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.

Accepted Answer

This statement accurately describes the concept of consistency for an estimator. Consistency means that as the sample size increases, the estimator becomes increasingly accurate, and the difference between the estimated value and the true value of the parameter approaches zero. This is an important property for an estimator to have, as it ensures that with enough data, the estimator will converge on the true value of the parameter.

Question 2

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

Accepted Answer

An unbiased estimator is defined as an estimator whose expected value is equal to the true population parameter. Therefore, on average, across all possible samples, an unbiased estimator will have a value equal to the population parameter.

Question 3

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

Accepted Answer

A good estimator should be unbiased, which means it has zero bias. It should also be consistent, meaning that as the sample size increases, the estimator converges to the true value of the parameter being estimated. Finally, a good estimator should have high relative efficiency, which means it produces estimates with smaller variance than other competing estimators. Therefore, all of the choices listed are important characteristics of a good estimator.

Question 4

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

Accepted Answer

A specific confidence interval obtained from data provides an estimate of the population parameter, but there is a chance (determined by the level of confidence chosen) that the interval may not contain the true population parameter. In other words, there is a level of uncertainty associated with any estimate obtained from a sample of data.

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.

Accepted Answer

The statement is true. The sample proportion is an unbiased estimator of the population proportion, which means that on average, it will equal the true population proportion. Additionally, the variance of the sample proportion is p(1-p)/n, which means that the variance decreases as sample size increases. Both of these properties make the sample proportion a consistent estimator of the population proportion.

Question 6

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

Accepted Answer

This is the definition of an unbiased estimator - its expected value equals the population parameter.

Question 7

An unbiased estimator of a population parameter is defined as:

Accepted Answer

An unbiased estimator is one whose expected value is equal to the parameter being estimated. This means that, on average, the estimator will produce an estimate that is exactly equal to the true value of the parameter. This is the most desirable property for an estimator to have.

Question 8

A point estimator is defined as:

Accepted Answer

A point estimator is a single value that estimates an unknown population parameter. It is used to make inferences about the population based on information gathered from a sample. Examples of point estimators include the sample mean and the sample proportion.

Question 9

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

Accepted Answer

The sample variance is calculated from a sample of data and is used as a point estimate for the population variance. However, since it is only based on a sample, it may not perfectly represent the true population variance.

Question 10

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

Accepted Answer

The sample variance $s^2$ is an unbiased estimator of the population variance $\sigma^2$ when the denominator is $n-1$, not $n$, to correct for the bias introduced by estimating the population mean from the sample.

Question 11

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

Accepted Answer

An interval estimate is an estimate of the range within which a population parameter is expected to lie, not a sample statistic.

Question 12

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

Accepted Answer

A sample mean is considered a consistent estimator of the population mean because as the sample size increases, the sample mean converges to the population mean. Therefore, the statement is true.

Question 13

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

Accepted Answer

This is true. The confidence interval estimate of the population mean is calculated using the sample mean and the sample standard deviation, and is used to estimate the range of values within which the true population mean is likely to fall.

Question 14

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

Accepted Answer

A relatively efficient estimator is the one with the smaller variance. This means that it is more likely to be closer to the true value of the parameter than the other estimator, making it a better choice for estimation. An unbiased estimator with a smaller variance is always relatively efficient compared to other unbiased estimators of the same parameter.

Question 15

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

Accepted Answer

A point estimate is indeed a single sample statistic 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.

Accepted Answer

An unbiased estimator is said to be consistent if the difference between the estimator and the parameter approaches zero as the sample size grows larger, not 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.

Accepted Answer

Unbiasedness only ensures that, on average, the estimator is "correct" and does not have any systematic bias. It is possible for an unbiased estimator to have high variance or be very far off from the true value in some cases. Therefore, unbiasedness does not necessarily imply closeness to the parameter.

Question 18

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

Accepted Answer

This statement is true. The sample variance using n-1 as the denominator is an unbiased estimator of the population variance.

Question 19

An estimator is said to be consistent if:

Accepted Answer

Consistency is the property of an estimator that tends to get closer to the true value of the population parameter as the sample size increases. Option A states this property correctly. Option B, C, and D are incorrect as unbiasedness and zero variance do not imply consistency, and an estimator that stays the same with growing sample size is not consistent.

Question 20

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

Accepted Answer

This is true. An interval estimate provides a range of values within which it is estimated that the population parameter falls. The estimate includes a point estimate (a single value) and a margin of error (the range of values around the point estimate).

Quiz 10: Introduction to Estimation