Deck 8: Confidence Interval Estimation

Question 1

A sample size of 5 provides a sample mean of 9.6.If the population variance is known to be 5 and the population distribution is assumed to be normal,the lower limit for a 90% confidence interval is 7.96.

Answer 1

True

Question 2

A population parameter is used to estimate a confidence interval.

Answer 2

False

Question 3

A point estimate consists of a single sample statistic that is used to estimate the true population parameter.

Answer 3

True

Question 4

Instruction 8-1
A random sample of 100 stores from a large chain of 1,000 garden supply stores was selected to determine the average number of lawnmowers sold at an end-of-season clearance sale.The sample results indicated an average of 6 and a standard deviation of 2 lawnmowers sold.A 95% confidence interval (5.623 to 6.377)was established based on these results.
Referring to Instruction 8-1,95% of the stores have sold between 5.623 and 6.377 lawnmowers.

Question 5

Given a sample mean of 2.1 and a population standard deviation of 0.7 from a sample of 10 data points,a 90% confidence interval will have a width of 2.36.

Question 6

Instruction 8-3
A quality control engineer is interested in the mean length of timber poles being cut automatically by machine.The desired length of the insulation is 12 metres.It is known that the standard deviation in the cutting length is 0.15 metres.A sample of 70 cut poles yields a mean length of 12.14 metres.This sample will be used to obtain a 99% confidence interval for the mean length cut by machine.
Referring to Instruction 8-3,the critical value to use in obtaining the confidence interval is ________.

Question 7

Other things being equal,the confidence interval for the mean will be wider for 95% confidence than for 90% confidence.

Question 8

The t distribution approaches the standardised normal distribution when the number of degrees of freedom increases.

Question 9

Instruction 8-1
A random sample of 100 stores from a large chain of 1,000 garden supply stores was selected to determine the average number of lawnmowers sold at an end-of-season clearance sale.The sample results indicated an average of 6 and a standard deviation of 2 lawnmowers sold.A 95% confidence interval (5.623 to 6.377)was established based on these results.
Referring to Instruction 8-1,there are 10 possible samples of 100 stores that can be selected out of the population of 1,000 stores.

Question 10

Instruction 8-3
A quality control engineer is interested in the mean length of timber poles being cut automatically by machine.The desired length of the insulation is 12 metres.It is known that the standard deviation in the cutting length is 0.15 metres.A sample of 70 cut poles yields a mean length of 12.14 metres.This sample will be used to obtain a 99% confidence interval for the mean length cut by machine.
Referring to Instruction 8-3,the confidence interval is valid only if the lengths cut are normally distributed.

Question 11

Instruction 8-1
A random sample of 100 stores from a large chain of 1,000 garden supply stores was selected to determine the average number of lawnmowers sold at an end-of-season clearance sale.The sample results indicated an average of 6 and a standard deviation of 2 lawnmowers sold.A 95% confidence interval (5.623 to 6.377)was established based on these results.
Referring to Instruction 8-1,we do not know for sure whether the true population mean is between 5.623 and 6.377 lawnmowers sold.

Question 12

Instruction 8-2
The managers of a company are worried about the morale of their employees.In order to determine if a problem in this area exists,they decide to evaluate the attitudes of their employees with a standardised test.They select the Fortunato test of job satisfaction,which has a known standard deviation of 24 points.
Referring to Instruction 8-2,this confidence interval is only valid if the scores on the Fortunato test are normally distributed.

Question 13

Other things being equal,as the confidence level for a confidence interval increases,the width of the interval increases.

Question 14

Instruction 8-1
A random sample of 100 stores from a large chain of 1,000 garden supply stores was selected to determine the average number of lawnmowers sold at an end-of-season clearance sale.The sample results indicated an average of 6 and a standard deviation of 2 lawnmowers sold.A 95% confidence interval (5.623 to 6.377)was established based on these results.
Referring to Instruction 8-1,of all possible samples of 100 stores taken from the population of 1,000 stores,95% of the confidence intervals developed will contain the true population mean within the interval.

Question 15

The confidence interval estimate of the population mean is constructed around the sample mean.

Question 16

The confidence interval obtained will always correctly estimate the population parameter.

Question 17

The sample mean is a point estimate of the population mean.

Question 18

Instruction 8-2
The managers of a company are worried about the morale of their employees.In order to determine if a problem in this area exists,they decide to evaluate the attitudes of their employees with a standardised test.They select the Fortunato test of job satisfaction,which has a known standard deviation of 24 points.
Referring to Instruction 8-2,due to financial limitations,the managers decide to take a sample of 45 employees.This yields a mean score of 88.0 points.A 90% confidence interval would go from ________ to ________.

Question 19

Instruction 8-1
A random sample of 100 stores from a large chain of 1,000 garden supply stores was selected to determine the average number of lawnmowers sold at an end-of-season clearance sale.The sample results indicated an average of 6 and a standard deviation of 2 lawnmowers sold.A 95% confidence interval (5.623 to 6.377)was established based on these results.
Referring to Instruction 8-1,of all possible samples of 100 stores drawn from the population of 1,000 stores,95% of the sample means will fall between 5.623 and 6.377 lawnmowers.

Question 20

Instruction 8-1
A random sample of 100 stores from a large chain of 1,000 garden supply stores was selected to determine the average number of lawnmowers sold at an end-of-season clearance sale.The sample results indicated an average of 6 and a standard deviation of 2 lawnmowers sold.A 95% confidence interval (5.623 to 6.377)was established based on these results.
Referring to Instruction 8-1,if the population had consisted of 10,000 stores,the confidence interval estimate of the mean would have been wider in range.

Question 21

Instruction 8-6
A sample of salary offers (in thousands of dollars)given to students majoring in management is: 28,31,26,32,27,28,27,30,31,and 29.Using this data to obtain a 95% confidence interval resulted in an interval from 27.5 to 30.3.
Referring to Instruction 8-6,a 99% confidence interval for the mean of the population from the same sample would be wider than 27.5 to 30.3.

Question 22

Instruction 8-4
To become an actuary,it is necessary to pass a series of 10 exams,including the most important one,an exam in probability and statistics.An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program.They take a sample of 8 actuarial students in this program and determine that their scores are: 2,5,8,8,7,6,5,and 7.This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program.
Referring to Instruction 8-4,if we use the same sample information to obtain a 95% confidence interval,the resulting interval would be narrower than the one obtained here with 90% confidence.

Question 23

Instruction 8-6
A sample of salary offers (in thousands of dollars)given to students majoring in management is: 28,31,26,32,27,28,27,30,31,and 29.Using this data to obtain a 95% confidence interval resulted in an interval from 27.5 to 30.3.
Referring to Instruction 8-6,95% of all confidence intervals constructed similarly to this one will contain the mean of the population.

Question 24

Instruction 8-5
The actual voltages of power packs labelled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
Referring to Instruction 8-5,it is possible that the 99% confidence interval calculated from the data will not contain the mean voltage for the sample.

Question 25

Instruction 8-8
A hotel chain wants to estimate the average number of rooms rented daily in each month.The population of rooms rented daily is assumed to be normally distributed for each month with a standard deviation of 24 rooms.
Referring to Instruction 8-8,during January,a sample of 16 days has a sample mean of 48 rooms.This information is used to calculate an interval estimate for the population mean to be from 40 to 56 rooms.What is the level of confidence of this interval?

Question 26

Instruction 8-3
A quality control engineer is interested in the mean length of timber poles being cut automatically by machine.The desired length of the insulation is 12 metres.It is known that the standard deviation in the cutting length is 0.15 metres.A sample of 70 cut poles yields a mean length of 12.14 metres.This sample will be used to obtain a 99% confidence interval for the mean length cut by machine.
Referring to Instruction 8-3,the confidence interval indicates that the machine is not working properly.

Question 27

Instruction 8-6
A sample of salary offers (in thousands of dollars)given to students majoring in management is: 28,31,26,32,27,28,27,30,31,and 29.Using this data to obtain a 95% confidence interval resulted in an interval from 27.5 to 30.3.
Referring to Instruction 8-6,95% of the salary offers are between 27.5 and 30.3.

Question 28

Instruction 8-4
To become an actuary,it is necessary to pass a series of 10 exams,including the most important one,an exam in probability and statistics.An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program.They take a sample of 8 actuarial students in this program and determine that their scores are: 2,5,8,8,7,6,5,and 7.This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program.
Referring to Instruction 8-4,it is possible that the confidence interval obtained will not contain the mean score for all actuarial students in the special class.

Question 29

Instruction 8-5
The actual voltages of power packs labelled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
Referring to Instruction 8-5,a 95% confidence interval for the mean voltage of the power pack is wider than a 99% confidence interval.

Question 30

Instruction 8-8
A hotel chain wants to estimate the average number of rooms rented daily in each month.The population of rooms rented daily is assumed to be normally distributed for each month with a standard deviation of 24 rooms.
Referring to Instruction 8-8,during February,a sample of 25 days has a sample mean of 37 rooms.Use this information to calculate a 92% confidence interval for the population mean.

Question 31

Instruction 8-6
A sample of salary offers (in thousands of dollars)given to students majoring in management is: 28,31,26,32,27,28,27,30,31,and 29.Using this data to obtain a 95% confidence interval resulted in an interval from 27.5 to 30.3.
Referring to Instruction 8-6,95% of the time,the sample mean salary offer to students majoring in management will fall between 27.5 and 30.3.

Question 32

Instruction 8-12
A poll was conducted by the marketing department of a video game company to determine the popularity of a new game that was targeted to be launched in three months.Telephone interviews with 1,500 young adults were conducted which revealed that 49% said they would purchase the new game.The margin of error was ±3 percentage points.
Referring to Instruction 8-12,you are 99% confident that the percentage of the targeted young adults who will purchase the new game is somewhere between 46% and 52%.

Question 33

Instruction 8-6
A sample of salary offers (in thousands of dollars)given to students majoring in management is: 28,31,26,32,27,28,27,30,31,and 29.Using this data to obtain a 95% confidence interval resulted in an interval from 27.5 to 30.3.
Referring to Instruction 8-6,it is possible that the mean of the population is between 27.5 and 30.3.

Question 34

Instruction 8-3
A quality control engineer is interested in the mean length of timber poles being cut automatically by machine.The desired length of the insulation is 12 metres.It is known that the standard deviation in the cutting length is 0.15 metres.A sample of 70 cut poles yields a mean length of 12.14 metres.This sample will be used to obtain a 99% confidence interval for the mean length cut by machine.
Referring to Instruction 8-3,the confidence interval goes from ________ to ________.

Question 35

Instruction 8-5
The actual voltages of power packs labelled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
Referring to Instruction 8-5,it is possible that the 99% confidence interval calculated from the data will not contain the mean voltage for the entire population.

Question 36

Instruction 8-6
A sample of salary offers (in thousands of dollars)given to students majoring in management is: 28,31,26,32,27,28,27,30,31,and 29.Using this data to obtain a 95% confidence interval resulted in an interval from 27.5 to 30.3.
Referring to Instruction 8-6,it is possible that the mean of the population is not between 27.5 and 30.3.

Question 37

Instruction 8-4
To become an actuary,it is necessary to pass a series of 10 exams,including the most important one,an exam in probability and statistics.An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program.They take a sample of 8 actuarial students in this program and determine that their scores are: 2,5,8,8,7,6,5,and 7.This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program.
Referring to Instruction 8-4,the mean of the sample is ________,while the standard deviation of the sample is ________.

Question 38

Instruction 8-6
A sample of salary offers (in thousands of dollars)given to students majoring in management is: 28,31,26,32,27,28,27,30,31,and 29.Using this data to obtain a 95% confidence interval resulted in an interval from 27.5 to 30.3.
Referring to Instruction 8-6,95% of the sample means will fall between 27.5 and 30.3.

Question 39

Instruction 8-5
The actual voltages of power packs labelled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
Referring to Instruction 8-5,a 90% confidence interval calculated from the same data would be narrower than a 99% confidence interval.

Question 40

Instruction 8-5
The actual voltages of power packs labelled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
Referring to Instruction 8-5,a 99% confidence interval will contain 99% of the voltages for all such power packs.

Question 41

The t distribution allows the calculation of confidence intervals for means when the actual standard error is not known.

Question 42

For a t distribution with 12 degrees of freedom,the area between -2.6810 and 2.1788 is 0.980.

Question 43

The t distribution allows the calculation of confidence intervals for means for small samples when the population variance is not known,regardless of the shape of the distribution in the population.

Question 44

Suppose a 95% confidence interval for μ has been constructed.If it is decided to take a larger sample and to decrease the confidence level of the interval,then the resulting interval width would ________.(Assume that the sample statistics gathered would not change very much for the new sample. )

A)be the same as the current interval width
B)be narrower than the current interval width
C)be unknown until actual sample sizes and reliability levels were determined
D)be larger than the current interval width

Question 45

It is desired to estimate 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).Which of the following interpretations is correct?

A)95% of the sampled total compensation values fell between $2,181,260 and $5,836,180.
B)In the population of Service industry CEOs,95% of them will have total compensations that fall in the interval $2,181,260 to $5,836,180.
C)We are 95% confident that the mean of the sampled CEOs falls in the interval $2,181,260 to $5,836,180.
D)We are 95% confident that the average total compensation of all CEOs in the Service industry falls in the interval $2,181,260 to $5,836,180.

Question 46

Given a sample mean of 2.1 and a sample standard deviation of 0.7 from a sample of 10 data points,a 90% confidence interval will have a width of 2.36.

Question 47

An economist is interested in studying the incomes of consumers in a particular region.The population standard deviation is known to be $2,000.A random sample of 50 individuals resulted in an average income of $30,000.What is the width of the 90% confidence interval?

A)$930.56
B)$1,457.20
C)$465.20
D)$728.60

Question 48

It is desired to estimate 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 $3,000,000?

A)I am 95% confident that the average compensation is $3,000,000.
B)Yes,and I am 95% confident of it.
C)Yes,and I am 75% confident of it.
D)I cannot conclude that the average exceeds $3,000,000 at the 95% confidence level.

Question 49

An economist is interested in studying the incomes of consumers in a particular region.The population standard deviation is known to be $2,000.A random sample of 50 individuals resulted in an average income of $30,000.What is the upper end point in a 99% confidence interval for the average income?

A)$30,282
B)$30,1042
C)$30,728
D)$30,660

Question 50

A random sample of 50 provides a sample mean of 31 with a standard deviation of S =14.The upper bound of a 90% confidence interval estimate of the population mean is 34.32.

Question 51

The head librarian at the National Library of Australia has asked her assistant for an interval estimate of the mean number of books checked out each day.The assistant provides the following interval estimate: from 740 to 920 books per day.What is an efficient,unbiased point estimate of the number of books checked out each day at the National Library of Australia?

A)830
B)1,660
C)740
D)920

Question 52

Suppose a 95% confidence interval for μ turns out to be (1,000,2,100).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 population.
B)Increase the sample size.
C)Increase the confidence level.
D)Increase the sample mean.

Question 53

Suppose a 95% confidence interval for μ turns out to be (1,000,2,100).What type of reasoning would you make when defining what "95% confident" means about the population?

A)Abductive.
B)Deductive.
C)Inductive.
D)None of the above.

Question 54

In forming a 90% confidence interval for a population mean from a sample size of 22,the number of degrees of freedom from the t distribution equals 22.

Question 55

In the construction of confidence intervals,if all other quantities are unchanged,an increase in the sample size will lead to a _________ interval.

A)less significant
B)narrower
C)biased
D)wider

Question 56

If you were constructing a 99% confidence interval of the population mean based on a sample of n=25 where the standard deviation of the sample S = 0.05,the critical value of t will be

A)2.7874.
B)2.4851.
C)2.4922.
D)2.7969.

Question 57

Private colleges rely on money contributed by individuals for much of their major building expenses.Much of this money is put into a fund called a trust,and the college spends only the interest earned by the fund.A recent survey of 8 private colleges in Australia revealed the following trusts (in millions of dollars): 60.2,47.0,235.1,490.0,122.6,177.5,95.4,and 220.0.What value will be used as the point estimate for the mean trust of all private colleges in Australia?

A)$180.975
B)$8
C)$143.042
D)$1,447.8

Question 58

Suppose a 95% confidence interval for μ turns out to be (1,000,2,100).Give a definition of what it means to be "95% confident" in an inference.

A)95% of the observations in the entire population fall in the given interval.
B)In repeated sampling,the population parameter would fall in the given interval 95% of the time.
C)95% of the observations in the sample fall in the given interval.
D)In repeated sampling,95% of the intervals constructed would contain the population mean.

Question 59

A 99% confidence interval estimate can be interpreted to mean that

A)if all possible samples are taken and confidence interval estimates are developed,99% of them would include the true population mean somewhere within their interval.
B)we have 99% confidence that we have selected a sample whose interval does include the population mean.
C)Both of the above.
D)None of the above.

Question 60

A race car driver tested his car for time from 0 to 100 km/h,and in 20 tests obtained an average of 4.85 seconds with a standard deviation of 1.47 seconds.A 95% confidence interval for the 0 to 100 time is 4.52 seconds to 5.18 seconds.

Question 61

The standardised normal distribution is used to develop a confidence interval estimate of the population proportion when the sample size is sufficiently large.

Question 62

Instruction 8-9
A wealthy real estate investor wants to decide whether it is a good investment to build a high-end shopping complex in a suburb of Sydney.His main concern is the total market value of the 3,605 houses in the suburb.From past experience,the standard deviation of market housing prices is estimated to be $81,000.He commissioned a statistical consulting group to take a sample of 200 houses and obtained a sample average market price of $450,000 and a sample standard deviation of $77,400.The consulting group also found out that the average differences between market prices and appraised prices was $250,000 with a standard deviation of $6,800.Also the proportion of houses in the sample that are appraised for higher than the market prices is 0.24.
Referring to Instruction 8-9,what will be the 90% confidence interval for the average market price of the houses in the suburb constructed by the consulting group?

Question 63

Instruction 8-4
To become an actuary,it is necessary to pass a series of 10 exams,including the most important one,an exam in probability and statistics.An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program.They take a sample of 8 actuarial students in this program and determine that their scores are: 2,5,8,8,7,6,5,and 7.This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program.
Referring to Instruction 8-4,a 90% confidence interval for the mean score of actuarial students in the special program is from ________ to ________.

Question 64

Instruction 8-5
The actual voltages of power packs labelled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
Referring to Instruction 8-5,the critical value for a 99% confidence interval for this sample is ________.

Question 65

Instruction 8-4
To become an actuary,it is necessary to pass a series of 10 exams,including the most important one,an exam in probability and statistics.An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program.They take a sample of 8 actuarial students in this program and determine that their scores are: 2,5,8,8,7,6,5,and 7.This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program.
Referring to Instruction 8-4,the confidence interval will be based on ________ degrees of freedom.

Question 66

Instruction 8-6
A sample of salary offers (in thousands of dollars)given to students majoring in management is: 28,31,26,32,27,28,27,30,31,and 29.Using this data to obtain a 95% confidence interval resulted in an interval from 27.5 to 30.3.
Referring to Instruction 8-6,the confidence interval obtained is valid only if the distribution of the population of salary offers is normal.

Question 67

The t distribution is used to develop a confidence interval estimate of the population proportion when the population standard deviation is unknown.

Question 68

The t distribution is used to develop a confidence interval estimate of the population mean when the population standard deviation is unknown.

Question 69

In estimating the population mean with the population standard deviation unknown,if the sample size is 12,there will be 6 degrees of freedom.

Question 70

Instruction 8-4
To become an actuary,it is necessary to pass a series of 10 exams,including the most important one,an exam in probability and statistics.An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program.They take a sample of 8 actuarial students in this program and determine that their scores are: 2,5,8,8,7,6,5,and 7.This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program.
Referring to Instruction 8-4,for the confidence interval to be valid,it is necessary that test scores of students in the special study program on the actuarial exam be normally distributed.

Question 71

Instruction 8-5
The actual voltages of power packs labelled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
Referring to Instruction 8-5,a 99% confidence interval for the mean voltage of the power packs is from ________ to ________.

Question 72

Instruction 8-5
The actual voltages of power packs labelled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
Referring to Instruction 8-5,a confidence interval estimate of the population mean would only be valid if the distribution of voltages is normal.

Question 73

Instruction 8-4
To become an actuary,it is necessary to pass a series of 10 exams,including the most important one,an exam in probability and statistics.An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program.They take a sample of 8 actuarial students in this program and determine that their scores are: 2,5,8,8,7,6,5,and 7.This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program.
Referring to Instruction 8-4,the critical value used in constructing a 90% confidence interval is ________.

Question 74

The standardised normal distribution is used to develop a confidence interval estimate of the population proportion regardless of whether the population standard deviation is known.

Question 75

Instruction 8-5
The actual voltages of power packs labelled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
Referring to Instruction 8-5,a confidence interval for this sample would be based on the t distribution with ________ degrees of freedom.

Question 76

The confidence interval estimate of the population proportion is constructed around the sample proportion.

Question 77

The t distribution is used to develop a confidence interval estimate of the population mean when the population standard deviation is unknown.

Question 78

Which of the following is NOT true about the Student's t distribution?

A)It is bell shaped and symmetrical.
B)It has more area in the tails and less in the centre than does the normal distribution.
C)It is used to construct confidence intervals for the population mean when the population standard deviation is known.
D)As the number of degrees of freedom increases,the t distribution approaches the normal distribution.

Question 79

The t distribution

A)approaches the normal distribution as the sample size increases.
B)assumes the population is normally distributed.
C)has more area in the tails than does the normal distribution.
D)All of the above.

Question 80

A sample of 100 fuses from a very large shipment is found to have 10 that are defective.The 95% confidence interval would indicate that,for this shipment,the proportion of defective fuses is between 0 and 0.28.