Question 1

Increasing the sample size, n, will result in a point estimate that is closer to the true value of the population parameter.

Accepted Answer

The answer of Increasing the sample size, n, will result...

Question 2

In developing an interval estimate for the population mean  , the population standard deviation  was assumed to be 6. The interval estimate was 45.0  1.5. Had  equalled 12, the interval estimate would be 90  3.

Accepted Answer

The interval estimate would not be 90 11eab386_b92d_d6fb_aec0_e559aff462cb_TB4962_11 3. The interval estimate would be affected but cannot be determined without recalculating the interval estimate with the new population standard deviation.

Question 3

If a store manager has recently stated that she estimates the mean amount spent per customer per visit to be between $38.75 and $72.23, the numbers $38.75 and $72.23 are considered point estimates for the true population mean.

Accepted Answer

The numbers $38.75 and $72.23 are not point estimates for the true population mean. The statement made by the store manager is an interval estimate or a range of values within which the true population mean is estimated to lie with a certain level of confidence.

Question 4

Given that n = 49,  = 75, and  = 7, the lower and upper limits of the 68.26% confidence interval for the population mean  are 74 and 76, respectively.

Accepted Answer

The given information implies that we have a sample mean of 75 with a sample size of 49, and we want to construct a 68.26% confidence interval for the population mean. Since the sample size is large (n≥30), we can use the z-distribution for the confidence interval. The formula for the confidence interval is:

x̄ ± z* (σ/√n)

where x̄ is the sample mean (75), σ is the population standard deviation (unknown in this case), n is the sample size (49), and z* is the critical value that corresponds to the desired confidence level (68.26% in this case).

Using a z-table, we find that the critical value for a 68.26% confidence interval is approximately 1. Therefore, the confidence interval is:

75 ± 1*(σ/√49)
= 75 ± σ/7

We are given that 11eab386_b92d_88d3_aec0_17845e56f1da_TB4962_11 = 7, which corresponds to the sample standard deviation of the data set. Therefore, we can estimate σ as 7/0.9949 ≈ 7.04. Substituting this value into the formula, we get:

75 ± 7.04/7
= 75 ± 1.006

The lower and upper limits of the confidence interval are therefore 74 and 76, respectively. Since these are the same as the ones given in the statement, the answer is true.

Question 5

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

Accepted Answer

The answer of An interval estimate is an estimate of...

Question 6

A point estimate is a single value estimate of the value of a population parameter.

Accepted Answer

A point estimate is a single value estimate of the value of a population parameter, such as the sample mean being used to estimate the population mean.

Question 7

As the sample size increases and other factors remain the same, the width of a confidence interval for a population mean tends to decrease.

Accepted Answer

This is because as the sample size increases, the standard error of the mean decreases as well, resulting in a smaller margin of error and a narrower confidence interval.

Question 8

A point estimate is an estimate of the range of a population parameter.

Accepted Answer

A point estimate is an estimate of a single value of a population parameter, not the range. The range of the parameter would be estimated using interval estimates, such as confidence intervals.

Question 9

The margin of error is a half-width of an interval estimate, equal to the difference between the point estimate on the one hand and either the lower or the upper limit of the interval on the other hand.

Accepted Answer

This is the correct definition of a margin of error.

Question 10

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

Accepted Answer

The value of p is not needed in order to construct a confidence interval estimate of the population proportion. It can be estimated using the sample proportion and the sample size.

Question 11

The unknown parameter of a population is presumed to lie at the centre of the interval that the point estimate and margin of error create.

Accepted Answer

The unknown parameter is not presumed to lie at the center of the confidence interval; it is simply within the interval with a certain level of confidence. The point estimate is at the center, but the true parameter could be anywhere within the interval.

Question 12

The concept of margin of error applies directly when estimating the population mean,  , but is not applicable when estimating the population proportion, p.

Accepted Answer

Margin of error can be used for estimating population proportion as well as population mean.

Question 13

The sample proportion  is an unbiased estimator of the population proportion, p.

Accepted Answer

The sample proportion is an unbiased estimator of the population proportion, which means that on average, the sample proportion equals the population proportion.

Question 14

A point estimate of a population parameter will likely be different from the corresponding population value due to the fact that point estimates are subject to sampling error.

Accepted Answer

Point estimates are based on a sample of individuals, and therefore, the estimate will never be exactly the same as the entire population value. This difference is known as sampling error.

Question 15

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

Accepted Answer

The answer of In the formula  , the ...

Question 16

The confidence coefficient is the probability that a confidence interval will enclose the estimated parameter.

Accepted Answer

The confidence coefficient is a measure of the probability that a given confidence interval will contain the true population parameter.

Question 17

A point estimate is subject to sampling error and will almost always be different from the true value of the population parameter.

Accepted Answer

A point estimate is a single value that is used to estimate the population parameter. It is based on a sample and is subject to sampling error, which means it may not be exactly equal to the true value of the parameter. This is why it is important to calculate a confidence interval around the point estimate to get an idea of how much sampling error is present and to have an idea of the range of values that the true parameter value could fall within.

Question 18

Suppose a 95% confidence interval for the mean height of a 12-year-old male in Canada is 137 to 165 cm. It can be said that 95% of 12-year-old males in Canada have height greater than or equal to 137 cm and less than or equal to 165 cm.

Accepted Answer

The statement is incorrect. A 95% confidence interval means that there is a 95% chance that the true population mean falls within the interval, not that 95% of the individual values fall within the interval. It is possible that some 12-year-old males in Canada have heights outside of this interval, but the interval provides a range of plausible values for the population mean height.

Question 19

Statisticians routinely construct interval estimates by setting the point estimate as the centre of the interval and then creating a range of other possible values, known as the margin of error, below and above the centre.

Accepted Answer

This is a correct description of how interval estimates are constructed in statistics.

Question 20

If the population variance is increased and other factors remain the same, the width of a confidence interval for the population mean tends to increase.

Accepted Answer

This is true because as the population variance increases, the data points are more spread out and thus there is more variability in the sample means. This leads to a wider confidence interval for the population mean.

Quiz 8: Large-Sample Estimation