Q: Suppose a poll finds that 40% of voters believe scientists disagree about global warming. If a random sample of 1000 voters is taken, what is the standard deviation of the sample proportion?

$$\sqrt{\frac{0.4 \times 0.6}{1000}}$$

Q: A survey of 1019 US adults found that 36.1% update their Facebook status daily. What is the 95% confidence interval for the true proportion of US adults who update their status daily?

$$0.361 \pm 1.96 \times \sqrt{\frac{0.361 \times 0.639}{1019}}$$

Q: A sample of 1000 US adults finds that 15% smoke. What is the standard error of the sample proportion?

$$\sqrt{\frac{0.15 \times 0.85}{1000}}$$

Q: A poll finds that 23% of US adults believe climate change is a hoax. If the sample size is 2000, what is the margin of error for a 95% confidence interval?

$$1.96 \times \sqrt{\frac{0.23 \times 0.77}{2000}}$$

Q: If the sample proportion is 0.4 and the sample size is 500, what is the 99% confidence interval for the population proportion?

$$0.4 \pm 2.576 \times \sqrt{\frac{0.4 \times 0.6}{500}}$$

Question 1

Suppose a poll finds that 40% of voters believe scientists disagree about global warming. If a random sample of 1000 voters is taken, what is the standard deviation of the sample proportion?

Accepted Answer

$$\sqrt{\frac{0.4 	imes 0.6}{1000}}$$

Question 2

Which of the following is a necessary condition for using the normal model to approximate the sampling distribution of a proportion?

Accepted Answer

Both $$np \geq 10$$ and $$n(1-p) \geq 10$$

Question 3

A survey of 1019 US adults found that 36.1% update their Facebook status daily. What is the 95% confidence interval for the true proportion of US adults who update their status daily?

Accepted Answer

$$0.361 \pm 1.96 	imes \sqrt{\frac{0.361 	imes 0.639}{1019}}$$

Question 4

If the margin of error for a 95% confidence interval is 2.5%, what is the corresponding critical value from the normal distribution?

Accepted Answer

$$z^* = 1.96$$

Question 5

A random sample of 200 students finds that 17% are left-handed. What is the probability that in a class of 99 students, there will be at least 20 left-handed students?

Accepted Answer

Use $$P(\hat{p} \geq \frac{20}{99})$$ where $$\hat{p} is $$approximately normal with mean $$0.17$$ and standard deviation $$\sqrt{\frac{0.17 	imes 0.83}{99}}$$

Question 6

Which statement best describes the meaning of a 95% confidence interval for a population proportion?

Accepted Answer

If we repeated the sampling process many times, about 95% of the intervals would contain the true population proportion.

Question 7

If a researcher wants to reduce the margin of error in a confidence interval for a proportion, which of the following should they do?

Accepted Answer

Increase the sample size $$n$$

Question 8

A sample of 1000 US adults finds that 15% smoke. What is the standard error of the sample proportion?

Accepted Answer

$$\sqrt{\frac{0.15 	imes 0.85}{1000}}$$

Question 9

Which of the following is NOT a valid interpretation of a confidence interval for a proportion?

Accepted Answer

The interval contains the true proportion with 95% certainty for this sample.

Question 10

A poll finds that 23% of US adults believe climate change is a hoax. If the sample size is 2000, what is the margin of error for a 95% confidence interval?

Accepted Answer

$$1.96 	imes \sqrt{\frac{0.23 	imes 0.77}{2000}}$$

Question 11

If the sample proportion is 0.4 and the sample size is 500, what is the 99% confidence interval for the population proportion?

Accepted Answer

$$0.4 \pm 2.576 	imes \sqrt{\frac{0.4 	imes 0.6}{500}}$$

Question 12

Which of the following would make the sampling distribution of the sample proportion less normal?

Accepted Answer

A small sample size with $$np \u003c 10 or$$ $$n(1-p) \u003c 10$$

Question 13

A confidence interval for a proportion is given by $$\hat{p} \pm z^* 	imes SE(\hat{p}). $$What does $$SE(\hat{p})$$ represent?

Accepted Answer

The standard error of the sample proportion

Question 14

If a 95% confidence interval for a proportion is (0.23, 0.27), what can you say about the true proportion?

Accepted Answer

We are 95% confident that the true proportion is between 0.23 and 0.27.

Question 15

Which of the following best describes the margin of error in a confidence interval for a proportion?

Accepted Answer

It is the maximum expected difference between the sample proportion and the true population proportion due to sampling variability.

Question 16

A researcher wants a margin of error less than 2% for a 95% confidence interval. What should they do?

Accepted Answer

Increase the sample size until $$1.96 	imes \sqrt{\frac{p(1-p)}{n}} \u003c 0.02$$

Question 17

If the sample size is doubled, what happens to the standard error of the sample proportion?

Accepted Answer

It decreases by a factor of $$\sqrt{2}$$

Question 18

Which of the following is a correct formula for the standard error of a sample proportion?

Accepted Answer

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

Question 19

A sample of 1500 US adults finds that 41% think the death penalty is applied fairly. What is the 95% confidence interval for the proportion who think so?

Accepted Answer

$$0.41 \pm 1.96 	imes \sqrt{\frac{0.41 	imes 0.59}{1500}}$$

Sampling Distribution Models & Confidence Intervals for Proportions

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