Q: Suppose you want to estimate the mean weight of a population with a margin of error $$E = 0.5 kg at 95$$% confidence. If the population standard deviation is $$\sigma = 4 kg$$, what is the minimum sample size required?

$$n = \left( \frac{1.96 \times 4}{0.5} \$$right)^2$$ = 245.86$$, so round up to $$n = 246$$

Q: A hospital wants to estimate the mean recovery time for a new treatment with a margin of error of 2 days at 99% confidence. If $$\sigma = 6$$ days, what is the minimum sample size required? (Use $$z_{\alpha/2} = 2.576$$ for 99% confidence)

$$n = \left( \frac{2.576 \times 6}{2} \$$right)^2$$ = 119.52$$, so round up to $$n = 120$$

Q: A scientist wants to estimate the mean concentration of a pollutant in a river with a margin of error of $$0.05 mg/L at 90$$% confidence. If $$\sigma = 0.2 mg/L $$and $$z_{\alpha/2} = 1.645$$, what sample size is required?

$$n = \left( \frac{1.645 \times 0.2}{0.05} \$$right)^2$$ = 43.24$$, so round up to $$n = 44$$

Question 1

Which of the following best describes the meaning of a 95% confidence interval for the mean height of all adult American men being (68.91, 69.69)?

Accepted Answer

We are 95% confident that the mean height of all adult American men is between 68.91 and 69.69 inches.

Question 2

Suppose you want to estimate the mean weight of a population with a margin of error $$E = 0.5 kg at 95$$% confidence. If the population standard deviation is $$\sigma = 4 kg$$, what is the minimum sample size required?

Accepted Answer

$$n = \left( \frac{1.96 	imes 4}{0.5} \$$right)^2$$ = 245.86$$, so round up to $$n = 246$$

Question 3

If you increase the sample size $$n in a $$confidence interval calculation, what happens to the margin of error $$E$$?

Accepted Answer

The margin of error $$E$$ decreases.

Question 4

Given the formula for margin of error $$E = \frac{z_{\alpha/2} \cdot \sigma}{\sqrt{n}}$$, which variable can you control to reduce $$E if$$ $$z_{\alpha/2}$$ and $$\sigma$$ are fixed?

Accepted Answer

$$n$$

Question 5

A researcher calculates a required sample size as $$n = 95.3$$ for a desired margin of error. What should the researcher do?

Accepted Answer

Round up to $$n = 96.$$

Question 6

If you want to decrease the margin of error in your confidence interval but cannot change the confidence level or population standard deviation, what should you do?

Accepted Answer

Increase the sample size.

Question 7

Which of the following is the correct formula for calculating the minimum sample size $$n$$ needed for a desired margin of error $$E$$?

Accepted Answer

$$n = \left( \frac{z_{\alpha/2} \cdot \sigma}{E} \$$right)^2$$

Question 8

A hospital wants to estimate the mean recovery time for a new treatment with a margin of error of 2 days at 99% confidence. If $$\sigma = 6$$ days, what is the minimum sample size required? (Use $$z_{\alpha/2} = 2.576$$ for 99% confidence)

Accepted Answer

$$n = \left( \frac{2.576 	imes 6}{2} \$$right)^2$$ = 119.52$$, so round up to $$n = 120$$

Question 9

Why is it important to always round up the calculated sample size when it is not a whole number?

Accepted Answer

Rounding up ensures the margin of error is not larger than desired.

Question 10

If a sample size is too small, what is the most likely effect on the margin of error in a confidence interval?

Accepted Answer

The margin of error will be larger than desired.

Question 11

A scientist wants to estimate the mean concentration of a pollutant in a river with a margin of error of $$0.05 mg/L at 90$$% confidence. If $$\sigma = 0.2 mg/L $$and $$z_{\alpha/2} = 1.645$$, what sample size is required?

Accepted Answer

$$n = \left( \frac{1.645 	imes 0.2}{0.05} \$$right)^2$$ = 43.24$$, so round up to $$n = 44$$

Question 12

Which of the following statements is TRUE regarding confidence intervals?

Accepted Answer

A confidence interval gives a range where we are confident the population mean lies.

Question 13

If the population standard deviation $$\sigma$$ increases, what happens to the margin of error $$E$$ for a fixed sample size and confidence level?

Accepted Answer

The margin of error $$E$$ increases.

Question 14

A company wants to estimate the mean time employees spend on training with a margin of error of 0.25 hours at 95% confidence. If $$\sigma = 1.5$$ hours, what is the minimum sample size required?

Accepted Answer

$$n = \left( \frac{1.96 	imes 1.5}{0.25} \$$right)^2$$ = 138.24$$, so round up to $$n = 139$$

Question 15

Which of the following would NOT reduce the margin of error in a confidence interval for a mean?

Accepted Answer

Increasing the confidence level.

Question 16

If you want to be more confident (e.g., 99% instead of 95%) in your estimate, what happens to the required sample size for the same margin of error?

Accepted Answer

The required sample size increases.

Question 17

A biologist wants to estimate the mean length of a species of fish with a margin of error of 0.2 cm at 95% confidence. If $$\sigma = 1 cm$$, what is the minimum sample size required?

Accepted Answer

$$n = \left( \frac{1.96 	imes 1}{0.2} \$$right)^2$$ = 96.04$$, so round up to $$n = 97$$

Question 18

Which of the following is a common misconception about confidence intervals?

Accepted Answer

A confidence interval means that 95% of the population values are within the interval.

Estimation of a Population Mean: Margin of Error and Sample Size Determination

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