Question 1

A random sample of 144 observations has a mean of 20, a median of 21, and a mode of 22. The population standard deviation is known to equal 4.8. The 95.44% confidence interval for the population mean is
A) 15.2 to 24.8
B) 19.200 to 20.800
C) 19.216 to 20.784
D) 21.2 to 22.8

Accepted Answer

The answer of A random sample of 144 observations has...

Question 2

Whenever using the t distribution for interval estimation when the sample size is very small), we must assume that
A) the sample has a mean of at least 30
B) the sampling distribution is not normal
C) the population is approximately normal
D) the finite population correction factor is necessary

Accepted Answer

When using the t distribution for interval estimation with a small sample size, it is necessary to assume that the population is approximately normal. This is because the t distribution assumes normality of the population. Choice A is incorrect because the mean of the sample is irrelevant when using the t distribution. Choice B is also incorrect because the t distribution is used specifically when the sampling distribution is not normal. Choice D is not relevant to the use of the t distribution.

Question 3

For which of the following values of P is the value of P1 - P) maximized?
A) P = 0.99
B) P = 0.90
C) P = 0.01
D) P = 0.50

Accepted Answer

To maximize the value of P1 - P, we need P1 to be as close to 1 as possible and P to be as close to 0 as possible. Therefore, the best choice is when P = 0.50 since it is the farthest away from both 0 and 1, which maximizes the difference between P1 and P.

Question 4

The sample size needed to provide a margin of error of 2 or less with a .95 probability when the population standard deviation equals 11 is
A) 10
B) 11
C) 116
D) 117

Accepted Answer

The formula for calculating sample size with known population standard deviation is:
n = ((z * σ) / E) ^ 2
Where:
z = z-score for desired level of confidence (in this case, z = 1.96 for .95 probability)
σ = population standard deviation
E = margin of error desired

Plugging in the values given:
n = ((1.96 * 11) / 2) ^ 2
n = (21.56) ^ 2
n ≈ 466.20

Round up to the nearest integer, since sample size must be a whole number:
n = 467

Therefore, the best choice is D, 117 since we cannot have half a participant.

Question 5

A random sample of 64 students at a university showed an average age of 25 years and a sample standard deviation of 2 years. The 98% confidence interval for the true average age of all students in the university is
A) 20.5 to 26.5
B) 24.4 to 25.6
C) 23.0 to 27.0
D) 20.0 to 30.0

Accepted Answer

The answer of A random sample of 64 students at...

Question 6

A sample of 225 elements from a population with a standard deviation of 75 is selected. The sample mean is 180. The 95% confidence interval for μ is
A) 105.0 to 225.0
B) 175.0 to 185.0
C) 100.0 to 200.0
D) 170.2 to 189.8

Accepted Answer

The formula for a 95% confidence interval is:

x̄ ± Z*(σ/√n)

where x̄ is the sample mean, σ is the population standard deviation, n is the sample size, and Z* is the critical value from the standard normal distribution corresponding to a 95% level of confidence (which is 1.96).

Plugging in the values given in the problem, we get:

180 ± 1.96*(75/√225)

= 180 ± 1.96*(5)

= 180 ± 9.8

The 95% confidence interval for μ is therefore (180 - 9.8) to (180 + 9.8), or 170.2 to 189.8. Therefore, the best choice is D.

Question 7

The ability of an interval estimate to contain the value of the population parameter is described by the
A) confidence level
B) degrees of freedom
C) precise value of the population mean μ
D) degrees of freedom minus 1

Accepted Answer

The ability of an interval estimate to contain the value of the population parameter is described by the confidence level. The confidence level is the probability that a randomly selected interval estimate will contain the true population parameter. Therefore, a higher confidence level will provide a greater assurance that the interval estimate contains the population parameter. The other options (B, C, and D) are not related to the concept of confidence level.

Question 8

In general, higher confidence levels provide
A) wider confidence intervals
B) narrower confidence intervals
C) a smaller standard error
D) unbiased estimates

Accepted Answer

Higher confidence levels correspond to a higher level of certainty in the estimate, which requires a wider interval to encapsulate a larger range of possible values. Thus, wider confidence intervals are associated with higher confidence levels.

Question 9

A 95% confidence interval for a population mean is determined to be 100 to 120. If the confidence coefficient is reduced to 0.90, the interval for μ
A) becomes narrower
B) becomes wider
C) does not change
D) becomes 0.1

Accepted Answer

Reducing the confidence level from 95% to 90% decreases the margin of error, resulting in a narrower confidence interval.

Question 10

An interval estimate is a range of values used to estimate
A) the shape of the population's distribution
B) the sampling distribution
C) a sample statistic
D) a population parameter

Accepted Answer

Interval estimate is used to estimate a population parameter with a certain level of confidence.

Question 11

It is known that the population variance equals 484. With a 0.95 probability, the sample size that needs to be taken if the desired margin of error is 5 or less is
A) 25
B) 74
C) 189
D) 75

Accepted Answer

The answer of It is known that the population variance...

Question 12

A random sample of 49 statistics examinations was taken. The average score, in the sample, was 84 with a variance of 12.25. The 95% confidence interval for the average examination score of the population of the examinations is
A) 76.00 to 84.00
B) 77.40 to 86.60
C) 83.00 to 85.00
D) 68.00 to 100.00

Accepted Answer

The answer of A random sample of 49 statistics examinations...

Question 13

As the sample size increases, the margin of error
A) increases
B) decreases
C) stays the same
D) increases or decreases depending on the size of the mean

Accepted Answer

As the sample size increases, the margin of error decreases. This is because a larger sample size provides more accurate data, resulting in a smaller margin of error.

Question 14

It is known that the variance of a population equals 1,936. A random sample of 121 has been taken from the population. There is a .95 probability that the sample mean will provide a margin of error of
A) 7.84
B) 31.36
C) 344.96
D) 1,936

Accepted Answer

The answer of It is known that the variance of...

Question 15

In determining the sample size necessary to estimate a population proportion, which of the following information is not needed?
A) the maximum margin of error that can be tolerated
B) the confidence level required
C) a preliminary estimate of the true population proportion P
D) the mean of the population

Accepted Answer

The mean of the population is not relevant in determining the sample size necessary to estimate a population proportion. The maximum margin of error, confidence level, and preliminary estimate of the true population proportion are all important factors in determining the sample size.

Question 16

A sample of 20 items from a population with an unknown σ is selected in order to develop an interval estimate of μ. Which of the following is not necessary?
A) We must assume the population has a normal distribution.
B) We must use a t distribution.
C) Sample standard deviation must be used to estimate σ.
D) The sample must have a normal distribution.

Accepted Answer

The sample itself does not need to have a normal distribution; however, the population from which the sample is drawn should be approximately normal, especially for small sample sizes, to justify using the t-distribution.

Question 17

If we change a 95% confidence interval estimate to a 99% confidence interval estimate, we can expect
A) the size of the confidence interval to increase
B) the size of the confidence interval to decrease
C) the size of the confidence interval to remain the same
D) the sample size to increase

Accepted Answer

Increasing the confidence level will increase the size of the interval because we need to be more certain about our estimate, which will require a wider range of possible values. This is because the margin of error, which is based on the standard error of the mean and the critical value from the t-distribution, will increase as the confidence level increases. Therefore, a 99% confidence interval will have a larger width than a 95% confidence interval for the same sample data.

Question 18

Using an α = 0.04 a confidence interval for a population proportion is determined to be 0.65 to 0.75. If the level of significance is decreased, the interval for the population proportion
A) becomes narrower
B) becomes wider
C) does not change
D) remains the same

Accepted Answer

Decreasing the level of significance (α) increases the confidence level, which results in a wider confidence interval.

Question 19

After computing a confidence interval, the user believes the results are meaningless because the width of the interval is too large. Which one of the following is the best recommendation?
A) Increase the level of confidence for the interval.
B) Decrease the sample size.
C) Increase the sample size.
D) Reduce the population variance.

Accepted Answer

Increasing the sample size will reduce the standard error of the mean, which will result in a narrower confidence interval. Therefore, increasing the sample size is the best option to reduce the width of the confidence interval.

Question 20

When the level of confidence decreases, the margin of error
A) stays the same
B) becomes smaller
C) becomes larger
D) becomes smaller or larger, depending on the sample size

Accepted Answer

The answer of When the level of confidence decreases, the...

Quiz 8: Interval Estimation

A random sample of 144 observations has a mean of 20, a median of 21, and a mode of 22. The population standard deviation is known to equal 4.8. The 95.44% confidence interval for the population mean is

Whenever using the t distribution for interval estimation when the sample size is very small) , we must assume that

For which of the following values of P is the value of P1 - P) maximized?

The sample size needed to provide a margin of error of 2 or less with a .95 probability when the population standard deviation equals 11 is

A random sample of 64 students at a university showed an average age of 25 years and a sample standard deviation of 2 years. The 98% confidence interval for the true average age of all students in the university is

A sample of 225 elements from a population with a standard deviation of 75 is selected. The sample mean is 180. The 95% confidence interval for μ is

The ability of an interval estimate to contain the value of the population parameter is described by the

In general, higher confidence levels provide

A 95% confidence interval for a population mean is determined to be 100 to 120. If the confidence coefficient is reduced to 0.90, the interval for μ

An interval estimate is a range of values used to estimate

It is known that the population variance equals 484. With a 0.95 probability, the sample size that needs to be taken if the desired margin of error is 5 or less is

A random sample of 49 statistics examinations was taken. The average score, in the sample, was 84 with a variance of 12.25. The 95% confidence interval for the average examination score of the population of the examinations is

As the sample size increases, the margin of error

It is known that the variance of a population equals 1,936. A random sample of 121 has been taken from the population. There is a .95 probability that the sample mean will provide a margin of error of

In determining the sample size necessary to estimate a population proportion, which of the following information is not needed?

A sample of 20 items from a population with an unknown σ is selected in order to develop an interval estimate of μ. Which of the following is not necessary?

If we change a 95% confidence interval estimate to a 99% confidence interval estimate, we can expect

Using an α = 0.04 a confidence interval for a population proportion is determined to be 0.65 to 0.75. If the level of significance is decreased, the interval for the population proportion

After computing a confidence interval, the user believes the results are meaningless because the width of the interval is too large. Which one of the following is the best recommendation?

When the level of confidence decreases, the margin of error