Q: A clinical trial of 800 patients found that 32 experienced a side effect. Construct the point estimate $$\hat{p}$$ and calculate the margin of error $$E$$ for a 95% confidence interval. Use $$z_{\alpha/2} = 1.96.$$

$$\hat{p} = 0.04$$, $$E = 1.96 \sqrt{\dfrac{0.04 \times 0.96}{800}} \approx 0.0136$$

Q: If you want to estimate a population proportion with a margin of error no greater than 0.02 at a 95% confidence level, and you have no prior estimate for $$\hat{p}$$, what sample size $$n$$ should you use?

$$n = \left[ \dfrac{z_{\alpha/2}^2 \times 0.25}{E^2$} \right] = \left[ \dfrac{1.96^2$ \times 0.25}{0.02^2$} \right] = 2401$$

Q: Which formula should be used to determine the minimum sample size needed to estimate a population proportion when no prior estimate is available?

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

Question 1

Which of the following best describes a point estimate?

Accepted Answer

A single value used to estimate a population parameter.

Question 2

What is the correct interpretation of a 95% confidence interval for a population proportion?

Accepted Answer

We are 95% confident that the interval actually does contain the true value of the population proportion.

Question 3

Given a 99% confidence level, what is the corresponding value of $$\alpha$$?

Accepted Answer

$$\alpha = 0.01$$

Question 4

Which formula is used to calculate the margin of error $$E$$ for proportions?

Accepted Answer

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

Question 5

A survey of 1200 people finds that 48% support a new policy. What is the point estimate $$\hat{p} of $$the population proportion?

Accepted Answer

$$\hat{p} = 0.48$$

Question 6

If a 95% confidence interval for a population proportion is $$(0.40, 0.50)$$, what is the margin of error $$E$$?

Accepted Answer

$$E = 0.05$$

Question 7

Which of the following statements about the critical value $$z_{\alpha/2} is $$correct?

Accepted Answer

It is a z-score on the borderline separating significantly low and significantly high values in the standard normal distribution.

Question 8

A clinical trial of 800 patients found that 32 experienced a side effect. Construct the point estimate $$\hat{p}$$ and calculate the margin of error $$E$$ for a 95% confidence interval. Use $$z_{\alpha/2} = 1.96.$$

Accepted Answer

$$\hat{p} = 0.04$$, $$E = 1.96 \sqrt{\dfrac{0.04 	imes 0.96}{800}} \approx 0.0136$$

Question 9

If you want to estimate a population proportion with a margin of error no greater than 0.02 at a 95% confidence level, and you have no prior estimate for $$\hat{p}$$, what sample size $$n$$ should you use?

Accepted Answer

$$n = \left[ \dfrac{z_{\alpha/2}^2 	imes 0.25}{E^2$} ight] = \left[ \dfrac{1.96^2$ 	imes 0.25}{0.02^2$} ight] = 2401$$

Question 10

A 99% confidence interval for a population proportion is wider than a 90% confidence interval for the same data. Why?

Accepted Answer

A higher confidence level requires a wider interval to be more certain of capturing the true parameter.

Question 11

Given a sample proportion $$\hat{p} = 0.6$$, sample size $$n = 400$$, and $$z_{\alpha/2} = 1.96$$, what is the 95% confidence interval for the population proportion $$p$$?

Accepted Answer

$$0.55 \u003c p \u003c 0.65$$

Question 12

Which formula should be used to determine the minimum sample size needed to estimate a population proportion when no prior estimate is available?

Accepted Answer

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

Question 13

A poll reports a 95% confidence interval for the proportion of voters supporting a candidate as $$(0.47, 0.53). $$What is the best point estimate for the population proportion?

Accepted Answer

$$\hat{p} = 0.50$$

Question 14

Which of the following is NOT a requirement for constructing a confidence interval for a population proportion?

Accepted Answer

The sample size must be larger than 30.

Question 15

A survey of 500 people finds that 120 are regular users of a product. Construct a 90% confidence interval for the population proportion. Use $$z_{\alpha/2} = 1.645.$$

Accepted Answer

$$0.208 \u003c p \u003c 0.272$$

Question 16

If the upper and lower bounds of a confidence interval are 0.32 and 0.44, respectively, what is the margin of error $$E$$?

Accepted Answer

$$E = 0.06$$

Estimating Population Proportions and Determining Sample Sizes

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