and constructed the following 95 percent confidence interval for the mean cost of the term life insurance: ($850.00, $1050.00). What value of alpha was used to create this confidence interval?
A)0.10B)0.05C)0.025D)0.01
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
10)Suppose (1,000, 2,100) is a 95% confidence interval for μ. To make more useful inferences from the data, it is desired to reduce the width of the confidence interval. Explain why an increase in sample size will lead to a narrower interval of the estimate of μ.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question True or False.
11)One way of reducing the width of a confidence interval is to reduce the confidence level.
12)The Central Limit Theorem guarantees an approximately normal sampling distribution for the sample mean for large sample sizes, so no knowledge about the distribution of the population is necessary for the corresponding interval to be valid.
13)Since the population standard deviation σ is almost always known, we use it instead of the sample standard deviation s when finding a confidence interval.
14)The confidence coefficient is the relative frequency with which the interval estimator encloses the population parameter when the estimator is used repeatedly a very large number of times.
15)The confidence level is the confidence coefficient expressed as a percentage.
Solve the problem.
16)What is the confidence coefficient in a 95% confidence interval for μ? A).95B).05C).025D).475
17)Which information is not shown on the screen below?
A)the confidence levelB)the sample mean
C)the sample standard deviationD)the sample size
168
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem.
18)Find z α/2 for the given value of α
α = 0.14
A)1.48B)1.08C)0.14D)1.58
19)Determine the confidence level for the given confidence interval for μ
x ± 1.48 σ
A)86%B)93%C)7%D)96.5%
20)A random sample of n measurements was selected from a population with unknown mean μ and known standard deviation σ. Calculate a 95% confidence interval for μ for the given situation. Round to the nearest hundredth when necessary.
n = 100, x = 58, σ = 20
A)58 ± 3.92B)58 ± 3.29C)58 ± 39.2D)58 ± 0.39
21)A 90% confidence interval for the average salary of all CEOs in the electronics industry was constructed using the results of a random survey of 45 CEOs. The interval was ($139,048, $154,144). Give a practical interpretation of the interval.
A)We are 90% confident that the mean salary of all CEOs in the electronics industry falls in the interval $139,048 to $154,144.
B)90% of all CEOs in the electronics industry have salaries that fall between $139,048 to $154,144.
C)We are 90% confident that the mean salary of the sampled CEOs falls in the interval $139,048to $154,144.
D)90% of the sampled CEOs have salaries that fell in the interval $139,048to $154,144.
22)A random sample of 250 students at a university finds that these students take a mean of 15.6 credit hours per quarter with a standard deviation of 2.2 credit hours. Estimate the mean credit hours taken by a student each quarter using a 98% confidence interval. Round to the nearest thousandth.
A)15.6 ± .324B)15.6 ± .219C)15.6
23)A random sample of 250 students at a university finds that these students take a mean of 15.4 credit hours per quarter with a standard deviation of 2.2 credit hours. The 99% confidence interval for the mean is 15.4 ± 0.358. Interpret the interval.
A)We are 99% confident that the average number of credit hours per quarter of students at the university falls in the interval 15.042 to 15.758 hours.
B)99% of the students take between 15.042to 15.758credit hours per quarter.
C)We are 99% confident that the average number of credit hours per quarter of the sampled students falls in the interval 15.042 to 15.758 hours.
D)The probability that a student takes 15.042to 15.758credit hours in a quarter is 0.99.
24)The director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24- hour period. The director randomly selects 36 different 24- hour periods and determines the number of admissions for each. For this sample, x = 17.3 and s2 = 16. Estimate the mean number of admissions per 24- hour period with a 99% confidence interval.
A)17.3 ± 1.717B)17.3 ± 6.867C)17.3 ± .286D)17.3 ± .660 Page 169
25)Suppose a large labor union wishes to estimate the mean number of hours per month a union member is absent from work. The union decides to sample 406 of its members at random and monitor the working time of each of them for 1 month. At the end of the month, the total number of hours absent from work is recorded for each employee. If the mean and standard deviation of the sample are x = 8.1 hours and s = 2.3 hours, find a 99% confidence interval for the true mean number of hours a union member is absent per month. Round to the nearest thousandth.
26)Parking at a large university can be extremely difficult at times. One particular university is trying to determine the location of a new parking garage. As part of their research, officials are interested in estimating the average parking time of students from within the various colleges on campus. A survey of 338 College of Business (COBA) students yields the following descriptive information regarding the length of time (in minutes) it took them to find a parking spot. Note that the "Lo 95%" and "Up 95%" refer to the endpoints of the desired confidence interval.
Give a practical interpretation for the 95% confidence interval given above.
A)95% of the COBA students had parking times of 10.466 minutes.
B)95% of the COBA students had parking times that fell between 9.19 and 11.74 minutes.
C)We are 95% confident that the average parking time of all COBA students falls between 9.19 and 11.74 minutes.
D)We are 95% confident that the average parking time of the 338 COBA students surveyed falls between 9.19 and 11.74 minutes.
27)Parking at a large university can be extremely difficult at times. One particular university is trying to determine the location of a new parking garage. As part of their research, officials are interested in estimating the average parking time of students from within the various colleges on campus. A survey of 338 College of Business (COBA) students yields the following descriptive information regarding the length of time (in minutes) it took them to find a parking spot. Note that the "Lo 95%" and "Up 95%" refer to the endpoints of the desired confidence interval.
Explain what the phrase "95% confident" means when working with a 95% confidence interval.
A)In repeated sampling, 95% of the sample means will fall within the interval created.
B)95% of the observations in the population will fall within the endpoints of the interval.
C)In repeated sampling, 95% of the population means will fall within the interval created.
D)In repeated sampling, 95% of the intervals created will contain the population mean.
28)A retired statistician was interested in determining the average cost of a $200,000.00 term life insurance policy for a 60 - year- old male non - smoker. He randomly sampled 65 subjects (60 - year- old male non - smokers) and constructed the following 95 percent confidence interval for the mean cost of the term life insurance: ($850.00, $1050.00). State the appropriate interpretation for this confidence interval. Note that all answers begin with "We are 95 percent confidence that…"
A)The average term life insurance cost for sampled 65 subjects falls between $850.00 and $1050.00
B)The term life insurance cost of the retired statistician's insurance policy falls between $850.00 and $1050.00
C)The term life insurance cost for all 60- year- old male non - smokers' insurance policies falls between $850.00 and $1050.00
D)The average term life insurance costs for all 60 - year- old male non - smokers falls between $850.00 and $1050.00 Page 170
29)A retired statistician was interested in determining the average cost of a $200,000.00 term life insurance policy for a 60 - year- old male non - smoker. He randomly sampled 65 subjects (60 - year- old male non - smokers) and constructed the following 95 percent confidence interval for the mean cost of the term life insurance: ($850.00, $1050.00). Explain what the phrase "95 percent confident" means in this situation.
A)In repeated sampling, the mean of the population will fall within the specified intervals 95 percent of the time.
B)In repeated sampling, 95 percent of the intervals constructed would contain the value of the true population mean.
C)95 percent of all retired statisticians are underinsured.
D)95 percent of all the life insurance costs will fall within the specified interval.
30)How much money does the average professional football fan spend on food at a single football game? That question was posed to 60 randomly selected football fans. The sampled results show that the sample mean was $70.00 and prior sampling indicated that the population standard deviation was $17.50. Use this information to create a 95 percent confidence interval for the population mean.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
31)How much money does the average professional football fan spend on food at a single football game? That question was posed to 40 randomly selected football fans. The sample results provided a sample mean and standard deviation of $11.00 and $2.80, respectively. Find and interpret a 99% confidence interval for μ
32)To help consumers assess the risks they are taking, the Food and Drug Administration (FDA) publishes the amount of nicotine found in all commercial brands of cigarettes. A new cigarette has recently been marketed. The FDA tests on this cigarette yielded a mean nicotine content of 24.5 milligrams and standard deviation of 2.3 milligrams for a sample of n = 82 cigarettes. Find a 95% confidence interval for μ
33)The following data represent the scores of a sample of 50 randomly chosen students on a standardized test.
a.Write a 95% confidence interval for the mean score of all students who took the test.
b.Identify the target parameter and the point estimator.
34)Suppose that 100 samples of size n = 50 are independently chosen from the same population and that each sample is used to construct its own 95% confidence interval for an unknown population mean μ. How many of the 100 confidence intervals would you expect to actually contain μ?
35)A random sample of n = 100 measurements was selected from a population with unknown mean μ and standard deviation σ. Calculate a 95% confidence interval if x = 26 and s2 = 16.
36)A random sample of n = 144 measurements was selected from a population with unknown mean μ and standard deviation σ. Calculate a 90% confidence interval if x = 3.55 and s = .49.
37)A random sample of 80 observations produced a mean x = 35.4 and a standard deviation s = 3.1.
a. Find a 90% confidence interval for the population mean μ
b. Find a 95% confidence interval for μ
c. Find a 99% confidence interval for μ
d. What happens to the width of a confidence interval as the value of the confidence coefficient is increased while the sample size is held fixed?
1Compare t -Distribution to Normal Distribution
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
1)Suppose you selected a random sample of n = 7 measurements from a normal distribution. Compare the standard normal z value with the corresponding t value for a 90% confidence interval.
2)Suppose you selected a random sample of n = 29 measurements from a normal distribution. Compare the standard normal z value with the corresponding t value for a 95% confidence interval.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
3)An educator wanted to look at the study habits of university students. As part of the research, data was collected for three variables - the amount of time (in hours per week) spent studying, the amount of time (in hours per week) spent playing video games and the GPA - for a sample of 20 male university students. As part of the research, a 95% confidence interval for the average GPA of all male university students was calculated to be: (2.95, 3.10). Which of the following statements is true?
A)In construction of the confidence interval, a t - value with 19 degrees of freedom was used.
B)In construction of the confidence interval, a t - value with 20 degrees of freedom was used.
C)In construction of the confidence interval, a z - value was used.
D)In construction of the confidence interval, a z - value with 20 degrees of freedom was used.
2Use t-Distribution
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem.
4)Find the value of t0 such that the following statement is true: P(- t0 ≤ t ≤ t0 ) = .99 where df = 9.
A)3.250B)2.2821C)2.262D)1.833
5)Find the value of t0 such that the following statement is true: P(- t0 ≤ t ≤ t0 ) = .95 where df = 15.
A)2.131B)1.753C)2.602D)2.947
6)Find the value of t0 such that the following statement is true: P(- t0 ≤ t ≤ t0 ) = .90 where df = 14.
A)1.761B)1.345C)2.145D)2.624
7)Let t0 be a specific value of t. Find t0 such that the following statement is true: P(t ≥ t0 ) = .025 where df = 20.
A)2.086B) - 2.086C)2.093D) - 2.093
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
8)Let t0 be a particular value of t. Find a value of t0 such that P(t ≤ t0 or t ≥ t0 ) = .1 where df = 14. Page 172
9)Let t0 be a particular value of t. Find a value of t0 such that P(
where
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
10)Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is invested in a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of eight private colleges in the United States revealed the following endowments (in millions of dollars): 64.5, 55.1, 232.8, 496.1, 127.6, 186.4, 104.7, and 212.2. What value will be used as the point estimate for the mean endowment of all private colleges in the United States?
A)184.925B)1479.4C)211.343D)8
11)Fifteen SmartCars were randomly selected and the highway mileage of each was noted. The analysis yielded a mean of 47 miles per gallon and a standard deviation of 5 miles per gallon. Which of the following would represent a 90% confidence interval for the average highway mileage of all SmartCars?
12)How much money does the average professional football fan spend on food at a single football game? That question was posed to ten randomly selected football fans. The sampled results show that the sample mean and sample standard deviation were $70.00 and $17.50, respectively. Use this information to create a 95 percent confidence interval for the population mean.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem.
13)You are interested in purchasing a new car. One of the many points you wish to consider is the resale value of the car after 5 years. Since you are particularly interested in a certain foreign sedan, you decide to estimate the resale value of this car with a 99% confidence interval. You manage to obtain data on 17 recently resold 5 - year- old foreign sedans of the same model. These 17 cars were resold at an average price of $12,380 with a standard deviation of $800. What is the 99% confidence interval for the true mean resale value of a 5 - year- old car of this model?
A)12,380 ± 2.921(800/17)B)12,380 ± 2.898(800/17)
C)12,380 ± 2.575(800/17)D)12,380 ± 2.921(800/16)
14)You are interested in purchasing a new car. One of the many points you wish to consider is the resale value of the car after 5 years. Since you are particularly interested in a certain foreign sedan, you decide to estimate the resale value of this car with a 90% confidence interval. You manage to obtain data on 17 recently resold 5 - year- old foreign sedans of the same model. These 17 cars were resold at an average price of $12,320 with a standard deviation of $600. Suppose that the interval is calculated to be ($12,065.92, $12,574.08). How could we alter the sample size and the confidence coefficient in order to guarantee a decrease in the width of the interval?
A)Increase the sample size but decrease the confidence coefficient.
B)Increase the sample size and increase the confidence coefficient.
C)Keep the sample size the same but increase the confidence coefficient.
D)Decrease the sample size but increase the confidence coefficient. Page 173
15)How much money does the average professional football fan spend on food at a single football game? That question was posed to 10 randomly selected football fans. The sample results provided a sample mean and standard deviation of $19.00 and $2.95, respectively. Use this information to construct a 98% confidence interval for the mean.
A)19 ± 2.821(2.95/10)B)19 ± 2.764(2.95/10)
C)19 ± 2.718(2.95/10)D)19 ± 2.262(2.95/10)
16)A marketing research company is estimating the average total compensation of CEOs in the service industry. Data were randomly collected from 18 CEOs and the 90% confidence interval for the mean was calculated to be ($2,181,260, $5,836,180). Explain what the phrase "90% confident" means.
A)In repeated sampling, 90% of the intervals constructed would contain μ
B)90% of the population values will fall within the interval.
C)90% of the sample means from similar samples fall within the interval.
D)90% of the similarly constructed intervals would contain the value of the sample mean.
17)A marketing research company is estimating the average total compensation of CEOs in the service industry. Data were randomly collected from 18 CEOs and the 98% confidence interval for the mean was calculated to be ($2,181,260, $5,836,180). What additional assumption is necessary for this confidence interval to be valid?
A)The population of total compensations of CEOs in the service industry is approximately normally distributed.
B)None. The Central Limit Theorem applies.
C)The sample standard deviation is less than the degrees of freedom.
D)The distribution of the sample means is approximately normal.
18)A marketing research company is estimating the average total compensation of CEOs in the service industry. Data were randomly collected from 18 CEOs and the 97% confidence interval for the mean was calculated to be ($2,181,260, $5,836,180). What would happen to the confidence interval if the confidence level were changed to 95%?
A)The interval would get narrower.
B)The interval would get wider.
C)There would be no change in the width of the interval.
D)It is impossible to tell until the 95% interval is constructed.
19)A computer package was used to generate the following printout for estimating the mean sale price of homes in a particular neighborhood.
At what level of reliability is the confidence interval made?
A)98%B)2%C)49%D)51%
20)A computer package was used to generate the following printout for estimating the mean sale price of homes in a particular neighborhood.
SAMPLE
Which of the following is a practical interpretation of the interval above?
A)We are 90% confident that the mean sale price of all homes in this neighborhood falls between $40,349.40 and $52,850.60.
B)We are 90% confident that the true sale price of all homes in this neighborhood fall between $40,349.40 and $52,850.60.
C)90% of the homes in this neighborhood have sale prices that fall between $40,349.40 and $52,850.60.
D)All are correct practical interpretations of this interval.
21)A computer package was used to generate the following printout for estimating the mean sale price of homes in a particular neighborhood.
A friend suggests that the mean sale price of homes in this neighborhood is $44,000. Comment on your friend's suggestion.
A)Based on this printout, all you can say is that the mean sale price might be $44 ,000.
B)Your friend is wrong, and you are 95% certain.
C)Your friend is correct, and you are 95% certain.
D)Your friend is correct, and you are 100% certain.
22)To help consumers assess the risks they are taking, the Food and Drug Administration (FDA) publishes the amount of nicotine found in all commercial brands of cigarettes. A new cigarette has recently been marketed. The FDA tests on this cigarette yielded mean nicotine content of 28.4 milligrams and standard deviation of 2.2 milligrams for a sample of n = 9 cigarettes. Construct a 98% confidence interval for the mean nicotine content of this brand of cigarette.
A)28.4 ± 2.124B)28.4 ± 2.069C)28.4 ± 2.253D)28.4 ± 2.194
23)Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is invested in a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of eight private colleges in the United States revealed the following endowments (in millions of dollars): 60.2, 47.0, 235.1, 490.0, 122.6, 177.5, 95.4, and 220.0. Summary statistics yield x = 180.975 and s = 143.042. Calculate a 99% confidence interval for the mean endowment of all private colleges in the United States.
24)An educator wanted to look at the study habits of university students. As part of the research, data was collected for three variables - the amount of time (in hours per week) spent studying, the amount of time (in hours per week) spent playing video games and the GPA - for a sample of 20 male university students. As part of the research, a 95% confidence interval for the average GPA of all male university students was calculated to be: (2.95, 3.10). The researcher claimed that the average GPA of all male students exceeded 2.94. Using the confidence interval supplied above, how do you respond to this claim?
A)We are 95% confident that the researcher is correct.
B)We are 100% confident that the researcher is incorrect.
C)We are 95% confident that the researcher is incorrect.
D)We cannot make any statement regarding the average GPA of male university students at the 95% confidence level.
25)An educator wanted to look at the study habits of university students. As part of the research, data was collected for three variables - the amount of time (in hours per week) spent studying, the amount of time (in hours per week) spent playing video games and the GPA - for a sample of 20 male university students. As part of the research, a 95% confidence interval for the average GPA of all male university students was calculated to be: (2.95, 3.10). What assumption is necessary for the confidence interval analysis to work properly?
A)The Central Limit theorem guarantees that no assumptions about the population are necessary.
B)The population that we are sampling from needs to be a t - distribution with n - 1 degrees of freedom.
C)The sampling distribution of the sample mean needs to be approximately normally distributed.
D)The population that we are sampling from needs to be approximately normally distributed.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
26)You are interested in purchasing a new car. One of the many points you wish to consider is the resale value of the car after 5 years. Since you are particularly interested in a certain foreign sedan, you decide to estimate the resale value of this car with a 95% confidence interval. You manage to obtain data on 17 recently resold 5 - year- old foreign sedans of the same model. These 17 cars were resold at an average price of $12,700 with a standard deviation of $700. Create a 95% confidence interval for the true mean resale value of a 5 - year- old car of that model.
27)A marketing research company is estimating the average total compensation of CEOs in the service industry. Data were randomly collected from 18 CEOs and the 97% confidence interval was calculated to be ($2,181,260, $5,836,180). Give a practical interpretation of the confidence interval.
28)A marketing research company is estimating the average total compensation of CEOs in the service industry. Data were randomly collected from 18 CEOs and the 95% confidence interval was calculated to be ($2,181,260, $5,836,180). Based on the interval above, do you believe the average total compensation of CEOs in the service industry is more than $1,500,000? Page 176
29)A computer package was used to generate the following printout for estimating the mean sale price of homes in a particular neighborhood.
SAMPLE MEAN OF X = 46300
SAMPLE STANDARD DEV = 13747
SAMPLE SIZE OF X = 25
CONFIDENCE = 90
UPPER LIMIT = 51003.90
SAMPLE MEAN OF X = 46300
LOWER LIMIT = 41596.10
A friend suggests that the mean sale price of homes in this neighborhood is $42,000. Comment on your friend's suggestion.
30)The following random sample was selected from a normal population: 9, 11, 8, 10, 14, 8. Construct a 95% confidence interval for the population mean μ
31)The following sample of 16 measurements was selected from a population that is approximately normally distributed.
6185927783817578
9587697476848083
Construct a 90% confidence interval for the population mean.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem.
1)A marketing research company is estimating which of two soft drinks college students prefer. A random sample of 157 college students produced the following confidence interval for the proportion of college students who prefer drink A: (.344, .494). Is this a large enough sample for this analysis to work?
A)Yes, since both np ^ ≥ 15 and nq ^ ≥ 15.
B)Yes, since n = 157 (which is 30 or more).
C)No.
D)It is impossible to say with the given information.
2)A marketing research company is estimating which of two soft drinks college students prefer. A random sample of 341 college students produced the following 95% confidence interval for the proportion of college students who prefer one of the colas: (.341, .451). What additional assumptions are necessary for the interval to be valid?
A)No additional assumptions are necessary.
B)The sample proportion equals the population proportion.
C)The sample was randomly selected from an approximately normal population.
D)The population proportion has an approximately normal distribution. Page 177
3)What type of car is more popular among college students, American or foreign? One hundred fifty - nine college students were randomly sampled and each was asked which type of car he or she prefers. A computer package was used to generate the printout below for the proportion of college students who prefer American automobiles.
SAMPLE PROPORTION = .396226
SAMPLE SIZE = 159
UPPER LIMIT = .46418
LOWER LIMIT = .331125
Is the sample large enough for the interval to be valid?
A)Yes, since np ^ and nq ^ are both greater than 15.
B)Yes, since n > 30.
C)No, the sample size should be at 10% of the population.
D)No, the population of college students is not normally distributed.
4)A study was conducted to determine what proportion of all college students considered themselves as full - time students. A random sample of 300 college students was selected and 210 of the students responded that they considered themselves full- time students. A computer program was used to generate the following 95% confidence interval for the population proportion: (0.64814, 0.75186). The sample size that was used in this problem is considered a large sample. What criteria should be used to determine if n is large?
A)If n > 30, then n is considered large.
B)Both np ^ ≥ 15 and nq ^ ≥ 15.
C)If n > 25, then n is considered large.
D)When working with proportions, any n is considered large.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
5)For n = 40 and p ^ = .35, is the sample size large enough to construct a confidence for p ?
6)For n = 40 and p ^ = .45, is the sample size large enough to construct a confidence for p ?
7)For n = 800 and p ^ = .99, is the sample size large enough to construct a confidence for p ?
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question True or False.
8)The sampling distribution for p ^ is approximately normal for a large sample size n, where n is considered large if both n p ^ ≥ 15 and n(1 - p ^ ) ≥ 15.
A)TrueB)False
9)When the sample size is small, confidence intervals for a population proportion are more reliable when the population proportion p is near 0 or 1.
A)TrueB)False Page 178
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem.
10)A marketing research company is estimating which of two soft drinks college students prefer. A random sample of n college students produced the following 99% confidence interval for the proportion of college students who prefer drink A: (.265, .545). Identify the point estimate for estimating the true proportion of college students who prefer that drink.
A).405B).14C).265D).545
11)What type of car is more popular among college students, American or foreign? One hundred fifty - nine college students were randomly sampled and each was asked which type of car he or she prefers. A computer package was used to generate the printout below for the proportion of college students who prefer American automobiles.
SAMPLE PROPORTION = .394141
SAMPLE SIZE = 159
UPPER LIMIT = .464240
LOWER LIMIT = .331153
What proportion of the sampled students prefer foreign automobiles?
A).605859B).394141C).464240D).331153
12)What type of car is more popular among college students, American or foreign? One hundred fifty - nine college students were randomly sampled and each was asked which type of car he or she prefers.. A computer package was used to generate the printout below of a 90% confidence interval for the proportion of college students who prefer American automobiles.
SAMPLE PROPORTION = .396
SAMPLE SIZE = 159
UPPER LIMIT = .472
LOWER LIMIT = .320
Which of the following is a correct practical interpretation of the interval?
A)We are 95% confident that the proportion of all college students who prefer American cars falls between .320 and .472.
B)We are 95% confident that the proportion of the 159 sampled students who prefer American cars falls between .320 and .472.
C)95% of all college students prefer American cars between .320and .472of the time.
D)We are 95% confident that the proportion of all college students who prefer foreign cars falls between .320 and .472.
13)What type of car is more popular among college students, American or foreign? One hundred fifty - nine college students were randomly sampled and each was asked which type of car he or she prefers. A computer package was used to generate the printout below of a 90% confidence interval for the proportion of college students who prefer American automobiles.
SAMPLE PROPORTION = .396
SAMPLE SIZE = 159
UPPER LIMIT = .460
LOWER LIMIT = .332
Based on the interval above, do you believe that 21% of all college students prefer American automobiles?
A)No, and we are 90% confident of it.B)Yes, and we are 100 %sure of it.
C)Yes, and we are 90% confident of it.D)No, and we are 100% sure of it.
14)A newspaper reported on the topics that teenagers most want to discuss with their parents. The findings, the results of a poll, showed that 46% would like more discussion about the family's financial situation, 37% would like to talk about school, and 30% would like to talk about religion. These and other percentages were based on a national sampling of 512 teenagers. Estimate the proportion of all teenagers who want more family discussions about school. Use a 90% confidence level.
A).37 ± .035B).37 ± .002C).63 ± .035D).63 ± .002
15)A newspaper reported on the topics that teenagers most want to discuss with their parents. The findings, the results of a poll, showed that 46% would like more discussion about the family's financial situation, 37% would like to talk about school, and 30% would like to talk about religion. These and other percentages were based on a national sampling of 549 teenagers. Using 99% reliability, can we say that more than 30% of all teenagers want to discuss school with their parents?
A)Yes, since the values inside the 99% confidence interval are greater than .30.
B)No, since the value .30 is not contained in the 99% confidence interval.
C)Yes, since the value .30 falls inside the 99% confidence interval.
D)No, since the value .30 is not contained in the 99% confidence interval.
16)A random sample of 4000 U.S. citizens yielded 2280who are in favor of gun control legislation. Find the point estimate for estimating the proportion of all Americans who are in favor of gun control legislation.
A).5700B)2280C)4000D).4300
17)A random sample of 4000 U.S. citizens yielded 2250 who are in favor of gun control legislation. Estimate the true proportion of all Americans who are in favor of gun control legislation using a 95% confidence interval.
A).5625 ± .0154B).5625 ± .4823C).4375 ± .0154D).4375 ± .4823
18)A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. Use a 90% confidence interval to estimate the true proportion of students who receive financial aid.
A).59 ± .057B).59 ± .002C).59 ± .398D).59 ± .004
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19)A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. The 99% confidence interval for p is 59 ± .09. Interpret this interval.
A)We are 99% confident that the true proportion of all students receiving financial aid is between .50and .68.
B)99% of the students receive between 50% and 68% of their tuition in financial aid.
C)We are 99% confident that between 50% and 68% of the sampled students receive some sort of financial aid.
D)We are 99% confident that 59% of the students are on some sort of financial aid.
20)A confidence interval was used to estimate the proportion of statistics students who are female. A random sample of 72 statistics students generated the following 99% confidence interval: (.438, .642). State the level of reliability used to create the confidence interval.
A)99%B)72%
C)64.2%D) between 43.8% and 64.2%
21)A confidence interval was used to estimate the proportion of statistics students who are female. A random sample of 72 statistics students generated the following 90% confidence interval: (.438, .642). Based on the interval, is the population proportion of females equal to 59%?
A)Maybe. 59% is a believable value of the population proportion based on the information above.
B)No, and we are 90% sure of it.
C)Yes, and we are 90% sure of it.
D)No, the proportion is 54%.
22)A study was conducted to determine what proportion of all college students considered themselves as full - time students. A random sample of 300 college students was selected and 210 of the students responded that they considered themselves full- time students. A computer program was used to generate the following 95% confidence interval for the population proportion: (0.64814, 0.75186). Which of the following practical interpretations is correct for this confidence interval?
A)We are 95% confident that the percentage of all college students who consider themselves full - time students was 0.700.
B)We are 95% confident that the percentage of the 300 students who responded that they considered themselves full- time students falls between 0.648 and 0.752.
C)We are 95% confident that the percentage of all college students who consider themselves full - time students falls between 0.648 and 0.752.
D)We are 95% confident that the percentage of the 300 students who responded that they considered themselves full- time students was 0.700.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
23)The U.S. Commission on Crime randomly selects 600 files of recently committed crimes in an area and finds 380 in which a firearm was reportedly used. Find a 99% confidence interval for p , the true fraction of crimes in the area in which some type of firearm was reportedly used.
24)A newspaper reports on the topics that teenagers most want to discuss with their parents. The findings, the results of a poll, showed that 46% would like more discussion about the family's financial situation, 37% would like to talk about school, and 30% would like to talk about religion. These and other percentages were based on a national sampling of 505 teenagers. Estimate the proportion of all teenagers who want more family discussions about religion. Use a 99% confidence level.
Page 181
25)A random sample of 50 employees of a large company was asked the question, "Do you participate in the company's stock purchase plan?" The answers are shown below.
yesnonoyesnonoyesyesnono noyesyesyesnoyesnonoyesyes noyesyesnoyesyesnoyesyesyes yesnonoyesyesyesyesyesnoyes no yesyesnoyesyesyesyesyesyes
Use a 90% confidence interval to estimate the proportion of employees who participate in the company's stock purchase plan.
1Determine Sample Size
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem.
1)We intend to estimate the average driving time of a group of commuters. From a previous study, we believe that the average time is 42 minutes with a standard deviation of 12 minutes. We want our 98 percent confidence interval to have a margin of error of no more than plus or minus 3 minutes. What is the smallest sample size that we should consider?
A)87B)10C)8D)261
2)A local men's clothing store is being sold. The buyers are trying to estimate the percentage of items that are outdated. They will choose a random sample from the 100,000 items in the store's inventory in order to determine the proportion of merchandise that is outdated. The current owners have never determined the percentage of outdated merchandise and cannot help the buyers. How large a sample do the buyers need in order to be 99% confident that the margin of error of their estimate is about 3%?
A)1842B)3684C)7368D)716
3)A confidence interval was used to estimate the proportion of statistics students who are female. A random sample of 72 statistics students generated the following confidence interval: (.438, .642). Using the information above, what sample size would be necessary if we wanted to estimate the true proportion to within 1% using 95% reliability?
A)9543B)9604C)9989D)9220
4)Sales of a new line of athletic footwear are crucial to the success of a company. The company wishes to estimate the average weekly sales of the new footwear to within $500 with 90% reliability. The initial sales indicate that the standard deviation of the weekly sales figures is approximately $1200. How many weeks of data must be sampled for the company to get the information it desires?
A)16 weeksB)4 weeksC)10weeksD)7794weeks
5)The director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24- hour period. The director randomly selects 64 different 24 - hour periods and determines the number of admissions for each. For this sample, x = 19.8 and s2 = 9. If the director wishes to estimate the mean number of admissions per 24- hour period to within 1 admission with 95% reliability, what is the minimum sample size she should use?
A)35B)312C)18D)159
6)A previous random sample of 4000 U.S. citizens yielded 2250 who are in favor of gun control legislation. How many citizens would need to be sampled for a 99% confidence interval to estimate the true proportion within 5%?
A)653B)690C)664D)611
7)A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. If the dean wanted to estimate the proportion of all students receiving financial aid to within 3% with 90% reliability, how many students would need to be sampled?
A)728B)176C)443D)22
8)After elections were held, it was desired to estimate the proportion of voters who regretted that they did not vote. How many voters must be sampled in order to estimate the true proportion to within 2% (e.g., + 0.02) at the 90% confidence level? Assume that we believe this proportion lies close to 30%.
A)n = 2017
B)n = 2401
C)n = 1692
D)n = 1421
E)Cannot determine because no estimate of p or q exists in this problem.
9)Suppose it is desired to estimate the average time a customer spends in a particular store to within 5 minutes (e.g., + 5 minutes) at 97% reliability. It is estimated that the standard deviation of the times is 15 minutes. How large a sample should be taken to get the desired interval?
A)n = 6.51B)n = 150C)n = 43D)n = 7
10)It is desired to estimate the average time it takes Statistics students to finish a computer project to within two hours at 97% reliability. It is estimated that the standard deviation of the times is 14 hours. How large a sample should be taken to get the desired interval?
A)n = 17B)n = 189C)n = 231D)n = 133
11)It is desired to estimate the proportion of college students who feel a sudden relief now that their statistics class is over. How many students must be sampled in order to estimate the true proportion to within 2% at the 90% confidence level?
A)n = 189
B)n = 2401
C)n = 1692
D)n = 133
E)Cannot determine because no estimate of p or q exists in this problem
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
12)A local men's clothing store is being sold. The buyers are trying to estimate the percentage of items that are outdated. They will choose a random sample from the 100,000 items in the store's inventory in order to determine the proportion of merchandise that is outdated. The current owners have never determined the percentage of outdated merchandise and cannot help the buyers. How large a sample do the buyers need in order to be 90% confident that the margin of error of their estimate is within 5%?
13)Suppose you wanted to estimate a binomial proportion, p , correct to within .05with probability 0.98. What size sample would need to be selected if p is known to be approximately 0.7?
14)The standard deviation of a population is estimated to be 345units. To estimate the population mean to within 50 units with 98% reliability, what size sample should be selected? Page 183
15)Sales of a new line of athletic footwear are crucial to the success of a newly formed company. The company wishes to estimate the average weekly sales of the new footwear to within $150 with 90% reliability. The initial sales indicate that the standard deviation of the weekly sales figures is approximately $1650. How many weeks of data must be sampled for the company to get the information it desires?
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
16)In the construction of confidence intervals, if all other quantities are unchanged, an increase in the sample size will lead to a __________ interval.
A)narrowerB)widerC)less significantD) biased
Answer the question True or False.
17)One way of reducing the width of a confidence interval is to reduce the size of the sample taken.
18)If no estimate of p exists when determining the sample size for a confidence interval for a proportion, we can use .5 in the formula to get a value for n
1Construct, Interpret Confidence Interval
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem.
1)For the given combination of α and degrees of freedom (df), find the value of χ
/2 that would be used to find the lower endpoint of a confidence interval for σ2 .
α = 0.01, df = 30
2)For the given combination of α and degrees of freedom (df), find the value of χ 2 (1 - α/2) that would be used to find the upper endpoint of a confidence interval for σ2 .
= 0.05, df = 25
3)Given the values of x , s, and n, form a 99% confidence interval for σ2 .
= 10.2, s = 8.6, n = 8
, 523.34)B)(28.02 , 417.84)C)(29.18 , 598.1)D)(3.26 , 48.59)
4)Given the values of x , s, and n, form a 99% confidence interval for σ x = 11.8, s = 9.8, n = 8
, 26.07)B)(6.03 , 23.29)C)(33.15 , 679.58)D)(3.38 , 69.34)
5)The daily intakes of milk (in ounces) for ten five - yearold children selected at random from one school were: 28.3 16.6 23.8 25.7 23.0
Find a 99% confidence interval for the standard deviation, σ, of the daily milk intakes of all five - year olds at this school. Round to the nearest hundredth when necessary.
, 13.65)B)(3.58 , 12.27)C)(3.70 , 12.27)D)(0.87 , 3.36) Page 184
6)The mean systolic blood pressure for a random sample of 28 women aged 18 - 24 is 112.4 mm Hg and the standard deviation is 12.8 mm Hg. Construct a 90% confidence interval for the standard deviation σ, of the systolic blood pressures of all women aged 18 - 24. Round to the nearest hundredth when necessary.
7)The mean replacement time for a random sample of 12 CD players is 8.6 years with a standard deviation of 5.8 years. Construct the 98% confidence interval for the population variance, σ2 . Assume the data are normally distributed, and round to the nearest hundredth when necessary.
8)A random sample of 15 crates have a mean weight of 165.2 pounds and a standard deviation of 14.9pounds. Construct a 95% confidence interval for the population standard deviation σ. Assume the population is normally distributed, and round to the nearest hundredth when necessary.
9)The volumes (in ounces) of juice in eight randomly selected juice bottles are as follows:
15.8 15.9 15.1 15.8 15.4 15.6 15.3 15.6 Find a 98% confidence interval for the standard deviation, σ, of the volumes of juice in all such bottles. Round to the nearest hundredth when necessary.
Ch. 7Inferences Based on a Single Sample: Estimation with Confidence Intervals
Answer Key
7.1Identifying and Estimating the Target Parameter
1Define
7.2Confidence Interval for a Population Mean: Normal (z) Statistic
1Understand Confidence Level, Confidence Interval, α
10)An increase in the sample size reduces the sampling variation of the point estimate as it is calculated as σ/ n. The larger the sample size, the smaller this variation which leads to narrower intervals.
We are 99% confident that the average amount a fan spends on food at a single professional football game is between $9.86 and $12.14. Page 186
b.The target parameter is the mean score of all students who took the test, and the point estimator is the sample mean 79.98. 34)95% of the 100 samples, or 95, are expected to produce a confidence interval that contains μ
d. increases
1Compare t -Distribution to Normal Distribution
1) z : 1.645 and t: 1.943; The t value is considerably bigger than the z value.
2) z : 1.96 and t: 2.048; The t value is a little bigger than the z value.
2Use t-Distribution
8) t0 = 1.761; Use table for t.050 with 14 degrees of freedom. 9) t0 = 3.250; Use table for t.005 with 9 degrees of freedom.
3Calculate, Interpret Confidence Interval
For this interval to be valid, we must assume that the population of resale values for all 5 - year- old cars of this model follows an approximately normal distribution.
27)We are 97% confident that the average total compensation of CEOs in the service industry is contained in the interval $2,181,260 to $5,836,180.
28)Since all of the values in the interval are greater than $1,500,000, it seems very likely that the mean is greater than $1,500,000, but we can't be 100% certain.
29)Your friend could be correct. $42 ,000 is contained in the 90% confidence interval. It cannot be ruled out as a possible value for the mean sales price.
23)Let p = the true fraction of crimes in the area in which some type of firearm was reportedly used.
7.6Confidence