Question 1

From a population that is normally distributed with an unknown standard deviation, a sample of 25 elements is selected. For the interval estimation of μ, the proper distribution to use is the _____.
A) standard normal distribution
B) z distribution
C) t distribution with 26 degrees of freedom
D) t distribution with 24 degrees of freedom

Accepted Answer

When estimating the population mean μ from a normally distributed population with an unknown standard deviation, and the sample size is n, the appropriate distribution to use is the t distribution with n-1 degrees of freedom. Here, n = 25, so n-1 = 24.

Question 2

A 95% confidence interval for a population mean is determined to be 100 to 120. If the confidence coefficient is reduced to .90, the interval for μ _____.
A) becomes narrower
B) becomes wider
C) does not change
D) becomes .1

Accepted Answer

When the confidence level decreases (from 95% to 90% in this case), the confidence interval becomes narrower because a lower confidence level means we are accepting a higher level of uncertainty, thus requiring a smaller range to capture the population mean.

Question 3

A random sample of 36 students at a community college showed an average age of 25 years. Assume the ages of all students at the college are normally distributed with a standard deviation of 1.8 years. The 98% confidence interval for the average age of all students at this college is _____.
A) 24.301 to 25.699
B) 24.385 to 25.615
C) 23.200 to 26.800
D) 23.236 to 26.764

Accepted Answer

The correct answer is A because the 98% confidence interval for the average age of all students at this college is 24.301 to 25.699. This is calculated using the formula: sample mean +/- (z-score * standard deviation / square root of sample size). The z-score for a 98% confidence interval is 2.326.

Question 4

A bank manager wishes to estimate the average waiting time for customers in line for tellers. A random sample of 50 times is measured and the average waiting time is 5.7 minutes. The population standard deviation of waiting time is 2 minutes. Which Excel function would be used to construct a confidence interval estimate?
A) CONFIDENCE.NORM
B) NORM.INV
C) T.INV
D) INT

Accepted Answer

The CONFIDENCE.NORM function is used to construct a confidence interval estimate for the population mean when the population standard deviation is known.

Question 5

Whenever using the t distribution in interval estimation, we must assume that _____.
A) the sample size is less than 30
B) a random sample was selected
C) the population is approximately normal
D) the finite population correction factor is necessary

Accepted Answer

The answer of Whenever using the t distribution in interval...

Question 6

The t distribution should be used whenever _____.
A) the sample size is less than 30
B) the sample standard deviation is used to estimate the population standard deviation
C) the population is not normally distributed
D) None of the answers is correct.

Accepted Answer

The t distribution should be used whenever the sample standard deviation is used to estimate the population standard deviation.

Question 7

A sample of 26 elements from a normally distributed population is selected. The sample mean is 10 with a standard deviation of 4. The 95% confidence interval for μ is _____.
A) 6.000 to 14.000
B) 9.846 to 10.154
C) 8.384 to 11.616
D) 8.462 to 11.538

Accepted Answer

The formula for a confidence interval for the population mean is:

CI = x̄ ± z*(σ/√n)

where:
x̄ = sample mean
z = z-score for the desired confidence level (1.96 for 95%)
σ = population standard deviation (unknown, so we use the sample standard deviation, s)
n = sample size

Plugging in the values from the problem, we get:

CI = 10 ± 1.96*(4/√26)
CI = 10 ± 1.96*(0.78)
CI = 10 ± 1.53
CI = (8.47, 11.53)

Therefore, the 95% confidence interval for μ is 8.384 to 11.616, which is option C.

Question 8

When the level of confidence increases, the confidence interval _____.
A) stays the same
B) becomes wider
C) becomes narrower
D) cannot be determined from the information given

Accepted Answer

When the level of confidence increases, the confidence interval becomes wider because a higher confidence level means you want to be more certain about capturing the true parameter, which requires a broader range.

Question 9

Whenever the population standard deviation is unknown, which distribution is used in developing an interval estimate for a population mean?
A) standard distribution
B) z distribution
C) binomial distribution
D) t distribution

Accepted Answer

When the population standard deviation is unknown, t distribution is used to develop an interval estimate for a population mean.

Question 10

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) 19.2 to 20.8
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 11

An auto manufacturer wants to estimate the annual income of owners of a particular model of automobile. A random sample of 200 current owners is selected. The population standard deviation is known. Which Excel function would NOT be appropriate to use to construct a confidence interval estimate?
A) NORM.S.INV
B) COUNTIF
C) AVERAGE
D) STDEV

Accepted Answer

COUNTIF is used to count the number of cells that meet a certain criteria. It is not relevant for constructing a confidence interval estimate of the population income. However, NORM.S.INV, AVERAGE and STDEV are appropriate functions to use in constructing a confidence interval. NORM.S.INV is used to find the z-score for a given confidence level, AVERAGE is used to find the sample mean, and STDEV is used to find the sample standard deviation.

Question 12

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

Accepted Answer

As we increase the level of confidence, we need to be more certain about the estimate. This requires including a larger range of values in the interval, leading to an increase in width. Conversely, if we decrease the level of confidence, we can include a smaller range of values in the interval, leading to a decrease in width. The sample size does not necessarily need to change when we change the level of confidence.

Question 13

As the number of degrees of freedom for a t distribution increases, the difference between the t distribution and the standard normal distribution _____.
A) becomes larger
B) becomes smaller
C) stays the same
D) None of the answers is correct.

Accepted Answer

As the number of degrees of freedom for a t distribution increases, the shape of the t distribution approaches the shape of the standard normal distribution. Therefore, the difference between the two becomes smaller.

Question 14

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

Accepted Answer

The margin of error (ME) in the context of a sample mean can be calculated using the formula ME = Z * (σ/√n), where Z is the Z-score corresponding to the desired confidence level (for a .95 probability, Z ≈ 1.96), σ is the population standard deviation, and n is the sample size. Given that the variance of the population is 1,936, the standard deviation σ is the square root of 1,936, which is 44. The sample size n is 121. Therefore, ME = 1.96 * (44/√121) = 1.96 * 4 = 7.84.

Question 15

A random sample of 25 employees of a local company has been measured. A 95% confidence interval estimate for the mean systolic blood pressure for all company employees is 123 to 139. Which of the following statements is valid?
A) 95% of the sample of employees has a systolic blood pressure between 123 and 139.
B) If the sampling procedure were repeated many times, 95% of the resulting confidence intervals would contain the population mean systolic blood pressure.
C) 95% of the population of employees has a systolic blood pressure between 123 and 139.
D) If the sampling procedure were repeated many times, 95% of the sample means would be between 123 and 139.

Accepted Answer

A 95% confidence interval means that if we were to repeat the sampling procedure many times, 95% of the resulting intervals would contain the true population mean. Statement A is incorrect because we cannot make statements about individual members of the sample based on the confidence interval. Statement C is incorrect because we cannot make statements about the population with complete certainty based on a sample. Statement D is incorrect because we cannot make statements about the sample mean based on a confidence interval for the population mean.

Question 16

From a population that is not normally distributed and whose standard deviation is not known, a sample of 50 items is selected to develop an interval estimate for μ. Which of the following statements is true?
A) The standard normal distribution can be used.
B) The t distribution with 50 degrees of freedom must be used.
C) The t distribution with 49 degrees of freedom must be used.
D) The sample size must be increased in order to develop an interval estimate.

Accepted Answer

When the population standard deviation is unknown and the sample size is relatively small (less than 30 is a common rule, but it applies more broadly), the t distribution is used instead of the normal distribution. The degrees of freedom for a sample of size n is n-1, so for a sample of 50, the degrees of freedom would be 49.

Question 17

The t value with a 95% confidence and 24 degrees of freedom is _____.
A) 1.711
B) 2.064
C) 2.492
D) 2.069

Accepted Answer

The t value with a 95% confidence and 24 degrees of freedom is 2.064 according to the t-distribution table. Therefore, the best choice is B.

Question 18

In developing an interval estimate of the population mean, if the population standard deviation is unknown _____.
A) it is impossible to develop an interval estimate
B) a sample proportion can be used
C) the sample standard deviation and t distribution can be used
D) None of the answers is correct.

Accepted Answer

When the population standard deviation is unknown, we can use the sample standard deviation and a t distribution to develop an interval estimate of the population mean. This is known as the t-interval or the confidence interval for the mean. The t distribution takes into account the uncertainty introduced by using the sample standard deviation instead of the population standard deviation.

Question 19

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 require capturing more of the data to be more certain of the estimate, which results in wider confidence intervals.

Question 20

A random sample of 25 statistics examinations was selected. The average score in the sample was 76 with a variance of 144. Assuming the scores are normally distributed, the 99% confidence interval for the population average examination score is _____.
A) 70.02 to 81.98
B) 69.82 to 82.18
C) 70.06 to 81.94
D) 69.29 to 82.71

Accepted Answer

The 99% confidence interval is calculated using the formula: mean ± (z * (standard deviation/sqrt(n))). Here, the mean is 76, the standard deviation is the square root of 144 (which is 12), n is 25, and the z-value for 99% confidence is approximately 2.576. This results in the interval 69.29 to 82.71.

Quiz 8: Interval Estimation

From a population that is normally distributed with an unknown standard deviation, a sample of 25 elements is selected. For the interval estimation of μ, the proper distribution to use is the _____.

A 95% confidence interval for a population mean is determined to be 100 to 120. If the confidence coefficient is reduced to .90, the interval for μ _____.

A random sample of 36 students at a community college showed an average age of 25 years. Assume the ages of all students at the college are normally distributed with a standard deviation of 1.8 years. The 98% confidence interval for the average age of all students at this college is _____.

Whenever using the t distribution in interval estimation, we must assume that _____.

The t distribution should be used whenever _____.

A sample of 26 elements from a normally distributed population is selected. The sample mean is 10 with a standard deviation of 4. The 95% confidence interval for μ is _____.

When the level of confidence increases, the confidence interval _____.

Whenever the population standard deviation is unknown, which distribution is used in developing an interval estimate for a population mean?

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 _____.

An auto manufacturer wants to estimate the annual income of owners of a particular model of automobile. A random sample of 200 current owners is selected. The population standard deviation is known. Which Excel function would NOT be appropriate to use to construct a confidence interval estimate?

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

As the number of degrees of freedom for a t distribution increases, the difference between the t distribution and the standard normal distribution _____.

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

A random sample of 25 employees of a local company has been measured. A 95% confidence interval estimate for the mean systolic blood pressure for all company employees is 123 to 139. Which of the following statements is valid?

From a population that is not normally distributed and whose standard deviation is not known, a sample of 50 items is selected to develop an interval estimate for μ. Which of the following statements is true?

The t value with a 95% confidence and 24 degrees of freedom is _____.

In developing an interval estimate of the population mean, if the population standard deviation is unknown _____.

In general, higher confidence levels provide _____.

A random sample of 25 statistics examinations was selected. The average score in the sample was 76 with a variance of 144. Assuming the scores are normally distributed, the 99% confidence interval for the population average examination score is _____.