Q: Suppose you flip a coin 100 times and get 56 heads. Construct a 90% confidence interval for the true proportion of heads. Use $$z_{\alpha/2} = 1.645$$ for 90% confidence.

$$0.56 \pm 1.645 \times \sqrt{\dfrac{0.56 \times 0.44}{100}}$$

Q: A researcher wants to estimate the proportion of students who own a car with a margin of error of 0.03 at a 95% confidence level. If a prior estimate of $$\hat{p} = 0.6 is $$available, what is the minimum sample size required? (Use $$z_{\alpha/2} = 1.96)$$

$$n = \dfrac{[1.96]^2 \times 0.6 \times 0.4}{0.03^2$}$$

Question 1

Which of the following best describes a point estimate in statistics?

Accepted Answer

A single value used to estimate a population parameter.

Question 2

A poll of 1,205 likely voters reports a margin of error of ±3% at the 95% confidence level. What does this margin of error represent?

Accepted Answer

The maximum likely difference between the sample proportion and the true population proportion with 95% confidence.

Question 3

Which formula gives the margin of error $$E$$ for a confidence interval for a population proportion?

Accepted Answer

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

Question 4

A 95% confidence interval for a population proportion $$p is $$calculated as $$0.42 \u003c p \u003c 0.48. $$Which of the following is the correct interpretation?

Accepted Answer

We are 95% confident that the interval from 0.42 to 0.48 actually does contain the true value of the population proportion $$p.$$

Question 5

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

Accepted Answer

The sample size must be greater than 100.

Question 6

If you want to estimate a population proportion with a margin of error $$E = 0.04 at a 95$$% confidence level and you have no prior estimate for $$\hat{p}$$, which formula should you use to determine the required sample size $$n$$?

Accepted Answer

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

Question 7

What is the critical value $$z_{\alpha/2}$$ for a 99% confidence interval?

Accepted Answer

$$2.575$$

Question 8

Suppose you flip a coin 100 times and get 56 heads. Construct a 90% confidence interval for the true proportion of heads. Use $$z_{\alpha/2} = 1.645$$ for 90% confidence.

Accepted Answer

$$0.56 \pm 1.645 	imes \sqrt{\dfrac{0.56 	imes 0.44}{100}}$$

Question 9

Which of the following statements about the confidence level is TRUE?

Accepted Answer

The confidence level is the probability that the confidence interval process will produce an interval containing the population parameter.

Question 10

A researcher wants to estimate the proportion of students who own a car with a margin of error of 0.03 at a 95% confidence level. If a prior estimate of $$\hat{p} = 0.6 is $$available, what is the minimum sample size required? (Use $$z_{\alpha/2} = 1.96)$$

Accepted Answer

$$n = \dfrac{[1.96]^2 	imes 0.6 	imes 0.4}{0.03^2$}$$

Question 11

Which of the following is a correct requirement for using the normal approximation to construct a confidence interval for a proportion?

Accepted Answer

$$np \geq 5$$ and $$nq \geq 5$$

Question 12

A news report states that 43% of likely voters support a candidate, with a margin of error of ±4% at the 95% confidence level. What is the confidence interval for the proportion of likely voters supporting the candidate?

Accepted Answer

$$0.39 \u003c p \u003c 0.47$$

Question 13

Which of the following best defines a critical value in the context of confidence intervals?

Accepted Answer

A z-score that separates significantly high or low sample statistics from those that are not significant.

Question 14

If a 95% confidence interval for a population proportion does not contain the value $$p = 0.50$$, what does this suggest?

Accepted Answer

There is evidence at the 95% confidence level that the population proportion is not 0.50.

Question 15

Which of the following is the correct formula for a confidence interval estimate of $$p$$?

Accepted Answer

$$\hat{p} - E \u003c p \u003c \hat{p} + E$$

Question 16

A sample proportion $$\hat{p} is $$considered an unbiased estimator of the population proportion $$p$$ because:

Accepted Answer

The mean of the sampling distribution of $$\hat{p}$$ equals $$p.$$

Estimating a Population Proportion and Confidence Intervals

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