Question 1

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

Accepted Answer

An unbiased estimator has an expected value that is equal to the true population parameter, which makes it a desirable property for an estimator.

Question 2

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:

Accepted Answer

The standard error of the sampling distribution of the sample proportion is calculated using the formula: SE = sqrt[(p(1-p))/n], where p is the population proportion and n is the sample size. Substituting the given values: SE = sqrt[(0.25(1-0.25))/250] = 0.0274.

Question 3

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.

Accepted Answer

This is the definition of a confidence interval.

Question 4

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

Accepted Answer

The sample proportion is an unbiased estimator of the population proportion, as long as the sample is randomly selected and the sample size is sufficiently large.

Question 5

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

Accepted Answer

A confidence interval is a range of values that is likely to include the true value of a parameter. The probability that a confidence interval includes the parameter of interest is the confidence level. The confidence level is typically set at 95%, which means that there is a 95% chance that the confidence interval will include the true value of the parameter.

Question 6

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

Accepted Answer

An interval estimate provides a range of values within which the population parameter is likely to fall.

Question 7

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:

Accepted Answer

The answer of If the standard error of the sampling...

Question 8

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

Accepted Answer

The point estimate of the population proportion defective is calculated by dividing the number of defective items by the total number of items in the sample, which is 15/300 = 0.05.

Question 9

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

Accepted Answer

The upper limit of a confidence interval is calculated using the formula: $\bar{X} + Z \frac{\sigma}{\sqrt{n}}$, where $\bar{X}$ is the sample mean, $Z$ is the Z-score corresponding to the desired confidence level, $\sigma$ is the population standard deviation, and $n$ is the sample size. Given $n = 64$, $\bar{X} = 70$, and $\sigma = 20$, for a 90% confidence level (where Z is approximately 1.645), the upper limit would be higher than 65.89.

Question 10

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.

Accepted Answer

The sample proportion $\hat{p}$ is considered a consistent estimator of the population proportion $p$ because it is unbiased (its expected value equals the parameter it estimates, $p$), and its variance, $p(1-p)/n$, decreases as the sample size $n$ increases, leading to increased precision in the estimate as more data is collected.

Question 11

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:

Accepted Answer

The sample proportion ($\hat{p}$) follows a normal distribution with mean $\mu_{\hat{p}} = p$ and standard deviation $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$, where $p = 0.70$ is the population proportion and $n = 300$ is the sample size. The Z-score for a sample proportion of 0.75 is calculated as $Z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}$. Substituting the values, we find the Z-score and then use the standard normal distribution to find the probability that $\hat{p} < 0.75$, which is approximately 0.9706.

Question 12

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:

Accepted Answer

To decrease the maximum allowable error from 0.05 to 0.025, the sample size needs to be doubled. Therefore, the new sample size will be 400 x 2 = 800.

Question 13

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

Accepted Answer

As the sample size increases, the sample mean $\bar{X}$ converges in probability to the population mean $\mu$, making it a consistent estimator.

Question 14

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

Being unbiased does not guarantee that the estimator is close to the parameter. There may be some variability or error associated with the estimator, even if its expected value is equal to the true parameter.

Question 15

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

Accepted Answer

The answer of An interval estimate is a range of...

Question 16

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 is the definition of consistency for an estimator.

Question 17

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:

Accepted Answer

The probability that the sample proportion will be greater than 0.58 is 1 - the probability that the sample proportion will be less than or equal to 0.58. The probability that the sample proportion will be less than or equal to 0.58 is the cumulative distribution function of the normal distribution with mean 0.60 and standard deviation sqrt(0.60 * 0.40 / 200) = 0.0387. The cumulative distribution function of the normal distribution with mean 0 and standard deviation 1 at -1.06 is 0.1479. Therefore, the probability that the sample proportion will be greater than 0.58 is 1 - 0.1479 = 0.8521.

Question 18

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.

Accepted Answer

The 68.26% confidence interval for the population mean µ, based on the empirical rule (or one standard deviation from the mean in a normal distribution), can be calculated as $\bar{X} \pm \frac{\sigma}{\sqrt{n}}$. Given $\bar{X} = 110$, $\sigma = 8$, and $n = 64$, the margin of error is $\frac{8}{\sqrt{64}} = 1$. Therefore, the confidence interval is $110 \pm 1$, which is $109$ to $111$, matching the given limits.

Question 19

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

Accepted Answer

As the level of significance increases, the confidence level decreases, leading to a narrower confidence interval.

Question 20

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

Accepted Answer

The sample standard deviation formula corrects for the degree of freedom and gives an unbiased estimator of the population standard deviation.

Quiz 10: Estimation: Describing a Single Population

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

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 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.

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

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

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

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:

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

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

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.

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:

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:

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

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.

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

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.

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:

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.

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

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

Quiz 10: Estimation: Describing a Single Population

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

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 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.

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

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

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

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:

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

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

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

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:

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:

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

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.

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

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.

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:

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

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

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

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

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.

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

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

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.