Q: A hospital administrator wants to estimate the average length of stay for patients. She wants to be 95% confident that the estimate is within 2 days of the true mean. If the population standard deviation is estimated to be 6 days, what is the minimum sample size required?

$$n = \left( \dfrac{1.96 \times 6}{2} \$$right)^2$$ = 34.6 \approx 35$$

Q: A pollster wants to estimate the proportion of voters who support a new policy. She wants a 95% confidence level and a margin of error of 0.04. If she has no prior estimate, what sample size should she use?

$$n = \left[ \dfrac{1.96^2$ \times 0.25}{0.04^2$} \right] = 601$$

Question 1

A researcher wants to estimate the proportion of college students who own a laptop. She surveys 500 students and finds that 350 own a laptop. Construct a 95% confidence interval for the population proportion. Which formula should she use for the margin of error?

Accepted Answer

$$E = z_{\alpha/2} \sqrt{\dfrac{\hat{p}\hat{q}}{n}}$$

Question 2

If a 95% confidence interval for a population proportion $$p is (0.40$$, 0.48), what is the point estimate $$\hat{p}$$ and the margin of error $$E$$?

Accepted Answer

$$\hat{p} = 0.44,\ E = 0.04$$

Question 3

Which of the following best describes the meaning of a 95% confidence level in the context of estimating a population proportion?

Accepted Answer

If we repeat the sampling process many times, about 95% of the confidence intervals constructed will contain the true population proportion.

Question 4

A company wants to estimate the proportion of customers who are satisfied with their service. They want a margin of error of no more than 0.02 at a 99% confidence level. No prior estimate is available. What formula should they use to determine the minimum sample size?

Accepted Answer

$$n = \left[ \dfrac{z_{\alpha/2}^2 \cdot 0.25}{E^2$} \right]$$

Question 5

Given a 90% confidence level, what is the corresponding value of $$\alpha$$?

Accepted Answer

$$\alpha = 0.10$$

Question 6

A sample of 100 batteries has a mean lifetime of 500 hours with a known population standard deviation of 50 hours. What is the 95% confidence interval for the population mean?

Accepted Answer

$$500 \pm 9.8$$

Question 7

Which formula should be used to calculate the margin of error when estimating a population mean with unknown $$\sigma$$ and $$n \u003e 30$$?

Accepted Answer

$$E = t_{\alpha/2} \dfrac{s}{\sqrt{n}}$$

Question 8

A 99% confidence interval for a population mean is calculated as $$(120, 140). $$What is the sample mean $$\bar{x}$$ and the margin of error $$E$$?

Accepted Answer

$$\bar{x} = 130,\ E = 10$$

Question 9

A hospital administrator wants to estimate the average length of stay for patients. She wants to be 95% confident that the estimate is within 2 days of the true mean. If the population standard deviation is estimated to be 6 days, what is the minimum sample size required?

Accepted Answer

$$n = \left( \dfrac{1.96 	imes 6}{2} \$$right)^2$$ = 34.6 \approx 35$$

Question 10

Which of the following statements is TRUE about the critical value $$z_{\alpha/2}$$ for a 99% confidence interval?

Accepted Answer

$$z_{\alpha/2} is $$approximately 2.576

Question 11

A pollster wants to estimate the proportion of voters who support a new policy. She wants a 95% confidence level and a margin of error of 0.04. If she has no prior estimate, what sample size should she use?

Accepted Answer

$$n = \left[ \dfrac{1.96^2$ 	imes 0.25}{0.04^2$} ight] = 601$$

Question 12

Which of the following is NOT a correct interpretation of a confidence interval for a population mean?

Accepted Answer

There is a 95% probability that the true mean is within the calculated interval for this sample.

Question 13

A sample of 25 students has a mean test score of 78 and a sample standard deviation of 10. What is the 95% confidence interval for the population mean?

Accepted Answer

$$78 \pm 4.08$$

Question 14

When estimating a population mean and the population standard deviation $$\sigma is $$unknown, which distribution should be used to construct the confidence interval?

Accepted Answer

Student's $$t$$ distribution

Question 15

A researcher calculates a 95% confidence interval for a population proportion as (0.52, 0.60). What is the correct way to express this interval?

Accepted Answer

$$0.56 \pm 0.04$$

Question 16

Which of the following is the correct formula for the margin of error when estimating a population mean with known $$\sigma$$?

Accepted Answer

$$E = z_{\alpha/2} \dfrac{\sigma}{\sqrt{n}}$$

Estimating Parameters and Determining Sample Sizes

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