Deck 10: Estimation: Describing a Single Population

Question 1

Which of the following statements are correct?

A)If there are two unbiased estimators of a parameter, the one whose variance is larger is said to be relatively efficient.
B)If there are two unbiased estimators of a parameter, the one whose mean is larger is said to be relatively efficient.
C)If there are two unbiased estimators of a parameter, the one whose mean is smaller is said to be relatively efficient.
D)If there are two unbiased estimators of a parameter, the one whose variance is smaller is said to be relatively efficient.

Answer 1

D

Question 2

In developing an interval estimate at 87.4% for a population mean, the value of z to use is:

A)1.15.
B)0.32.
C)1.53.
D)0.16.

Answer 2

C

Question 3

Which of the following is the width of the confidence interval for the population mean?

A)Upper confidence limit + Lower confidence limit
B)(Upper confidence limit - Lower confidence)/2
C)Upper confidence limit - Lower confidence limit
D)None of these choices are correct.

Answer 3

C

Question 4

If the Student t distribution is incorrectly used instead of the Standard normal distribution when finding the confidence interval for the population mean, and the population variance was known, what will happen to the width of the confidence interval?

A)The confidence interval would have the same width.
B)The confidence interval would be wider than it should be, as t is flatter than Z.
C)The confidence interval will be narrower than it should be as t is flatter than Z.
D)None of these choices are correct.

Question 5

Which of the following best describes an unbiased estimator?

A)Every estimator is an unbiased estimator.
B)An interval estimator is an unbiased estimator.
C)An unbiased estimator of a population parameter is an estimator whose expected value is equal to that parameter.
D)None of these choices are correct.

Question 6

A 90% confidence interval estimate of the population mean $\mu$ can be interpreted to mean that:

A)if we repeatedly draw samples of the same size from the same population, 90% of the values of the sample means $$\bar { x }$$ will result in a confidence interval that includes the population mean $\mu$ .
B)there is a 90% probability that the population mean $\mu$ will lie between the lower confidence limit (LCL) and the upper confidence limit (UCL).
C)we are 90% confident that we have selected a sample whose range of values does not contain the population mean $\mu$ .
D)We are 90% confident that 10% the values of the sample means $$\bar { x }$$ will result in a confidence interval that includes the population mean $\mu$ .

Question 7

A random sample of 64 observations has a mean of 30. The population variance is assumed to be 9. The 85.3% confidence interval estimate for the population mean (to the third decimal place) is:

A)[28.369, 31.631].
B)[29.456, 30.544].
C)[28.560, 31.440].
D)[29.383, 30.617].

Question 8

Which of the following statistical distributions is used when estimating the population mean when the population variance is known?

A)Student t distribution
B)Standard normal distribution
C)Chi-square distribution
D)None of these choices are correct.

Question 9

A 95% confidence interval estimate for a population mean $\mu$ is determined to be 43.78 to 52.19. If the confidence level is decreased to 90%, the confidence interval $\mu$ :

A)becomes wider.
B)remains the same.
C)becomes narrower.
D)None of these choices are correct.

Question 10

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

A)level of confidence increases.
B)sample size increases.
C)population standard deviation decreases.
D)Sample mean gets further from the population mean.

Question 11

Which of the following best describes an interval estimator?

A)An interval estimator is the same as a point estimator.
B)An interval estimator is an interval that draws inferences about a population based on a sample statistic.
C)An interval estimator can only be done for the population mean.
D)An interval estimator can only be done for the population proportion.

Question 12

Which of the following statistical distributions are used to find a confidence interval for the population proportion?

A)Student t distribution
B)Standard normal distribution
C)Chi-square distribution
D)None of these choices are correct.

Question 13

The degrees of freedom used to find the t/2 for a confidence interval for the population mean?

A)The degrees of freedom are n − 2.
B)The degrees of freedom are n − 3.
C)The degrees of freedom are n − 1.
D)The degrees of freedom are n.

Question 14

In developing an interval estimate for a population mean, the interval estimate 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)56.34.
B)62.96.
C)13.24.
D)66.15.

Question 15

Which of the following statements is (are) correct?

A)The sample mean is a biased estimator of the population mean.
B)The sample proportion is an unbiased estimator of the population proportion.
C)The sample mean is not a consistent estimator.
D)All of these choices are correct.

Question 16

The sample size needed to estimate a population mean to within 2 units with a 95% confidence when the population standard deviation equals 8 is:

A)9.
B)61.
C)62.
D)8.

Question 17

Which of the following statistical distributions is used when estimating the population mean when the population variance is unknown?

A)Student t distribution
B)Standard normal distribution
C)Chi-square distribution
D)None of these choices are correct.

Question 18

In developing an interval estimate for a population mean, the population standard deviation $\sigma$ was assumed to be 10. The interval estimate was 50.92 ± 2.14. Had the population standard deviation $\sigma$ equaled 20, the interval estimate would have been:

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

Question 19

A point estimate is defined as:

A)the average of the sample values.
B)the average of the population values.
C)a single value of an estimator.
D)an interval within which the population parameter is believed to lie .

Question 20

The z value for a 95% confidence interval estimate is:

A)2.12.
B)1.82.
C)2.00.
D)1.96.

Question 21

In constructing a confidence interval for the population mean when the population variance is unknown, which of the following assumptions is required when using the following formula? $$\bar { X } \pm t _ { a n , n - 1 } ( s / \sqrt { n } )$$

A)The sample size is greater than 30.
B)The population variance is known.
C)The population is normal.
D)The sample is drawn from a positively skewed distribution.

Question 22

For statistical inference about the mean of a single population when the population standard deviation is unknown, the number of degrees for freedom for the t-distribution is equal to n - 1 because we lose one degree of freedom by using the:

A)sample mean as an estimate of the population mean.
B)sample standard deviation as an estimate of the population standard deviation.
C)sample proportion as an estimate of the population proportion.
D)sample size as an estimate of the population size.

Question 23

The upper limit of a confidence interval at the 99% level of confidence for the population proportion if a sample of size 100 had 40 successes is:

A)0.3040.
B)0.4047.
C)0.5257.
D)0.4806.

Question 24

Which of the following is true about the t-distribution?

A)It approaches the normal distribution as the number of degrees of freedom increases.
B)It assumes that the population is normally distributed.
C)It is more spread out than the standard normal distribution.
D)All of these choices are correct.

Question 25

The objective of estimation is to determine the approximate value of:

A)a sample statistic.
B)a population parameter.
C)the sample mean.
D)the sample variance.

Question 26

As the number of degrees of freedom for a t-distribution increases:

A)the dispersion of the distribution decreases.
B)the shape of the distribution becomes narrower and taller.
C)the t-distribution becomes more and more similar to the standard normal distribution.
D)All of these choices are correct.

Question 27

Which of the following assumptions must be true in order to use the formula $$\bar { X } \pm z _ { a/2 } ( \sigma / \sqrt { n } )$$ to find a confidence interval estimate of the population mean?

A)The population variance is known.
B)The population mean is known.
C)The population is normally distributed.
D)The confidence level is greater than 90%.

Question 28

A random sample of size 15 taken from a normally distributed population revealed a sample mean of 75 and a sample variance of 25. The upper limit of a 95% confidence interval for the population mean would equal:

A)77.769.
B)72.231.
C)72.727.
D)77.273.

Question 29

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

A)it has no variance.
B)it might be unbiased.
C)it might not be relatively efficient.
D)it does not tell us how close or far the point estimate might be from the parameter.

Question 30

The smaller the level of confidence used in constructing a confidence interval estimate of the population mean, the:

A)more likely that the confidence interval will contain the population mean.
B)wider the confidence interval.
C)narrower the confidence interval.
D)larger the sample required to estimate the population mean to within a certain error bound.

Question 31

Under which of the following circumstances is it impossible to construct a confidence interval for the population mean?

A)A non-normal population with a large sample and an unknown population variance.
B)A normal population with a large sample and a known population variance.
C)A non-normal population with a small sample and an unknown population variance.
D)A normal population with a small sample and an unknown population variance.

Question 32

A random sample of size n has been selected from a normally distributed population whose standard deviation is s. In estimating an interval for the population mean, the t-distribution should be used instead of the z-test if:

A)n ≥ 30 and $\sigma$ is known.
B) $\sigma$ is unknown and estimated by s, and the population is normal.
C)Population is normal and $\sigma$ is known.
D)None of these choices are correct.

Question 33

For a sample of size 30 taken from a normally distributed population with standard deviation equal to 5, a 95% confidence interval for the population mean would require the use of:

A)z = 1.96.
B)z = 1.645.
C)t = 2.045.
D)t = 1.699.

Question 34

Which of the following statements is false?

A)The t-distribution is symmetric about zero.
B)The t-distribution is more spread out than the standard normal distribution.
C)As the number of degrees of freedom gets smaller, the t-distribution's dispersion gets smaller.
D)The t-distribution is mound-shaped.

Question 35

Which of the following is not a part of the formula for constructing a confidence interval estimate of the population mean?

A)A point estimate of the population mean.
B)The standard error of the sampling distribution of the sample mean.
C)The confidence level.
D)The value of the population mean.

Question 36

The use of the standard normal distribution for constructing a confidence interval estimate for the population proportion p requires that:

A)n $$\hat { p }$$ and n(1 - $$\hat { p }$$ ) are both greater than 5, where $$\hat { p }$$ denotes the sample proportion.
B)np and n(1 - p) are both greater than 5.
C)n $$\hat { p }$$ and n(p + $$\hat { p }$$ ) are both greater than 5.
D)the sample size is greater than 5.

Question 37

The sample variance s2 is an unbiased estimator of the population variance $\sigma$ 2 when the denominator of s2 is: $\begin{array}{|l|l|}\hline\text { A. } & n \\\hline \text { B. } & n-1 \\\hline \text { C. } & \sqrt{n} \\\hline \text { D. } & n+1 \\\hline\end{array}$

Question 38

In the formula $$\bar { X } \pm z _ { a/2 } ( \sigma / \sqrt { n } )$$ , the $\alpha$ /2 refers to:

A)the probability that the confidence interval will contain the population mean.
B)the probability that the confidence interval will not contain the population mean.
C)the area in the lower tail or upper tail of the sampling distribution of the sample mean.
D)the level of confidence.

Question 39

The student t-distribution approaches the normal distribution as the:

A)number of degrees of freedom increases.
B)number of degrees of freedom decreases.
C)sample size decreases.
D)population size increases.

Question 40

In selecting the sample size to estimate the population proportion p, if we have no knowledge of even the approximate values of the sample proportion $$\hat { p }$$ , we:

A)take another sample and estimate p.
B)take two more samples and find the average of their $$\hat { p }$$ values.
C)let $$\hat { p }$$ = 0.50.
D)let $$\hat { p }$$ = 0.95.

Question 41

An unbiased estimator of a population parameter is an estimator whose expected value is equal to the population parameter to be estimated.

Question 42

A sample of 250 observations is to be selected at random from an infinite population. Given that the population proportion is 0.25, the standard error of the sampling distribution of the sample proportion is:

A)0.0274.
B)0.50.
C)0.0316.
D)0.0548.

Question 43

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 44

The sample proportion is a biased estimator of the population proportion.

Question 45

The probability that a confidence interval includes the parameter of interest is either 1 or 0.

Question 46

An interval estimate is an estimate of the range for a population parameter.

Question 47

If the standard error of the sampling distribution of the sample proportion is 0.0229 for samples of size 400, then the population proportion must be either:

A)0.4 or 0.6.
B)0.5 or 0.5.
C)0.2 or 0.8.
D)0.3 or 0.7.

Question 48

From a sample of 300 items, 15 are defective. The point estimate of the population proportion defective will be:

A)0.05.
B)15.
C)0.20.
D)0.05-0.10.

Question 49

The upper limit of the 90% confidence interval for µ, given that n = 64, $$\bar { X }$$ = 70 and $\sigma$ = 20, is 65.89.

Question 50

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

Question 51

A sample of size 300 is to be taken at random from an infinite population. Given that the population proportion is 0.70, the probability that the sample proportion will be smaller than 0.75 is:

A)0.9706.
B)0.4772.
C)0.4706.
D)0.9772.

Question 52

After you calculate the sample size needed to estimate a population proportion to within 0.05, your statistics lecturer tells you the maximum allowable error must be reduced to just 0.025. If the original calculation led to a sample size of 400, the sample size will now have to be:

A)800.
B)200.
C)4000.
D)1600.

Question 53

The sample mean $$\bar { X }$$ is a consistent estimator of the population mean µ.

Question 54

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 55

An interval estimate is a range of values within which the actual value of a population parameter $\mu$ falls.

Question 56

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.

Question 57

A sample of size 200 is to be taken at random from an infinite population. Given that the population proportion is 0.60, the probability that the sample proportion will be greater than 0.58 is:

A)0.281.
B)0.719.
C)0.580.
D)0.762.

Question 58

The lower and upper limits of the 68.26% confidence interval for the population mean µ, given that n = 64, $$\bar { X }$$ = 110 and $\sigma$ = 8, are 109 and 111, respectively.

Question 59

The width of a confidence interval increases as the level of significance increases.

Question 60

The sample standard deviation is an unbiased estimator of the population standard deviation.

Question 61

In the formula $$\bar { X } \pm z _ { a/2 } ( \sigma / \sqrt { n } )$$ , the subscript $$\alpha / 2$$ refers to the area in the lower tail or upper tail of the sampling distribution of the sample mean.

Question 62

The percentage of the confidence interval relies on the significance level. $\alpha$

Question 63

At a given sample size and level of confidence, the smaller the population standard deviation $\sigma$ , the wider and thus the less precise the confidence interval estimate of $\mu$ is.

Question 64

In an effort to identify the true proportion of first-year university students who are under 18 years of age, a random sample of 500 first-year students was taken. Only 50 of them were under the age of 18. The value 0.10 would be used as a point estimate to the true proportion of first-year students aged under 18.

Question 65

When constructing a confidence interval estimate of $\mu$ , doubling the sample size n reduces the width %of the interval by half.

Question 66

You need four values to construct the confidence interval estimate of $\mu$ . These are the sample mean, sample size, population standard deviation and confidence level.

Question 67

In developing an interval estimate for a population mean, the population standard deviation $$\sigma$$ was assumed to be 8. The interval estimate was 50.0 ± 2.50. Had $\sigma$ equalled 16, the interval estimate would have been 100 ± 5.0.

Question 68

The width of the confidence interval estimate of the population mean $\mu$ is a function of only two quantities, the population standard deviation $\sigma$ and the sample size n.

Question 69

The sample variance $$s ^ { 2 }$$ is an unbiased estimator of the population variance $$\sigma ^ { 2 }$$ when the denominator of $$s ^ { 2 }$$ is n - 1.

Question 70

We cannot interpret the confidence interval estimate of $\mu$ as a probability statement about $\mu$ , simply because the population mean is a fixed but unknown quantity.

Question 71

In general, decreasing the confidence level ( $$1 - \alpha$$ ) will narrow the interval.

Question 72

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 narrower.

Question 73

A confidence interval becomes narrower as the sample size increases, for the same percentage of confidence.

Question 74

The larger the level of confidence used in constructing a confidence interval, the wider the confidence interval.

Question 75

Suppose that a 90% confidence interval for µ is given by $$\bar { X } \pm 0.75$$ . This notation means that we are 90% confident that µ falls between $$\bar { X } - 0.75$$ and $$\bar { X } + 0.75$$ .

Question 76

When constructing confidence interval for a parameter, we generally set the confidence level $$1 - \alpha$$ close to 1 (usually between 0.90 and 0.99) because we would like to be reasonably confident that the interval includes the actual value of the population parameter.

Question 77

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

Question 78

The sample mean is an unbiased estimator of the population mean µ, and (when sampling from a normal population) the sample median is also an unbiased estimator of µ. However, the sample mean is relatively more efficient than the sample median.

Question 79

It is possible to construct a confidence interval estimate of the population mean if the population variance is unknown.

Question 80

Suppose that a 95% confidence interval for $\mu$ is given by $$\bar { X } \pm 3.25$$ . This notation means that if we repeatedly draw samples of the same size from the same population, 95% of the values of $\bar { x }$ will be such that $\mu$ would lie somewhere between $\bar { X } - 3.25$ and $\bar { X } + 3.25$ .