Q: A company wants to estimate the proportion of unused gift vouchers. In a sample of 250 vouchers, 120 expired unused. What is the 95% confidence interval for the population proportion?

$$0.48 \pm 1.96 \sqrt{\frac{0.48 \times 0.52}{250}}$$

Q: In a survey of 300 people, 75 say they would vote for a particular party. What is the 99% confidence interval for the proportion?

$$0.25 \pm 2.58 \sqrt{\frac{0.25 \times 0.75}{300}}$$

Q: A sample proportion is calculated as $$\hat{p} = 0.62$$ from a sample of $$n = 40. $$What is the standard error of $$\hat{p}$$?

$$\sqrt{\frac{0.62 \times 0.38}{40}}$$

Question 1

Suppose a sample of 80 people is surveyed and 56 report a positive outcome with a new product. What is the sample proportion ($$\hat{p}$$)?

Accepted Answer

$$\hat{p} = \frac{56}{80} = 0.70$$

Question 2

Which formula correctly represents the standard deviation of the sampling distribution of a sample proportion?

Accepted Answer

$$\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 3

A company wants to estimate the proportion of unused gift vouchers. In a sample of 250 vouchers, 120 expired unused. What is the 95% confidence interval for the population proportion?

Accepted Answer

$$0.48 \pm 1.96 \sqrt{\frac{0.48 	imes 0.52}{250}}$$

Question 4

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

Accepted Answer

We are 95% confident that the interval contains the true population proportion.

Question 5

If you want to estimate a population proportion to within 2% with 99% confidence, what value of $$\hat{p}$$ should you use for the most conservative sample size estimate?

Accepted Answer

$$\hat{p} = 0.5$$

Question 6

Which z-value should be used for a 99% confidence interval for a population proportion?

Accepted Answer

$$2.58$$

Question 7

In a survey of 300 people, 75 say they would vote for a particular party. What is the 99% confidence interval for the proportion?

Accepted Answer

$$0.25 \pm 2.58 \sqrt{\frac{0.25 	imes 0.75}{300}}$$

Question 8

Which of the following best describes the shape of the sampling distribution of the sample proportion when $$n is $$large and $$p is $$not close to 0 or 1?

Accepted Answer

Approximately normal

Question 9

A researcher wants to reduce the margin of error in a confidence interval for a proportion. Which of the following will achieve this?

Accepted Answer

Increase the sample size

Question 10

If a 95% confidence interval for a proportion is $$(0.35, 0.45)$$, what can you conclude about the hypothesis that the population proportion is 0.5?

Accepted Answer

There is evidence against the hypothesis, since 0.5 is not in the interval.

Question 11

Which of the following is the correct formula for the required sample size $$n to $$estimate a population proportion with margin of error $$E at 95$$% confidence?

Accepted Answer

$$n = \frac{(1.96)^2$ \hat{p}(1-\hat{p})}{E^2$}$$

Question 12

A sample proportion is calculated as $$\hat{p} = 0.62$$ from a sample of $$n = 40. $$What is the standard error of $$\hat{p}$$?

Accepted Answer

$$\sqrt{\frac{0.62 	imes 0.38}{40}}$$

Question 13

Why is $$\hat{p} = 0.5$$ used for sample size calculations when the true proportion is unknown?

Accepted Answer

It maximizes the required sample size, ensuring the margin of error is not underestimated.

Question 14

Which of the following statements is TRUE about confidence intervals for proportions?

Accepted Answer

A wider interval results from a higher confidence level, all else equal.

Question 15

A 95% confidence interval for a proportion is $$(0.40, 0.60). $$What is the margin of error?

Accepted Answer

0.10

L10 Confidence Intervals for a Population Proportion

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