Deck 8: Confidence Intervals

Question 1

When the margin of error is added to and subtracted from the sample mean,an interval is formed that will contain μ with a probability of (1 - α).

Answer 1

True

Question 2

When the level of confidence and sample size remain the same,a confidence interval for a population proportion p will be narrower when $\hat { p } ( 1 - \hat { p } )$
is larger than when $\hat { p } ( 1 - \hat { p } )$
is smaller.

Answer 2

False

Question 3

When the level of confidence and the sample size remain the same,a confidence interval for a population mean μ will be wider when the sample standard deviation s is small than when s is large.

Answer 3

False

Question 4

When constructing a confidence interval for a sample proportion,the t distribution is appropriate if the sample size is small.

Question 5

If σ is unknown and there is no preliminary sample available to estimate σ,the range can be utilized to determine an estimate of σ when finding sample size.

Question 6

First,a 90 percent confidence interval is constructed from a sample size of 100.Then,for the same identical data,a 92 percent confidence interval is constructed.The width of the 90 percent interval is wider than the 92 percent confidence interval.

Question 7

When the population is normally distributed and the population standard deviation ? is unknown,then for any sample size n,the sampling distribution of $\bar { X }$
is based on the z distribution.

Question 8

Assuming the same level of significance α,as the sample size increases,the value of t_α/2 approaches the value of z_α/2.

Question 9

When determining the sample size n,if the value found for n is 79.2,we would choose to sample 79 observations.

Question 10

When the level of confidence and sample standard deviation remain the same,a confidence interval for a population mean based on a sample of n = 100 will be narrower than a confidence interval for a population mean based on a sample of n = 50.

Question 11

When solving for the sample size needed to compute a 95 percent confidence interval for a population proportion "p," having a given error bound "E," we choose a value of $\hat { p }$
that makes $\hat { p } ( 1 - \hat { p } )$
as small as reasonably possible.

Question 12

The more variable that the population measurements are,the larger the sample size that is needed to accurately estimate µ at a specific confidence level.

Question 13

When the level of confidence and sample proportion $\hat { p }$
remain the same,a confidence interval for a population proportion p based on a sample of n = 100 will be wider than a confidence interval for p based on a sample of n = 400.

Question 14

When determining the sample size (n)for a confidence interval for ρ,if you are using a previous sample,use the reasonable value of ρ that is closest to 0.25.

Question 15

The t distribution always has n degrees of freedom.

Question 16

The quantity √(N - n)/N in the confidence intervals for μ and τ is called the finite population correction and is always less than 1.

Question 17

When determining the sample size (n)for a confidence interval for ρ,if you have no idea what value p is (it could be any value between 0 and 1),use p = .5.

Question 18

When determining the sample size (n)for a confidence interval for ρ,if past experience tells us that p is at least 0.8,use p = .2.

Question 19

When the sample size and sample standard deviation remain the same,a 99 percent confidence interval for a population mean,μ,will be narrower than the 95 percent confidence interval for μ.

Question 20

The standard error of the sample mean is s/√n.

Question 21

When determining the sample size,if the value found is not an integer initially,you should ____________ choose the next highest integer value.

A)Always
B)Sometimes
C)Never

Question 22

The exact spread of the t distribution depends on the _________.

A)Standard deviation of the sample
B)Sample size n
C)Number of degrees of freedom
D)z distribution

Question 23

A confidence interval increases in width as

A)The level of confidence increases.
B)n decreases.
C)s increases.
D)All of these.

Question 24

When the level of confidence and sample standard deviation remain the same,a confidence interval for a population mean based on a sample of n = 100 will be ______________ a confidence interval for a population mean based on a sample of n = 50.

A)Wider than
B)Narrower than
C)Equal to

Question 25

As standard deviation increases,sample size _____________ to achieve a specified level of confidence.

A)Increases
B)Decreases
C)Remains the same

Question 26

The t distribution approaches the _______________ distribution as the sample size ___________.

A)Binomial,increases
B)Binomial,decreases
C)z,decreases
D)z,increases

Question 27

When solving for the sample size needed to compute a 95 percent confidence interval for a population proportion "p," having a given error bound "E," we choose a value of $\hat { p }$
That:

A)Makes
$\hat { p } ( 1 - \hat { p } )$
As small as reasonably possible.
B)Makes
$\hat { p } ( 1 - \hat { p } )$
As large as reasonably possible.
C)Makes
$\hat { p } ( 1 - \hat { p } )$
As close to .5 as reasonably possible.
D)Makes
E)Both Makes

Question 28

When the population is normally distributed,population standard deviation σ is unknown,and the sample size is n = 15;the confidence interval for the population mean μ is based on:

A)The z (normal)distribution.
B)The t distribution.
C)The binomial distribution.
D)The Poisson distribution.
E)None of thesE.

Question 29

The width of a confidence interval will be:

A)Narrower for 98 percent confidence than for 90 percent confidence.
B)Wider for a sample size of 64 than for a sample size of 36.
C)Wider for a 99 percent confidence than for 95 percent confidence
D)Narrower for sample size of 25 than for a sample size of 36.
E)None of thesE.

Question 30

When the sample size and sample standard deviation remain the same,a 99 percent confidence interval for a population mean,μ,will be _________________ the 95 percent confidence interval for μ.

A)Wider than
B)Narrower than
C)Equal to

Question 31

A tolerance interval is meant to contain a specified percentage of the individual population measurements.

Question 32

Which of the following is an advantage of a confidence interval estimate over a point estimate for a population parameter?

A)Interval estimates are more precise than point estimates.
B)Interval estimates are less accurate than point estimates.
C)Interval estimates are both more accurate and more precise than point estimates.
D)Interval estimates take into account the fact that the statistic being used to estimate the population parameter is a random variable.

Question 33

When constructing a confidence interval for a population mean,if a population is normally distributed and a small sample is taken,then the distribution of $\bar { X }$
Is based on the ____________ distribution.

A)z
B)t
C)Neither
D)Both z and t

Question 34

When the sample size and the sample proportion $\hat { p }$
Remain the same,a 90 percent confidence interval for a population proportion p will be ______________ the 99 percent confidence interval for p.

A)Wider than
B)Narrower than
C)Equal to

Question 35

A tolerance interval is always longer than a corresponding confidence interval.

Question 36

When the level of confidence and sample proportion $\hat { p }$
Remain the same,a confidence interval for a population proportion p based on a sample of n = 100 will be ______________ a confidence interval for p based on a sample of n = 400.

A)Wider than
B)Narrower than
C)Equal to

Question 37

When the level of confidence and sample size remain the same,a confidence interval for a population proportion p will be ______________ when $\hat { p } ( 1 - \hat { p } )$
Is larger than when $\hat { p } ( 1 - \hat { p } )$
Is smaller.

A)Wider
B)Narrower
C)Neither Wider nor Narrower,they will be the same.

Question 38

The width of a confidence interval will be:

A)Narrower for 99 percent confidence than 95 percent confidence.
B)Wider for a sample size of 100 than for a sample size of 50.
C)Narrower for 90 percent confidence than 95 percent confidence.
D)Wider when the sample standard deviation (s)is small than when s is large.

Question 39

When the level of confidence and the sample size remain the same,a confidence interval for a population mean μ will be ________________ when the sample standard deviation s is small than when s is large.

A)Wider
B)Narrower
C)Neither Wider nor Narrower,they will be the same.

Question 40

A confidence interval for the population mean is meant to contain a specified percentage of the individual population measurements.

Question 41

If everything else is held constant,decreasing the margin of error causes the required sample size to ____________.

A)Stay the same
B)Decrease
C)Increase

Question 42

When a confidence interval for a population proportion is constructed for a sample size n = 30 and the value of $\hat { p }$
= )4,the interval is based on:

A)The z distribution.
B)The t distribution.
C)The exponential distribution.
D)The Poisson distribution.
E)None of thesE.

Question 43

The tolerance interval of 95.44 percent is ________ a 95.44 percent confidence interval.

A)The same width as
B)Narrower than
C)Wider than

Question 44

As the margin of error decreases,the width of the confidence interval _______________.

A)Stays the same
B)Decreases
C)Increases

Question 45

A confidence interval for the population mean is an interval constructed around the ____________.

A)Sample mean
B)Population mean
C)z test statistic
D)t test statistic

Question 46

In a manufacturing process,a random sample of 9 bolts has a mean length of 3 inches with a variance of .09.What is the 90 percent confidence interval for the true mean length of the bolt?

A)2.8355 to 3.1645
B)2.5065 to 3.4935
C)2.4420 to 3.5580
D)2.8140 to 3.1860
E)2.9442 to 3.0558

Question 47

As the significance level α increases,the width of the confidence interval _______________.

A)Stays the same
B)Decreases
C)Increases

Question 48

In a manufacturing process a random sample of 9 bolts manufactured has a mean length of 3 inches with a standard deviation of .3 inches.What is the 95 percent confidence interval for the true mean length of the bolt?

A)2.804 to 3.196
B)2.308 to 3.692
C)2.769 to 3.231
D)2.412 to 3.588
E)2.814 to 3.186

Question 49

When constructing a confidence interval,as the confidence level required in estimating the mean increases,the width of the confidence interval ______________.

A)Stays the same
B)Decreases
C)Increases

Question 50

When establishing the confidence interval for the average weight of a cereal box,assume that the population standard deviation is known to be 2 ounces.Based on a sample,the average weight of a sample of 20 boxes is 16 ounces.The appropriate test statistics to use is ________.

A)t
B)z
C)χ
D)p

Question 51

In a manufacturing process,a random sample of 36 bolts has a mean length of 3 inches with a standard deviation of .3 inches.What is the 99 percent confidence interval for the true mean length of the bolt?

A)2.902 to 3.098
B)2.884 to 3.117
C)2.864 to 3.136
D)2.228 to 3.772
E)2.802 to 3.198

Question 52

Assuming the same level of significance α,as the sample size increases,the value of t_a/2 ___________ approaches the value of z_α/2.

A)Always
B)Sometimes
C)Never

Question 53

As the standard deviation (σ)decreases,the width of the confidence interval _______________.

A)Stays the same
B)Decreases
C)Increases

Question 54

The internal auditing staff of a local manufacturing company performs a sample audit each quarter to estimate the proportion of accounts that are current (between 0 and 60 days after billing).The historical records show that over the past 8 years 70 percent of the accounts have been current.Determine the sample size needed in order to be 99 percent confident that the sample proportion of the current customer accounts is within .03 of the true proportion of all current accounts for this company.

A)1842
B)1549
C)897
D)632
E)1267

Question 55

The internal auditing staff of a local manufacturing company performs a sample audit each quarter to estimate the proportion of accounts that are delinquent (more than 90 days overdue).For this quarter,the auditing staff randomly selected 400 customer accounts and found that 80 of these accounts were delinquent.What is the 95 percent confidence interval for the proportion of all delinquent customer accounts at this company?

A).1608 to .2392
B).1992 to .2008
C).1671 to .2329
D).1485 to .2515
E).1714 to .2286

Question 56

As the stated confidence level decreases,the width of the confidence interval _______________.

A)Stays the same
B)Decreases
C)Increases

Question 57

A researcher for a paint company is measuring the level of a certain chemical contained in a certain type of paint.If the paint contains too much of this chemical,the quality of the paint will be compromised.On average,each can of paint contains 10 percent of the chemical.How many cans of paint should the sample contain if the researcher wants to be 98 percent certain of being within 1 percent of the true proportion of this chemical?

A)4870
B)1107
C)26
D)645

Question 58

There is little difference between the values of t_α/2 and z_α/2 when:

A)The sample size is small.
B)The sample size is large.
C)The sample mean is small.
D)The sample mean is large.
E)The sample standard deviation is small.

Question 59

As the sample size n increases,the width of the confidence interval _______________.

A)Stays the same
B)Decreases
C)Increases

Question 60

In determining the sample size to estimate a population proportion,as p approaches .5,the calculated value of the sample size ______________.

A)Stays the same
B)Decreases
C)Increases

Question 61

In a random sample of 651 computer scientists who subscribed to a web-based daily news update,it was found that the average salary was $46,816 with a population standard deviation of $12,557.Calculate a 91 percent confidence interval for the mean salary of computer scientists.

A) $[ \$ 25,469 \quad \$ 68,163 ]$
B) $[ \$ 46,592 \quad \$ 47,040 ]$
C) $[ \$ 46,157 \quad \$ 47,475 ]$
D) $[ \$ 46,783 \quad \$ 46,849 ]$
E) $[ \$ 45,981$$\quad$$\$ 47,650 ]$

Question 62

Researchers have studied the role that the age of workers has in determining the hours per month spent on personal tasks.A sample of 1,686 adults were observed for one month.The data are: $\begin{array} { l c c c } & & { \text { AGE GROUP } } \\& \underline { 18 - 24 } & \underline {25 - 44 }& \underline { 45 - 64 } \\\text { Mean } & 4.17 & 4.04 & 4.31\\\text { Std DeV } & 0.75 & 0.81 & 0.82 \\\text { N } & 241 & 768 & 677\end{array}$
Construct a 98 percent confidence interval for the mean hours spent on personal tasks for 25-44 year olds.

A) $\left[ \begin{array} { l l } 3.96 & 4.12\end{array} \right]$
B) $\left[ \begin{array} { l l } 3.97 & 4.11\end{array} \right]$
C) $[ 3.98$$\quad$$4.10 ]$
D) $[ 2.16 \quad 5.92 ]$
E) $\left[ \begin{array} { l l } 3.95 & 4.13\end{array} \right]$

Question 63

The coffee and soup machine at the local bus station is supposed to fill cups with 6 ounces of soup.Ten cups of soup are bought with results of a mean of 5.93 ounces and a standard deviation of 0.13 ounces.Construct a 99 percent confidence interval for the true machine-fill amount.

A) $\left[ \begin{array} { l l } 5.75 & 5.99\end{array} \right]$
B) $\left[ \begin{array} { l l } 5.85 & 6.05\end{array} \right]$
C) $\left[ \begin{array} { l l } 5.90 & 6.00\end{array} \right]$
D) $\left[ \begin{array} { l l } 5.70 & 6.16\end{array} \right]$
E) $\left[ \begin{array} { l l } 5.80 & 6.06\end{array} \right]$

Question 64

Unoccupied seats on flights cause airlines to lose revenue.Suppose a large airline wants to estimate its average number of unoccupied seats per flight over the past year.225 flight records are randomly selected and the number of unoccupied seats is noted,with a sample mean of 11.6 seats and a standard deviation of 4.1 seats.How many flights should we select if we wish to estimate μ to within 2 seats and be 95 percent confident?

A)130
B)65
C)33
D)17
E)12

Question 65

In a manufacturing process,we are interested in measuring the average length of a certain type of bolt.Past data indicate that the standard deviation is .25 inches.How many bolts should be sampled in order to make us 95 percent confident that the sample mean bolt length is within .02 inches of the true mean bolt length?

A)25
B)49
C)423
D)601
E)1225

Question 66

Health insurers and the federal government are both putting pressure on hospitals to shorten the average length of stay (LOS)of their patients.In 1996,the average LOS for non-heart patients was 4.6 days.A random sample of 20 hospitals in one state had a mean LOS for non-heart patients in 2000 of 3.8 days and a standard deviation of 1.2 days.How large a sample of hospitals would we need to be 99 percent confident that the sample mean is within 0.5 days of the population mean?

A)3
B)7
C)32
D)48
E)96

Question 67

The coffee and soup machine at the local bus station is supposed to fill cups with 6 ounces of soup.Ten cups of soup are bought with results of a mean of 5.93 ounces and a standard deviation of 0.13 ounces.How large a sample of soups would we need to be 95 percent confident that the sample mean is within 0.03 ounces of the population mean?

A)97
B)96
C)73
D)62
E)10

Question 68

The U.S.Department of Health and Human Services collected sample data for 772 males between the ages of 18 and 24.That sample group has a mean height of 69.7 inches with a standard deviation of 2.8 inches.Find the 99 percent confidence interval for the mean height of all males between the ages of 18 and 24.

A) $[ 63.19$$\quad$$76.21 ]$
B) $[ 62.49$$\quad$$76.91 ]$
C) $[ 69.65$$\quad$ $69.75 ]$
D) $[ 69.47 \quad 69.93 ]$
E) $[ 69.44$$\quad$$69.96 ]$

Question 69

A local company makes a candy that is supposed to weigh 1.00 ounces.A random sample of 25 pieces of candy produces a mean of 0.996 ounces with a standard deviation of 0.004 ounces.How many pieces of candy must we sample if we want to be 99 percent confident that the sample mean is within 0.001 ounces of the true mean?

A)126
B)124
C)107
D)12
E)6

Question 70

A sociologist develops a test designed to measure a person's attitudes about disabled people,and gives the test to 16 randomly selected subjects.Their mean score is 71.2 with a standard deviation of 10.5.Construct the 99 percent confidence interval for the mean score of all subjects.

A) $\left[ \begin{array} { l l } 40.26 & 102.14\end{array} \right]$
B) $\left[ \begin{array} { l l } 63.46 & 78.94\end{array} \right]$
C) $[ 60.66 \quad 81.74 ]$
D) $[ 64.44 \quad 77.96 ]$
E) $[ 63.21$$\quad$$\quad$$79.19 ]$

Question 71

At the end of 1990,1991,and 1992,the average prices of a share of stock in a portfolio were $34.83,$34.65,and $31.26 respectively.To investigate the average share price at the end of 1993,a random sample of 30 stocks was drawn and their closing prices on the last trading day of 1993 were observed with a mean of 33.583 and a standard deviation of 19.149.Estimate the average price of a share of stock in the portfolio at the end of 1993 with a 90 percent confidence interval.

A) $\left[ \begin{array} { l l } 27.646 & 39.523\end{array} \right.$
B) $\left[ \begin{array} { l l } 26.732 & 40.434\end{array} \right]$
C) $\left[ \begin{array} { l l l } 32.514 & 34.651\end{array} \right]$
D) $\left[ \begin{array} { l l } 32.533 & 34.633\end{array} \right]$
E) $\left[ \begin{array} { l l } 32.269 & 34.897 ]\end{array} \right.$

Question 72

A company is interested in estimating μ,the mean number of days of sick leave taken by its employees.The firm's statistician randomly selects 100 personnel files and notes the number of sick days taken by each employee.The sample mean is 12.2 days,and the sample standard deviation is 10 days.How many personnel files would the statistician have to select in order to estimate μ to within 2 days with a 99 percent confidence interval?

A)2
B)13
C)136
D)165
E)173

Question 73

A local company makes a candy that is supposed to weigh 1.00 ounces.A random sample of 25 pieces of candy produces a mean of 0.996 ounces with a standard deviation of 0.004 ounces.Construct a 98 percent confidence interval for the mean weight of all such candy.

A) $[ 0.9645 \quad 1.0275 ]$
B) $\left[ \begin{array} { l l } 0.9956 & 0.9964\end{array} \right]$
C) $\left[ \begin{array} { l l } 0.9860 & 1.0060\end{array} \right]$
D) $\left[ \begin{array} { l l } 0.9940 & 0.9980\end{array} \right]$
E) $\left[ \begin{array} { l l } 0.9942 & 0.9978\end{array} \right]$

Question 74

An environmental group at a local college is conducting independent tests to determine the distance a particular make of automobile will travel while consuming only 1 gallon of gas.They test a sample of five cars and obtain a mean of 28.2 miles.Assuming that the standard deviation is 2.7 miles,find the 95 percent confidence interval for the mean distance traveled by all such cars using 1 gallon of gas.

A) $\left[ \begin{array} { l l } 26.16 & 30.24\end{array} \right]$
B) $[ 20.70$$\quad$$\quad$$35.70 ]$
C) $\left[ \begin{array} { l l } 24.85 & 31.55\end{array} \right]$
D) $[ 26.70$$\quad$$\quad$$29.70 ]$
E) $\left[ \begin{array} { l l } 25.83 & 30.57\end{array} \right.$

Question 75

The state highway department is studying traffic patterns on one of the busiest highways in the state.As part of the study,the department needs to estimate the average number of vehicles that pass an intersection each day.A random sample of 64 days gives us a sample mean of 14,205 cars and a sample standard deviation of 1,010 cars.After calculating the confidence interval,the highway department officials decide that the precision is too low for their needs.They feel the precision should be 300 cars.Given this precision,and needing to be 99 percent confident,how many days do they need to sample?

A)109
B)80
C)79
D)62
E)9

Question 76

In a study of factors affecting soldiers' decisions to reenlist,320 subjects were measured for an index of satisfaction.The sample mean is 28.8 and the sample standard deviation is 7.3.Use the given sample data to construct the 98 percent confidence interval for the population mean.

A) $[ 27.85$$\quad$$29.75 ]$
B) $[ 27.96$$\quad$$29.64 ]$
C) $\left[ \begin{array} { l l } 11.82 & 45.78\end{array} \right]$
D) $\left[ \begin{array} { l l } 28.75 & 28.85\end{array} \right]$
E) $[ 28.60$$\quad$$29.00 ]$

Question 77

An environmental group at a local college is conducting independent tests to determine the distance a particular make of automobile will travel while consuming only 1 gallon of gas.They test a sample of five cars and obtain a mean of 28.2 miles.How many cars should the environmental group test if they wish to estimate μ,mean miles per 1 gallon,to within 0.5 miles and be 99 percent confident? Assume a population standard deviation of 2.7 miles.

A)25
B)124
C)194
D)618
E)619

Question 78

Health insurers and the federal government are both putting pressure on hospitals to shorten the average length of stay (LOS)of their patients.In 1996,the average LOS for non-heart patients was 4.6 days.A random sample of 20 hospitals in one state had a mean LOS for non-heart patients in 2000 of 3.8 days and a standard deviation of 1.2 days.Calculate a 95 percent confidence interval for the population mean LOS for non-heart patients in the state's hospitals in 2000.

A) $\left[ \begin{array} { l l } 3.24 & 4.36\end{array} \right]$
B) $\left[ \begin{array} { l l } 3.67 & 3.93\end{array} \right]$
C) $\left[ \begin{array} { l l } 3.34 & 4.26\end{array} \right]$
D) $\left[ \begin{array} { l l } 3.38 & 4.22\end{array} \right]$
E) $\left[ \begin{array} { l l } 3.27 & 4.33\end{array} \right]$

Question 79

Unoccupied seats on flights cause airlines to lose revenue.Suppose a large airline wants to estimate its average number of unoccupied seats per flight over the past year.225 flight records are randomly selected and the number of unoccupied seats is noted,with a sample mean of 11.6 seats and a standard deviation of 4.1 seats.Calculate a 90 percent confidence interval for ?,the mean number of unoccupied seats per flight during the past year.

A) $[ 4.86 \quad 18.34 ]$
B) $[ 11.25 \quad 11.95 ]$
C) $[ 11.57 \quad 11.63 ]$
D) $[ 11.15 \quad 12.05 ]$
E) $[ 11.30 \quad 12.20 ]$

Question 80

A psychologist is collecting data on the time it takes to learn a certain task.For 50 randomly selected adult subjects,the sample mean is 16.40 minutes and the sample standard deviation is 4.00 minutes.Construct the 95 percent confidence interval for the mean time required by all adults to learn the task.

A) $\left[ \begin{array} { l l } 8.56 & 24.24\end{array} \right]$
B) $[ 15.47 \quad 17.33 ]$
C) $[ 16.24 \quad 16.56 ]$
D) $\left[ \begin{array} { l l } 15.26 & 17.54\end{array} \right]$
E) $[ 17.12 \quad 48.48 ]$