Question 1

It is known that the population variance equals 484. With a .95 probability, the sample size that needs to be taken to estimate the population mean if the desired margin of error is 5 or less is
A) 25
B) 74
C) 189
D) 75

Accepted Answer

The formula for the sample size needed to estimate the population mean with a desired margin of error E is:

n = ((z-score * σ) / E)^2

Where z-score is the critical value for the desired level of confidence (in this case, 0.95), σ is the population standard deviation (the square root of the variance), and E is the desired margin of error (which is 5 in this case).

Since the population variance is given as 484, the population standard deviation is:

σ = sqrt(484) = 22

The critical value for a 0.95 confidence level is found using a standard normal distribution table or a calculator, and is approximately 1.96.

Plugging in the values, we get:

n = ((1.96 * 22) / 5)^2 = 74.12

Rounding up to the nearest integer, we get a sample size of 75, which is answer choice D.

Question 2

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

CONFIDENCE.NORM is the appropriate function to use because the sample size is larger than 30 and we know the population standard deviation. This formula uses a normal distribution to calculate the confidence interval.

Question 3

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 answer of A random sample of 25 statistics examinations...

Question 4

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

When we increase the level of confidence from 95% to 99%, we are increasing our certainty in the estimate. This requires a larger margin of error, which means that the width of the interval must increase. Conversely, if we decrease the level of confidence, the interval width would decrease. The sample size, while important to the accuracy of the estimate, is not directly related to the confidence level chosen.

Question 5

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 the population standard deviation is unknown and the sample size is used to estimate it, the t distribution is appropriate. The degrees of freedom for a sample of size n is n-1, so for a sample of 25, the degrees of freedom is 24.

Question 6

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 confidence interval is a range of values that is likely to contain the population parameter being estimated, in this case, the mean systolic blood pressure of all company employees. The 95% confidence interval means that if we repeated this sampling procedure many times, we would expect 95% of the resulting confidence intervals to contain the true population mean. Therefore, option B is correct. Option A is incorrect because we cannot make any statement about individual sample members based on the confidence interval. Option C is incorrect because we cannot make any claims about the proportion of the population within the confidence interval. Option D is incorrect because it only applies to the sample means, not the population mean, and is not relevant to the confidence interval.

Question 7

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 require less certainty about where the true population parameter lies, resulting in a smaller range of values.

Question 8

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

In this scenario, the best distribution to use is the t-distribution, as it allows for estimation of the population mean when the standard deviation is unknown. The t-distribution accounts for the increased uncertainty introduced by using the sample standard deviation to estimate the population standard deviation.

Question 9

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 indicate more certainty in the estimate, therefore a wider interval is needed to account for the increased likelihood of variability outside of the sample. Conversely, lower confidence levels provide narrower intervals as they allow for more uncertainty in the estimate.

Question 10

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, the sample standard deviation (s) can be used in conjunction with a t distribution to develop an interval estimate of the population mean.

Question 11

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 95% confidence interval for the population mean μ, when the population standard deviation is unknown and the sample size is small (n < 30), can be calculated using the t-distribution. The formula for the confidence interval is given by: $\bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $s$ is the sample standard deviation, $n$ is the sample size, and $t_{\alpha/2}$ is the t-score that corresponds to the desired level of confidence and degrees of freedom (df = n - 1). For a 95% confidence level and 25 degrees of freedom, the t-score is approximately 2.060. Plugging in the values: $10 \pm 2.060 \cdot \frac{4}{\sqrt{26}}$ gives the interval $8.384$ to $11.616$.

Question 12

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 increases, the t distribution approaches the standard normal distribution. Therefore, the difference between the two distributions becomes smaller.

Question 13

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 answer of A random sample of 36 students at...

Question 14

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, which corresponds to the two-tailed critical value. This value can be found in a t-table or calculated using a calculator or software.

Question 15

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 is not normally distributed and the standard deviation is not known, we use the t-distribution for the interval estimate. The correct degrees of freedom to use for the t-distribution in this case is n-1, where n is the sample size. Therefore, the t distribution with 49 degrees of freedom (50-1) should be used. Choice A is incorrect because the population is not normally distributed. Choice B is incorrect because the degrees of freedom should be based on the sample size, which is 50 in this case. Choice D is incorrect because increasing the sample size may not be necessary or feasible, and the t-distribution with 49 degrees of freedom is appropriate for the given sample size.

Question 16

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 cells within a range that meet a certain criterion. It is not relevant to constructing a confidence interval for estimating annual income. The other functions listed (NORM.S.INV, AVERAGE, STDEV) are commonly used in constructing confidence intervals.

Question 17

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 answer of The t distribution should be used whenever...

Question 18

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 19

To estimate a population mean, 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 answer of To estimate a population mean, the sample...

Question 20

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, thus requiring a broader range.

Quiz 8: Interval Estimation

It is known that the population variance equals 484. With a .95 probability, the sample size that needs to be taken to estimate the population mean if the desired margin of error is 5 or less is

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 _____.

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

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 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% 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 _____.

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

In general, higher confidence levels provide _____.

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

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 _____.

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

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 _____.

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

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?

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?

The t distribution should be used whenever _____.

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

To estimate a population mean, 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 _____.

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