Deck 10: Introduction to Estimation

Question 1

An unbiased estimator is said to be consistent if the difference between the estimator and the parameter grows smaller as the sample size grows larger.

Answer 1

True

Question 2

An unbiased estimator has an average value (across all samples)equal to the population parameter.

Answer 2

True

Question 3

Which of the following is a characteristic for a good estimator?

A)Being unbiased
B)Being consistent
C)Having relative efficiency
D)All of these choices are true.

Answer 3

All of these choices are true.

Question 4

A specific confidence interval obtained from data will always correctly estimate the population parameter.

Question 5

The sample proportion is a consistent estimator of the population proportion p because it is unbiased and the variance of is p(1 − p)/ n,which grows smaller as n grows larger.

Question 6

An unbiased estimator is a sample statistic whose expected value equals the population parameter.

Question 7

An unbiased estimator of a population parameter is defined as:

A)an estimator whose expected value is equal to the parameter.
B)an estimator whose variance is equal to one.
C)an estimator whose expected value is equal to zero.
D)an estimator whose variance goes to zero as the sample size goes to infinity.

Question 8

A point estimator is defined as:

A)a range of values that estimates an unknown population parameter.
B)a single value that estimates an unknown population parameter.
C)a range of values that estimates an unknown sample statistic.
D)a single value that estimates an unknown sample statistic.

Question 9

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

Question 10

The sample variance s² is an unbiased estimator of the population variance σ² when the denominator of s² is n.

Question 11

An interval estimate is an estimate of the range for a sample statistic.

Question 12

The sample mean is a consistent estimator of the population mean μ.

Question 13

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

Question 14

If there are two unbiased estimators of a parameter,the one whose variance is smaller is said to be relatively efficient.

Question 15

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

Question 16

An unbiased estimator is said to be consistent if the difference between the estimator and the parameter grows larger as the sample size grows larger.

Question 17

Knowing that an estimator is unbiased only assures us that its expected value equals the parameter,but it does not tell us how close the estimator is to the parameter.

Question 18

The sample variance (where you divide by n − 1)is an unbiased estimator of the population variance.

Question 19

An estimator is said to be consistent if:

A)the difference between the estimator and the population parameter grows smaller as the sample size grows larger.
B)it is an unbiased estimator.
C)the variance of the estimator is zero.
D)the difference between the estimator and the population parameter stays the same as the sample size grows larger.

Question 20

An interval estimate is a range of values within which the actual value of the population parameter,such as μ,may fall.

Question 21

Which of the following statements is true?

A)The sample mean is relatively more efficient than the sample median.
B)The version of the sample variance where you divide by n is biased.
C)The sample mean is consistent.
D)All of these choices are true.

Question 22

If there are two unbiased estimators of a population parameter available,the one that has the smallest variance is said to be:

A)a biased estimator.
B)relatively efficient.
C)consistent.
D)relatively unbiased.

Question 23

It is intuitively reasonable to expect that a larger sample will produce more ____________________ results.

Question 24

The sample ____________________ is relatively more efficient than the sample ____________________ when estimating the population mean.

Question 25

The librarian at the New York City Public Library has asked her assistant for an interval estimate of the mean number of books checked out each day.The assistant took a sample and found the mean to be 880 books.She provides the librarian with an interval estimate of between 790 and 970 books checked out per day.An efficient,unbiased point estimate of the number of books checked out each day at the New York City Public Library is:

A)790
B)880
C)90
D)None of these choices.

Question 26

____________________ estimators do not have the capacity to reflect the effects of larger sample sizes.

Question 27

Draw a sampling distribution of an unbiased estimator for .

Question 28

A(n)____________________ estimator of a population parameter is an estimator whose expected value is equal to that parameter.

Question 29

The problem with relying on a point estimate of a population parameter is that:

A)it is virtually certain to be wrong.
B)it doesn't have the capacity to reflect the effects of larger sample sizes.
C)it doesn't tell us how close or far the point estimate might be from the parameter.
D)All of these choices are true.

Question 30

If there are two unbiased estimators of the same parameter,the one whose variance is smaller is said to be relatively more ____________________.

Question 31

The sample variance s² is an unbiased estimator of the population variance σ² when the denominator of s² is

A)n + 1
B)n
C)n − 1
D)

Question 32

Which of the following statements is correct?

A)The sample mean is an unbiased estimator of the population mean.
B)The sample proportion is an unbiased estimator of the population proportion.
C)The difference between two sample means is an unbiased estimator of the difference between two population means.
D)All of these choices are true.

Question 33

____________________ estimators reflect the effects of larger sample sizes,but ____________________ estimators do not.

Question 34

Define unbiasedness.

Question 35

The version of the sample variance where you divide by ____________________ gives you an unbiased estimator of the population variance.

Question 36

An unbiased estimator is ____________________ if its variance gets smaller as n gets larger.

Question 37

Define consistency.

Question 38

An interval estimator estimates the value of an unknown ____________________.

Question 39

Is the sample mean a consistent estimator of the population mean? Explain

Question 40

The sample ____________________ is an unbiased estimator for the population mean.

Question 41

Draw a sampling distribution of a biased estimator for .

Question 42

The difference between the sample statistic and actual value of the population parameter is the confidence level of the estimate.

Question 43

The larger the confidence level used in constructing a confidence interval estimate of the population mean,the narrower the confidence interval.

Question 44

A random sample of 10 university students was surveyed to help estimate the average amount of time students spent per week on their computers.The student hours spent using a personal computer over a randomly selected week were 13,14,5,6,8,10,7,12,15,3.
a.Find an unbiased estimator of the average time per week for all university students.
b.Find an unbiased estimator of the variance.
c.Find a consistent estimator of the average time per week for all university students.

Question 45

When constructing confidence interval for a parameter,we generally set the confidence level 1 − α close to 1 (usually between 0.90 and 0.99)because it is the probability that the interval includes the actual value of the population parameter.

Question 46

In order to construct a confidence interval estimate of the population mean,the value of the population mean is needed.

Question 47

Explain briefly why interval estimators are preferred to point estimators.

Question 48

We cannot interpret the confidence interval estimate of μ as a probability statement about μ because the population mean is a fixed quantity.

Question 49

The width of the confidence interval estimate of the population mean μ is a function of only two quantities: the population standard deviation σ and the sample size n.

Question 50

Suppose that a 90% confidence interval for μ is given by .This notation means that we are 90% confident that μ falls between and and

Question 51

The width of a 95% confidence interval is 0.95.

Question 52

The term 1 − α refers to the probability that a confidence interval does not contain the population parameter.

Question 53

One can reduce the width of a confidence interval by taking a smaller sample size.

Question 54

A 95% confidence interval estimate for a population mean μ is determined to be 75 to 85.If the confidence level is reduced to 80%,the confidence interval for μ becomes wider.

Question 55

In developing an interval estimate for a population mean,the population standard deviation σ was assumed to be 8.The interval estimate was 50.0 ± 2.50.Had σ equaled 16,the interval estimate would be 100 ± 5.0.

Question 56

In the formula ,the subscript α / 2 refers to the area in the lower tail or upper tail of the sampling distribution of the sample mean.

Question 57

A confidence interval is an interval estimate for which there is a specified degree of certainty that the actual value of the population parameter will fall within the interval.

Question 58

Draw sampling distributions of a consistent estimator for μ where one sample mean is larger than the other.

Question 59

Draw the sampling distribution of two unbiased estimators for μ,one of which is relatively efficient.

Question 60

Define relative efficiency.

Question 61

The value for a 95% confidence interval estimate for a population mean μ is

A)0.95
B)0.025
C)1.65
D)1.96

Question 62

The width of a confidence interval estimate of the population mean increases when the:

A)level of confidence increases
B)sample size decreases
C)value of the population standard deviation increases
D)All of these choices are true.

Question 63

The letter α in the formula for constructing a confidence interval estimate of the population mean is:

A)the level of confidence.
B)the probability that a particular confidence interval will contain the population mean.
C)the area in the lower tail of the sampling distribution of the sample mean.
D)None of these choices.

Question 64

Suppose a sample size of 5 has mean 9.60.If the population variance is 5 and the population is normally distributed,the lower limit for a 92% confidence interval is 7.85.

Question 65

Which of the following is an incorrect statement about a 90% confidence interval?

A)If we repeatedly draw samples of the same size from the same population,90% of the resulting confidence intervals will include μ.
B)There is a 90% probability that the population mean μ will lie between the lower confidence limit (LCL)and the upper confidence limit (UCL).
C)We are 90% confident that our sample mean equals the population mean μ.
D)90% of the population values will lie within the confidence interval.

Question 66

Increasing the value of 1 − α narrows a confidence interval.

Question 67

In developing an interval estimate for a population mean,the population standard deviation σ was assumed to be 10.The interval estimate was 50.92 ± 2.14.Had σ equaled 20,the interval estimate would be

A)60.92 ± 2.14
B)50.92 ± 12.14
C)101.84 ± 4.28
D)50.92 ± 4.28

Question 68

If the confidence level is reduced,the confidence interval:

A)widens.
B)remains the same.
C)narrows.
D)disappears.

Question 69

When constructing confidence interval estimate of μ,doubling the sample size n decreases the width of the interval by half.

Question 70

Suppose an interval estimate for the population mean was 62.84 to 69.46.The population standard deviation was assumed to be 6.50,and a sample of 100 observations was used.The mean of the sample was:

A)6.62
B)56.34
C)62.96
D)66.15

Question 71

Other things being equal,the confidence interval for the mean will be wider for 99% confidence than for 95% confidence.

Question 72

After constructing a confidence interval estimate for a population mean,you believe that the interval is useless because it is too wide.In order to correct this problem,you need to:

A)increase the sample size.
B)increase the population standard deviation.
C)increase the level of confidence.
D)increase the sample mean.

Question 73

In developing an interval estimate for a population mean,a sample of 50 observations was used.The interval estimate was 19.76 ± 1.32.Had the sample size been 200 instead of 50,the interval estimate would have been:

A)19.76 ± .33
B)19.76 ± .66
C)19.76 ± 5.28
D)None of these choices.

Question 74

Doubling the population standard deviation σ has the effect of doubling the width of the confidence interval estimate of μ.

Question 75

The term 1 − α refers to:

A)the probability that a confidence interval does not contain the population parameter.
B)the confidence level.
C)the level of unbiasedness.
D)the level of consistency.

Question 76

The lower limit of the 90% confidence interval for μ,where n = 64, = 70,and σ = 20,is 65.89.

Question 77

In thisChapter you need four values to construct the confidence interval estimate of μ.They are the sample mean,the sample size,the population standard deviation,and the confidence level.

Question 78

Suppose that a 95% confidence interval for μ is given by .This notation means that,if we repeatedly draw samples of the same size from the same population,95% of the values of will be such that μ would lie somewhere between .

Question 79

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

Question 80

Given a mean of 2.1 and a standard deviation of 0.7,a 90% confidence interval will be 2.1 ± 0.7.