Question 1

Consider the factors:&#10; 1. the standard error of the estimate&#10; 2. the desired precision of the estimate&#10; 3. the desired degree of confidence&#10; 4. the size of the population&#10;&#10;-Which of these factors does one need to consider when estimating sample size?&#10;A) 1,2&#10;B) 2,3&#10;C) 1,2,3&#10;D) 2,3,4&#10;E) 1,2,3,4

Accepted Answer

Factors to consider when estimating sample size include the standard error of the estimate, the desired precision of the estimate, and the desired degree of confidence. The size of the population is less critical in many cases, especially for large populations.

Question 2

Which of the following is true?&#10;A) The particular sampling distribution of the statistic in question is of fundamental importance in the determination of sample size. For example, the sampling distribution of the sample means is used when estimating a population mean.&#10;B) It is impossible to specify both the degree of confidence and the degree of precision with a fixed size sample.&#10;C) When determining the sample size necessary to estimate a population mean, it is helpful to know the population variance, for if one knows the population variance, he or she also knows the standard error of the mean up to a proportionality constant.&#10;D) a and c&#10;E) a, b, and c are all true

Accepted Answer

a, b, and c are all true&#10;

Question 3

Which of the following would increase the sample size by the greatest amount?&#10;A) increasing the confidence from 68% to 95% confidence&#10;B) decreasing the confidence from 99% to 95% confidence&#10;C) increasing the precision from 30 to 10&#10;D) decreasing the precision from 10 to 40&#10;E) increasing the precision from 5 to 10

Accepted Answer

Increasing the precision (or decreasing the margin of error) from 30 to 10 requires a larger sample size to ensure that the sample mean is within 10 units of the population mean with the same level of confidence, compared to a precision of 30. This change demands the most significant increase in sample size among the options provided.

Question 4

A researcher wishes to estimate the average annual expenditures per household on gasoline. The estimate must be accurate within plus or minus $100 and the estimated standard deviation in such expenditures is $500.
Use this problem to answer the following two questions.

-For a 95% confidence interval (use z=2), the required sample size is

A) 10.
B) 100.
C) 1000.
D) 5000.
E) not enough information given.

Accepted Answer

The formula for determining sample size for estimating a mean with a specified margin of error (E) at a certain confidence level is $n = \left(\frac{z \cdot \sigma}{E}\right)^2$. Given $z = 2$, $\sigma = 500$, and $E = 100$, substituting these values gives $n = \left(\frac{2 \cdot 500}{100}\right)^2 = \left(\frac{1000}{100}\right)^2 = 10^2 = 100$.

Question 5

A researcher wishes to estimate the average annual expenditures per household on gasoline. The estimate must be accurate within plus or minus $100 and the estimated standard deviation in such expenditures is $500.
Use this problem to answer the following two questions.

-Given estimates and specifications that the population standard deviation is 250, total precision is 100 and the researcher wishes to be 95% confident in the result, the required sample size is

A) 10.
B) 50.
C) 100.
D) 500.
E) none of the above.

Accepted Answer

The answer of A researcher wishes to estimate the average...

Question 6

An estimate that is within plus or minus $3.00 of the median value is an example of&#10;A) relative precision.&#10;B) absolute precision.&#10;C) multiple precision.&#10;D) statistical precision.&#10;E) a and d.

Accepted Answer

Absolute precision refers to the exactness of a measurement relative to a specific value, in this case, being within plus or minus $3.00 of the median value.

Question 7

Which of the following is the most conservative estimate for determining the sample size required for a study to estimate a population proportion?&#10;A)  $\pi$ =.1,z=3, and H=.01&#10;B)  $\pi$ =.5,z=3, and H=.01&#10;C)  $\pi$=.1,z=1, and H=.01&#10;D)  $\pi$ =.1,z=1, and H=.5&#10;E)  $\pi$ =.5,z=1, and H=.5

Accepted Answer

The most conservative estimate for determining sample size for estimating a population proportion uses $\pi = .5$, which maximizes the variance ($\pi(1-\pi)$), and a higher z-value, which increases the confidence level. Thus, option B ($\pi = .5$, z=3, and H=.01) is the most conservative.

Question 8

Mary Marsh, head aerobic instructor at Tuff-it-Out Fitness Centers, wanted to estimate the proportion of all 19-39 year olds who aerobically exercise at least once each week. Mary would like the estimate to be within plus or minus 3 percentage points of the mean and she would like to have 90 percent confidence (z=1.645) in her results. Use this information to answer the next three questions.

-Suppose in the previous problem that a 99 percent confidence interval (z=3) were desired. The required sample size is

A) 24.
B) 240.
C) 1,600.
D) 2,400.
E) 3,200.

Accepted Answer

The sample size can be calculated using the formula for sample size estimation for proportions: n = (Z^2 * p * (1-p)) / E^2, where Z is the z-score for the desired confidence level, p is the estimated population proportion, and E is the margin of error. Plugging in the values: n = (1.645^2 * 0.40 * 0.60) / 0.03^2, which results in approximately 722.

Question 9

Mary Marsh, head aerobic instructor at Tuff-it-Out Fitness Centers, wanted to estimate the proportion of all 19-39 year olds who aerobically exercise at least once each week. Mary would like the estimate to be within plus or minus 3 percentage points of the mean and she would like to have 90 percent confidence (z=1.645) in her results. Use this information to answer the next three questions.

-Mary estimates the population proportion at .40. The required sample size is

A) 8.
B) 73.
C) 482.
D) 722.
E) 964.

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Accepted Answer

The sample size is inversely proportional to the square of the margin of error. So, if the margin of error is halved, the sample size must be quadrupled.

Question 10

Mary Marsh, head aerobic instructor at Tuff-it-Out Fitness Centers, wanted to estimate the proportion of all 19-39 year olds who aerobically exercise at least once each week. Mary would like the estimate to be within plus or minus 3 percentage points of the mean and she would like to have 90 percent confidence (z=1.645) in her results. Use this information to answer the next three questions.

-Suppose in the previous problem that a 99 percent confidence interval (z=3) were desired. The required sample size is

A) 24.
B) 240.
C) 1,600.
D) 2,400.
E) 3,200.

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Accepted Answer

To calculate the required sample size for estimating a population proportion with a specific margin of error (E) and confidence level (z), the formula is: $n = \frac{z^2 \cdot p \cdot (1-p)}{E^2}$. Without knowing the population proportion (p), a common approach is to use p = 0.5 for maximum variability, which gives the largest sample size. Given E = 0.05 (5%) and z = 1.96 for 90% confidence (not 1.645 as mentioned, which corresponds to approximately 90% confidence but the question states z=2), the calculation is $n = \frac{(1.96)^2 \cdot 0.5 \cdot (1-0.5)}{(0.05)^2} = 384.16$. However, since none of the provided options match this calculation and assuming a typo in the z value provided in the question, using z=2 for calculation simplifies to $n = \frac{(2)^2 \cdot 0.5 \cdot (1-0.5)}{(0.05)^2} = 400$. Given the options do not match this result either, and assuming a misunderstanding in the interpretation of the question's data or a mistake in the calculation process, the closest correct approach with the provided options and considering a standard approach to such problems would be to reevaluate the calculation with correct assumptions or provided details. The correct answer based on typical calculation methods and common assumptions (p=0.5, E=0.05, z=1.96) does not match the options, suggesting a need to reassess the question's details or the calculation process. Given the discrepancy, the selection of "E" is based on the assumption of a standard scenario with maximum variability and a common margin of error, but it's important to note the mismatch with standard calculation results.

Question 11

In order to calculate sample sizes for a stratified random sample, you need the following information:&#10;A) variability of each strata.&#10;B) cost per observation in each strata.&#10;C) size of each strata.&#10;D) a and b.&#10;E) a, b, and c.

Accepted Answer

The answer of In order to calculate sample sizes for...

Deck 12: Sample Size