Question 1

The  value for a 95% confidence interval estimate for a population mean μ is
A)0.95
B)0.025
C)1.65
D)1.96

Accepted Answer

The value 1.96 corresponds to a 95% confidence level for a two-tailed test.

Question 2

The width of a confidence interval estimate of the population mean increases when the:
A)level of confidence increases
B)sample size decreases
C)value of the population standard deviation increases
D)All of these choices are true.

Accepted Answer

The width of a confidence interval estimate of the population mean is calculated according to the level of confidence, sample size, and population standard deviation.

A) When the level of confidence increases, the width of the confidence interval also increases.

B) When the sample size decreases, the width of the confidence interval also increases because smaller sample sizes result in less precise estimates.

C) When the value of the population standard deviation increases, the width of the confidence interval also increases as it indicates more variability in the population.

Therefore, all of these choices are true, and the best answer is D.

Question 3

The letter α in the formula for constructing a confidence interval estimate of the population mean is:
A)the level of confidence.
B)the probability that a particular confidence interval will contain the population mean.
C)the area in the lower tail of the sampling distribution of the sample mean.
D)None of these choices.

Accepted Answer

The letter α in the formula for constructing a confidence interval estimate of the population mean represents the significance level, which is the probability of rejecting the null hypothesis when it is true. It is not directly the level of confidence, the probability that a particular confidence interval will contain the population mean, or the area in the lower tail of the sampling distribution of the sample mean. The level of confidence is actually 1 - α, not α itself.

Question 4

Suppose a sample size of 5 has mean 9.60.If the population variance is 5 and the population is normally distributed,the lower limit for a 92% confidence interval is 7.85.

Accepted Answer

Given the sample size (n=5), sample mean (x̄=9.60), population variance (σ^2=5), and assuming a normal distribution, the calculation for a confidence interval involves finding the standard error of the mean (σ/√n) and then using a Z or T distribution to find the critical value for the desired confidence level (92% in this case). Since the sample size is small (n<30), the T distribution is more appropriate. However, without the exact degrees of freedom and T value specified, we cannot calculate the exact confidence interval here. The statement claims the lower limit is 7.85, which could be true given the provided information and assuming the correct critical value from the T distribution was used.

Question 5

Which of the following is an incorrect statement about a 90% confidence interval?
A)If we repeatedly draw samples of the same size from the same population,90% of the resulting confidence intervals will include μ.
B)There is a 90% probability that the population mean μ will lie between the lower confidence limit (LCL)and the upper confidence limit (UCL).
C)We are 90% confident that our sample mean equals the population mean μ.
D)90% of the population values will lie within the confidence interval.

Accepted Answer

The answer of Which of the following is an incorrect...

Question 6

Increasing the value of 1 − α narrows a confidence interval.

Accepted Answer

Increasing the value of $1 - \alpha$ actually widens a confidence interval because it corresponds to increasing the confidence level, requiring a broader interval to capture the parameter with higher certainty.

Question 7

In developing an interval estimate for a population mean,the population standard deviation σ was assumed to be 10.The interval estimate was 50.92 ± 2.14.Had σ equaled 20,the interval estimate would be
A)60.92 ± 2.14
B)50.92 ± 12.14
C)101.84 ± 4.28
D)50.92 ± 4.28

Accepted Answer

The interval estimate depends on the standard deviation. Doubling σ from 10 to 20 would double the margin of error (the ± value) but not affect the estimate's center (50.92). Thus, the new interval is 50.92 ± 4.28.

Question 8

If the confidence level is reduced,the confidence interval:
A)widens.
B)remains the same.
C)narrows.
D)disappears.

Accepted Answer

Reducing the confidence level decreases the range of values within which the true parameter is expected to lie, thus narrowing the confidence interval.

Question 9

When constructing confidence interval estimate of μ,doubling the sample size n decreases the width of the interval by half.

Accepted Answer

Doubling the sample size decreases the standard error of the mean (since it is inversely proportional to the square root of the sample size), which in turn decreases the width of the confidence interval, but not by half. The decrease is less than half because the relationship is not linear but based on the square root of the sample size.

Question 10

Suppose an interval estimate for the population mean 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)6.62
B)56.34
C)62.96
D)66.15

Accepted Answer

The interval estimate is calculated as the sample mean plus or minus the margin of error. The midpoint of the interval (62.84 + 69.46) / 2 = 66.15, which is the sample mean.

Question 11

Other things being equal,the confidence interval for the mean will be wider for 99% confidence than for 95% confidence.

Accepted Answer

This is because a 99% confidence level requires a larger margin of error. Therefore, the range of values the true population mean may fall within is wider, resulting in a wider confidence interval.

Question 12

After constructing a confidence interval estimate for a population mean,you believe that the interval is useless because it is too wide.In order to correct this problem,you need to:
A)increase the sample size.
B)increase the population standard deviation.
C)increase the level of confidence.
D)increase the sample mean.

Accepted Answer

Increasing the sample size will decrease the width of the confidence interval. This is because the standard error of the mean (which is used to calculate the confidence interval) is inversely proportional to the sample size. Therefore, increasing the sample size will make the estimate more precise and reduce the width of the confidence interval. Increasing the population standard deviation, level of confidence, or sample mean will not necessarily decrease the width of the confidence interval.

Question 13

In developing an interval estimate for a population mean,a sample of 50 observations was used.The interval estimate was 19.76 ± 1.32.Had the sample size been 200 instead of 50,the interval estimate would have been:
A)19.76 ± .33
B)19.76 ± .66
C)19.76 ± 5.28
D)None of these choices.

Accepted Answer

The formula for calculating the margin of error is:
Margin of error = (critical value) * (standard error)
The critical value depends on the level of confidence and the sample size. As the sample size increases, the critical value decreases because the standard error decreases. Therefore, if the sample size had been 4 times larger (i.e. 200 instead of 50), the critical value would have been 2 times smaller (approximately). Since the margin of error is proportional to the critical value, it would also have been 2 times smaller (approximately). Therefore, the interval estimate would have been:
19.76 ± (1.32/2) = 19.76 ± 0.66
The correct choice is B.

Question 14

Doubling the population standard deviation σ has the effect of doubling the width of the confidence interval estimate of μ.

Accepted Answer

The width of the confidence interval estimate of μ is proportional to the standard deviation σ divided by the square root of the sample size n. Doubling σ would double the numerator of this proportion, thus doubling the width of the confidence interval.

Question 15

The term 1 − α refers to:
A)the probability that a confidence interval does not contain the population parameter.
B)the confidence level.
C)the level of unbiasedness.
D)the level of consistency.

Accepted Answer

The term 1-α represents the level of confidence or the probability that the confidence interval contains the true population parameter. Therefore, option B is the correct choice. Option A is incorrect because it refers to the complement of the confidence level. Options C and D are also unrelated to the concept of confidence level.

Question 16

The lower limit of the 90% confidence interval for μ,where n = 64,  = 70,and σ = 20,is 65.89.

Accepted Answer

Given n = 64, 11ea78b9_2e21_ef5f_8379_53feb8452376_TB4388_11 = 70, and σ = 20, the formula for the lower limit of the 90% confidence interval for μ is:

Lower limit = x̄ - 1.645(σ/√n)

Substituting the given values, we get:

Lower limit = 70 - 1.645(20/√64) = 65.89

Since the given value matches the calculated lower limit, the answer is true.

Question 17

In thisChapter you need four values to construct the confidence interval estimate of μ.They are the sample mean,the sample size,the population standard deviation,and the confidence level.

Accepted Answer

To construct a confidence interval for the population mean (μ), you need the sample mean, sample size, population standard deviation, and the confidence level to determine the margin of error.

Question 18

Suppose that a 95% confidence interval for μ is given by  .This notation means that,if we repeatedly draw samples of the same size from the same population,95% of the values of  will be such that μ would lie somewhere between  .

Accepted Answer

This statement accurately describes the interpretation of a 95% confidence interval.

Question 19

Other things being equal,as the confidence level increases,the width of the confidence interval increases.

Accepted Answer

When the confidence level increases, we need to be more certain about the estimate, which means we need to be more conservative in our assumptions. This results in a wider range of values that we consider plausible, leading to a wider confidence interval.

Question 20

Given a mean of 2.1 and a standard deviation of 0.7,a 90% confidence interval will be 2.1 ± 0.7.

Accepted Answer

A 90% confidence interval is typically calculated as the mean plus or minus 1.645 times the standard deviation, which in this case would result in a confidence interval of 2.1 ± 1.15, not 2.1 ± 0.7.

Quiz 10: Introduction to Estimation

The value for a 95% confidence interval estimate for a population mean μ is

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

The letter α in the formula for constructing a confidence interval estimate of the population mean is:

Suppose a sample size of 5 has mean 9.60.If the population variance is 5 and the population is normally distributed,the lower limit for a 92% confidence interval is 7.85.

Which of the following is an incorrect statement about a 90% confidence interval?

Increasing the value of 1 − α narrows a confidence interval.

In developing an interval estimate for a population mean,the population standard deviation σ was assumed to be 10.The interval estimate was 50.92 ± 2.14.Had σ equaled 20,the interval estimate would be

If the confidence level is reduced,the confidence interval:

When constructing confidence interval estimate of μ,doubling the sample size n decreases the width of the interval by half.

Suppose an interval estimate for the population mean 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:

Other things being equal,the confidence interval for the mean will be wider for 99% confidence than for 95% confidence.

After constructing a confidence interval estimate for a population mean,you believe that the interval is useless because it is too wide.In order to correct this problem,you need to:

In developing an interval estimate for a population mean,a sample of 50 observations was used.The interval estimate was 19.76 ± 1.32.Had the sample size been 200 instead of 50,the interval estimate would have been:

Doubling the population standard deviation σ has the effect of doubling the width of the confidence interval estimate of μ.

The term 1 − α refers to:

The lower limit of the 90% confidence interval for μ,where n = 64, = 70,and σ = 20,is 65.89.

In thisChapter you need four values to construct the confidence interval estimate of μ.They are the sample mean,the sample size,the population standard deviation,and the confidence level.

Suppose that a 95% confidence interval for μ is given by .This notation means that,if we repeatedly draw samples of the same size from the same population,95% of the values of will be such that μ would lie somewhere between .

Other things being equal,as the confidence level increases,the width of the confidence interval increases.

Given a mean of 2.1 and a standard deviation of 0.7,a 90% confidence interval will be 2.1 ± 0.7.