Deck 9: Introduction to Estimation

Question 1

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.

Answer 1

True

Question 2

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

Answer 2

True

Question 3

In determining the sample size needed to estimate the population proportion p, we let the sample proportion = 1 if we have no knowledge of even the approximate values of .

Answer 3

False

Question 4

The sample mean is a consistent estimator of the population mean $\mu$ .

Question 5

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

Question 6

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

Question 7

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

Question 8

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 9

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

Question 10

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

Question 11

The sample variance s² is an unbiased estimator of the population variance $\sigma$ ² when the denominator of s² is n.

Question 12

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 13

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

Question 14

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

Question 15

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

Question 16

The lower limit of the 90% confidence interval for the population proportion p, given that n = 400 and = 0.10, is 0.0247.

Question 17

The sampling distribution of is approximately normal if the sample size is more than 30.

Question 18

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

Question 19

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 20

In testing a hypothesis about a population proportion p, the z test statistic measures how close the computed sample proportion has come to the hypothesized population parameter.

Question 21

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

A) nand n(1 -) both greater than 5.
B) np and n(1 - p) both greater than 5.
C) n(1 +) and n(1 -) both greater than 5.
D) sample size greater than 5.

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

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 24

Under what condition(s) does the test statistic for p have an approximate normal distribution?

A) When np > 5.
B) When np and np(1 - p) are both > 5.
C) When n > 30.
D) When np and n(1 - p) are both > 5.

Question 25

Assuming that all necessary conditions are met, what needs to be changed in the formula so that we can use it to construct a (1 - $\alpha$ ) confidence interval estimate for the population proportion p?

A) should be replaced by p.
B) t _$\alpha$ should be replaced by z _$\alpha$ .
C) t _$\alpha$ should be replaced by t _{$\alpha$ / 2}.
D) t _$\alpha$ should be replaced by z _{$\alpha$ / 2}.

Question 26

Which of the following would be an appropriate null hypothesis?

A) The population proportion is equal to 0.60.
B) The sample proportion is equal to 0.60.
C) The population proportion is not equal to 0.60.
D) All of these choices are true.

Question 27

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 28

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 29

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 30

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 31

After calculating the sample size needed to estimate a population proportion to within 0.04, your statistics professor told you the maximum allowable error must be reduced to just .01. If the original calculation led to a sample size of 800, the sample size will now have to be:

A) 800
B) 3,200
C) 6,400
D) 12,800

Question 32

The width of a confidence interval estimate for a proportion will be:

A) narrower for 90% confidence than for 95% confidence.
B) wider for a sample size of 100 than for a sample size of 50.
C) narrower for 99% confidence than for 95% confidence.
D) narrower when the sample proportion if 0.50 than when the sample proportion is 0.20.

Question 33

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 , we:

A) take another sample and estimate.
B) take two more samples and find the average of their.
C) let= 0.50.
D) let= 0.95.

Question 34

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 35

Which of the following is not a characteristic for a good estimator?

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

Question 36

If we have some idea about the value of sample proportion , we use that value in determining the sample size needed to estimate the population proportion p.

Question 37

When determining the sample size needed for a proportion for a given level of confidence and sampling error, the closer to 0.50 that p is estimated to be:

A) the smaller the sample size required.
B) the larger the sample size required.
C) the sample size is not affected.
D) the effect cannot be determined from the information given.

Question 38

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 39

The 95% confidence interval would indicate that, for this shipment, the proportion of defective fuses is between 0 and 0.28.

Question 40

Which of the following would be an appropriate alternative hypothesis?

A) The population proportion is less than 0.65.
B) The sample proportion is less than 0.65.
C) The population proportion is equal to 0.65.
D) The sample proportion is equal to 0.65.

Question 41

When a population is small, it is necessary to include the ____________________ factor in our hypothesis tests and confidence interval estimators for p.

Question 42

A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim against the alternative that the actual proportion of doctors who recommend aspirin is less than 0.90, a random sample of 100 doctors' results in 83 who indicate that they recommend aspirin. The value of the test statistic in this problem is approximately equal to:

A) -1.67
B) -2.33
C) -1.86
D) -0.14

Question 43

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

Question 44

The sampling error for a confidence interval is also defined as the ____________________ of ____________________.

Question 45

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

Question 46

The formula is used to find the sample size needed to estimate a population proportion. In this formula, B represents the ____________________ on the error of estimation.

Question 47

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

Question 48

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

Question 49

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

Question 50

Estimating or testing for p is used in situations when the data are ____________________.

Question 51

To produce a confidence interval estimator for the total, we multiply the lower and upper confidence limits of the interval estimator of p by ____________________.

Question 52

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

Question 53

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

Question 54

The test statistic for p is approximately normal when np and n(1 - p) are both ____________________.

Question 55

The formula is used to find the ____________________ to estimate a population proportion.

Question 56

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

Question 57

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

Question 58

An interval estimator estimates the value of an unknown ____________________.

Question 59

The test statistic for p is approximately normal when ____________________ and ____________________ are both greater than 5.

Question 60

From a sample of 400 items, 14 are found to be defective. The point estimate of the population proportion defective will be:

A) 0.035
B) 0.05
C) 14
D) 28.57

Question 61

If the population standard deviation is guesstimated, and it turned out to be larger than you assumed, then the sample size you calculated is ____________________ than it needs to be.

Question 62

Draw a sampling distribution of an unbiased estimator for $\mu$ .

Question 63

Socialist Party Voters: A pollster in Italy wants to challenge a claim that 5% of the registered voters in his country are Socialists; he thinks the percentage is lower than that. In a test of hypothesis, H₀: p = 0.05 vs. H₁: p < 0.05, his random sample of size 1,000 registered voters revealed that the number of Socialists was 40.

-{Compute the p-value and explain how to use it to test the hypotheses.

Question 64

Define unbiasedness.

Question 65

Statisticians can control the ____________________ of a confidence interval by determining the sample size necessary to produce the desired results.

Question 66

When determining the required sample size for a confidence interval, you need to know the population ____________________, the confidence ____________________, and the ____________________ on the error of estimation.

Question 67

The error of estimation is the ____________________ between an estimator and the parameter.

Question 68

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

Question 69

Explain briefly why interval estimators are preferred to point estimators.

Question 70

Define relative efficiency.

Question 71

The bound on the error of estimation is the ____________________ amount of sampling error that we are willing to tolerate.

Question 72

Draw a sampling distribution of a biased estimator for $\mu$ .

Question 73

Socialist Party Voters: A pollster in Italy wants to challenge a claim that 5% of the registered voters in his country are Socialists; he thinks the percentage is lower than that. In a test of hypothesis, H₀: p = 0.05 vs. H₁: p < 0.05, his random sample of size 1,000 registered voters revealed that the number of Socialists was 40.

-Test the hypotheses at the 5% significance level.

Question 74

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

Question 75

If the population standard deviation is guesstimated, and it turned out to be smaller than you assumed, then the sample size you calculated is ____________________ than it needs to be.

Question 76

Define consistency.

Question 77

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

Question 78

Because n is an integer and we want the bound on the error of estimation to be no more than a given amount, any non-integer value found for n must always be rounded ____________________.

Question 79

As the bound on the error of estimation decreases, the sample size ____________________.

Question 80

If the bound on the error of estimation decreases, the sample size ____________________.