Q: If a confidence interval for a population proportion is (0.576, 0.644), what is the margin of error?

$$\text{Margin of error} = \frac{0.644 - 0.576}{2} = 0.034$$

Q: A company wants to estimate the proportion of defective products. In a sample of 500 products, 40 are defective. What is the 95% confidence interval for the proportion of defective products? (Use $$z_{0.025} = 1.96)$$

$$0.08 \pm 1.96 \times \sqrt{\frac{0.08 \times 0.92}{500}}$$

Q: If the sample proportion is 0.61 and the sample size is 784, what is the standard error of the proportion?

$$SE = \sqrt{\frac{0.61 \times 0.39}{784}}$$

Q: A poll found that 65% of 1000 respondents support a new policy. What is the 95% confidence interval for the proportion? (Use $$z_{0.025} = 1.96)$$

$$0.65 \pm 1.96 \times \sqrt{\frac{0.65 \times 0.35}{1000}}$$

Question 1

A researcher wants to estimate the proportion of students at a university who own a laptop. In a random sample of 1200 students, 900 report owning a laptop. What is the point estimate for the population proportion?

Accepted Answer

$$\frac{900}{1200} = 0.75$$

Question 2

In Example 1, a survey of 1002 people found that 701 said they voted. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{701}{1002} \approx 0.7006$$

Question 3

Which of the following best describes a confidence interval for a population proportion?

Accepted Answer

A range of values likely to contain the true population proportion with a specified level of confidence.

Question 4

In Example 2, the sample proportion of checks with leading digits of 5 is 0.61. If the sample size is 784, what is the number of checks with leading digits of 5?

Accepted Answer

$$x = 0.61 	imes 784 = 478$$

Question 5

A 99% confidence interval for the proportion of people who voted is calculated as (0.662, 0.737). What does this interval mean?

Accepted Answer

We are 99% confident that the true proportion of people who voted is between 0.662 and 0.737.

Question 6

Which formula is used to calculate the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{x}{n}$$

Question 7

If a confidence interval for a population proportion is (0.576, 0.644), what is the margin of error?

Accepted Answer

$$	ext{Margin of error} = \frac{0.644 - 0.576}{2} = 0.034$$

Question 8

A company wants to estimate the proportion of defective products. In a sample of 500 products, 40 are defective. What is the 95% confidence interval for the proportion of defective products? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$0.08 \pm 1.96 	imes \sqrt{\frac{0.08 	imes 0.92}{500}}$$

Question 9

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

Accepted Answer

Increasing the confidence level from 95% to 99%.

Question 10

A medical researcher wants to estimate the proportion of patients who respond to a new drug. In a sample of 200 patients, 120 respond. What is the sample proportion and its standard error?

Accepted Answer

$$\hat{p} = \frac{120}{200} = 0.6$$; $$SE = \sqrt{\frac{0.6 	imes 0.4}{200}}$$

Question 11

If the sample proportion is 0.61 and the sample size is 784, what is the standard error of the proportion?

Accepted Answer

$$SE = \sqrt{\frac{0.61 	imes 0.39}{784}}$$

Question 12

A poll found that 65% of 1000 respondents support a new policy. What is the 95% confidence interval for the proportion? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$0.65 \pm 1.96 	imes \sqrt{\frac{0.65 	imes 0.35}{1000}}$$

Question 13

Why is it important to use a random sample when estimating a population proportion?

Accepted Answer

To ensure the sample is representative of the population and reduce bias.

Question 14

A confidence interval for a population proportion is calculated as (0.576, 0.644). If the sample proportion is 0.61, what is the interpretation?

Accepted Answer

We are 95% confident that the true proportion is between 0.576 and 0.644.

Question 15

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

Accepted Answer

The population mean must be known.

Question 16

A survey of 800 people found that 480 support a new law. What is the 99% confidence interval for the proportion of supporters? (Use $$z_{0.005} = 2.576)$$

Accepted Answer

$$0.6 \pm 2.576 	imes \sqrt{\frac{0.6 	imes 0.4}{800}}$$

Question 17

If the sample size increases while the sample proportion remains the same, what happens to the margin of error for the confidence interval?

Accepted Answer

The margin of error decreases.

Question 18

A confidence interval for a population proportion is calculated using the formula $$\hat{p} \pm z 	imes SE. $$What does $$SE$$ represent?

Accepted Answer

The standard error of the sample proportion.

Question 19

A school administrator wants to estimate the proportion of students who participate in extracurricular activities. In a sample of 600 students, 360 participate. What is the 95% confidence interval for the proportion? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$0.6 \pm 1.96 	imes \sqrt{\frac{0.6 	imes 0.4}{600}}$$

Estimating a Population Proportion: Confidence Intervals

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