Q: Which of the following is the correct formula for a 95% confidence interval for a population proportion $$p$$?

$$\hat{p} \pm 1.96 \times SE(\hat{p})$$

Q: What is the margin of error (ME) for a confidence interval for a population proportion?

$$ME = z^* \times SE(\hat{p})$$

Question 1

Which formula gives the standard error for the sample proportion $$\hat{p}$$?

Accepted Answer

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

Question 2

A university dean randomly samples 200 students and finds that 92 receive financial aid. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{92}{200} = 0.46$$

Question 3

Which of the following is the correct formula for a 95% confidence interval for a population proportion $$p$$?

Accepted Answer

$$\hat{p} \pm 1.96 	imes SE(\hat{p})$$

Question 4

If the sample size $$n is $$tripled, how does the width of the 95% confidence interval for $$p$$ change?

Accepted Answer

It decreases by a factor of $$\sqrt{3}$$

Question 5

What does a 95% confidence interval for a population proportion $$p$$ mean?

Accepted Answer

We are 95% confident that the interval contains the true value of $$p$$

Question 6

Which assumption is NOT required for constructing a confidence interval for a population proportion?

Accepted Answer

The sample must be normally distributed

Question 7

What is the margin of error (ME) for a confidence interval for a population proportion?

Accepted Answer

$$ME = z^* 	imes SE(\hat{p})$$

Question 8

If no prior estimate of $$p is $$available, what value should be used for $$\hat{p}$$ when calculating the required sample size for a given margin of error?

Accepted Answer

$$\hat{p} = 0.5$$

Question 9

Which of the following increases the width of a confidence interval for a population proportion?

Accepted Answer

Increasing the confidence level

Question 10

A researcher wants to estimate a population proportion with a margin of error no greater than 0.04 at 99% confidence. What formula should be used to determine the required sample size?

Accepted Answer

$$n = \frac{(z^*)^2$ \hat{p}(1-\hat{p})}{ME^2$}$$

Question 11

Which statement best describes the interpretation of a confidence interval for $$p$$?

Accepted Answer

Over many samples, C% of the constructed intervals will contain the true $$p$$

Question 12

Why do we use $$\hat{p} = 0.5$$ when no prior information about $$p is $$available for sample size calculation?

Accepted Answer

It maximizes $$\hat{p}(1-\hat{p})$$, resulting in the largest required sample size

Question 13

A 99% confidence interval for $$p$$ will be ______ than a 95% confidence interval, assuming the same sample size.

Accepted Answer

Wider

Question 14

Which of the following is NOT a property of confidence intervals for population proportions?

Accepted Answer

They are constructed for statistics, not parameters

Question 15

Suppose a 95% confidence interval for $$p is (0.50$$, 0.65). Which statement is correct?

Accepted Answer

We are 95% confident that the interval (0.50, 0.65) contains the true $$p$$

Question 16

Which condition must be checked when constructing a confidence interval for a population proportion?

Accepted Answer

$$n\hat{p} \geq 10$$ and $$n(1-\hat{p}) \geq 10$$

Question 17

A sample of 400 patients shows that 120 have a certain condition. Construct a 90% confidence interval for the population proportion.

Accepted Answer

$$0.30 \pm 1.645 	imes \sqrt{\frac{0.3 	imes 0.7}{400}}$$

Question 18

Which real-world scenario best illustrates the use of a confidence interval for a population proportion?

Accepted Answer

Estimating the proportion of voters who support a candidate based on a random poll

Question 19

If a confidence interval for $$p is $$calculated as $$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$, what does $$z^*$$ represent?

Accepted Answer

The critical value from the standard normal distribution for the desired confidence level

Confidence Intervals for Proportions

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