Q: A poll finds that 420 out of 1000 randomly selected voters support a new policy. Construct a 95% confidence interval for the true proportion of supporters. Which formula should you use for the margin of error?

$$z^* \times \sqrt{\frac{\hat{p}\hat{q}}{n}}$$

Question 1

Which of the following best describes the sampling distribution of the sample proportion $$\hat{p}$$ when the sample size is large and the independence condition is met?

Accepted Answer

It is approximately Normal with mean $$p$$ and standard deviation $$\sqrt{\frac{pq}{n}}.$$

Question 2

A poll finds that 420 out of 1000 randomly selected voters support a new policy. Construct a 95% confidence interval for the true proportion of supporters. Which formula should you use for the margin of error?

Accepted Answer

$$z^* 	imes \sqrt{\frac{\hat{p}\hat{q}}{n}}$$

Question 3

Which of the following is NOT an assumption or condition required for constructing a confidence interval for a proportion?

Accepted Answer

Equal means Condition

Question 4

Suppose a company wants to estimate the proportion of customers who will buy a new product to within 4% with 95% confidence. What value of $$\hat{p}$$ should they use to ensure the largest required sample size?

Accepted Answer

$$\hat{p} = 0.50$$

Question 5

If a 95% confidence interval for a population proportion is (0.40, 0.48), what does this mean?

Accepted Answer

We are 95% confident that the true proportion is between 0.40 and 0.48.

Question 6

Which of the following would increase the margin of error for a confidence interval for a proportion?

Accepted Answer

Increasing the confidence level

Question 7

A seed packet claims a germination rate of 92%. For a sample of 160 seeds, which condition is checked by calculating $$np$$ and $$nq$$?

Accepted Answer

Success/Failure Condition

Question 8

What is the standard deviation of the sampling distribution of $$\hat{p}$$ for a population proportion $$p$$ and sample size $$n$$?

Accepted Answer

$$\sqrt{\frac{pq}{n}}$$

Question 9

Which of the following statements about confidence intervals is correct?

Accepted Answer

A 90% confidence interval has a smaller critical value $$z^*$$ than a 99% confidence interval.

Question 10

A survey of 1200 people finds that 720 support a new law. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$0.60$$

Question 11

Which of the following best explains why we use $$z^* in $$constructing confidence intervals for proportions?

Accepted Answer

Because the sampling distribution of $$\hat{p} is $$approximately Normal for large $$n.$$

Question 12

If you want to cut the margin of error in half for a confidence interval for a proportion, what must you do to the sample size?

Accepted Answer

Quadruple the sample size

Question 13

Which of the following is a correct interpretation of a 95% confidence interval for a proportion?

Accepted Answer

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

Question 14

A researcher wants to estimate the proportion of defective products with a margin of error of 2% at 99% confidence. Which critical value should be used?

Accepted Answer

$$z^* = 2.576$$

Question 15

Which of the following is a potential consequence of violating the independence assumption when constructing a confidence interval for a proportion?

Accepted Answer

The confidence interval may not have the stated confidence level.

Question 16

A pollster wants to estimate a proportion with a margin of error of 5% at 90% confidence. What is the critical value $$z^*$$?

Accepted Answer

$$1.645$$

Question 17

Which of the following best describes the effect of increasing the sample size on the standard deviation of the sampling distribution of $$\hat{p}$$?

Accepted Answer

The standard deviation decreases.

Question 18

A survey uses a sample size that is 15% of the population. Which condition for constructing a confidence interval for a proportion is violated?

Accepted Answer

10% Condition

Question 19

Which of the following is a correct formula for the standard error of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\sqrt{\frac{\hat{p}\hat{q}}{n}}$$

Question 20

Which of the following is NOT a recommended practice when interpreting a confidence interval for a proportion?

Accepted Answer

Claiming to know the exact value of the parameter.

Sampling Distributions and Confidence Intervals for Proportions

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