Q: A sample of 950 undergraduate students found that 53% take online courses. What is the margin of error $$E$$ for a 95% confidence interval? (Use $$z_{\alpha/2} = 1.96)$$

$$E = 1.96 \sqrt{\frac{0.53 \times 0.47}{950}}$$

Q: If a 2016 Pew Research Center survey showed that 79% of 4787 adults shop online, what is the minimum sample size required to estimate the proportion with a margin of error no more than 0.03 at 95% confidence? (Use $$z_{\alpha/2} = 1.96$$ and $$\hat{p} = 0.79)$$

$$n = \frac{0.79 \times 0.21 \times (1.96)^2$}{0.03^2$}$$

Question 1

Which of the following best describes the confidence level in the context of confidence intervals?

Accepted Answer

It is the probability that the confidence interval contains the population parameter if the estimation process is repeated many times.

Question 2

For a 95% confidence level, what is the corresponding value of $$\alpha$$?

Accepted Answer

$$\alpha = 0.05$$

Question 3

What is the critical value $$z_{\alpha/2}$$ for a 99% confidence interval?

Accepted Answer

$$z_{\alpha/2} = 2.575$$

Question 4

A sample of 950 undergraduate students found that 53% take online courses. What is the margin of error $$E$$ for a 95% confidence interval? (Use $$z_{\alpha/2} = 1.96)$$

Accepted Answer

$$E = 1.96 \sqrt{\frac{0.53 	imes 0.47}{950}}$$

Question 5

Which requirement must be met for the normal distribution to be a suitable approximation for the binomial distribution when estimating a population proportion?

Accepted Answer

There must be at least 5 successes and at least 5 failures ($$np \geq 5$$ and $$nq \geq 5).$$

Question 6

Given a 95% confidence interval for a population proportion is [0.58, 0.81], what is the point estimate $$\hat{p}$$?

Accepted Answer

$$\hat{p} = 0.695$$

Question 7

If a 2016 Pew Research Center survey showed that 79% of 4787 adults shop online, what is the minimum sample size required to estimate the proportion with a margin of error no more than 0.03 at 95% confidence? (Use $$z_{\alpha/2} = 1.96$$ and $$\hat{p} = 0.79)$$

Accepted Answer

$$n = \frac{0.79 	imes 0.21 	imes (1.96)^2$}{0.03^2$}$$

Question 8

Which of the following statements correctly interprets a 95% confidence interval for a population proportion?

Accepted Answer

If we repeat the sampling process many times, about 95% of the confidence intervals will contain the true population proportion.

Question 9

What is the formula for the margin of error $$E$$ when estimating a population proportion?

Accepted Answer

$$E = z_{\alpha/2} \sqrt{\frac{\hat{p}\hat{q}}{n}}$$

Question 10

A medical study reports that 70% of 71 subjects were abstinent from smoking at 8 weeks, with a 95% confidence interval of [58%, 81%]. What is the margin of error $$E$$?

Accepted Answer

$$E = 0.11$$

Question 11

If the confidence level is increased from 95% to 99%, what happens to the margin of error $$E$$?

Accepted Answer

The margin of error increases.

Question 12

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

Accepted Answer

The population must be normally distributed.

Question 13

A confidence interval for a population proportion is calculated as $$\hat{p} \pm E. If$$ $$\hat{p} = 0.53$$ and $$E = 0.032$$, what is the 95% confidence interval?

Accepted Answer

$$[0.498, 0.562]$$

Question 14

In a survey, 53% of 950 students take online courses. Can we conclude that more than 50% of all undergraduates take online courses at the 95% confidence level?

Accepted Answer

No, because the confidence interval may include values below 50%.

Question 15

If you have no prior information about the population proportion, what value should you use for $$\hat{p}$$ when calculating the required sample size?

Accepted Answer

$$\hat{p} = 0.5$$

Question 16

Which of the following best explains why the margin of error is important in reporting survey results?

Accepted Answer

It shows the range of possible values for the population parameter.

Question 17

What is the effect of increasing the sample size $$n on $$the margin of error $$E$$?

Accepted Answer

The margin of error decreases.

Question 18

Which formula would you use to calculate the minimum sample size needed to estimate a population proportion with a specified margin of error $$E$$ and confidence level?

Accepted Answer

$$n = \frac{\hat{p}\hat{q}(z_{\alpha/2})^2$}{E^2$}$$

Estimating Population Proportions: Confidence Intervals and Sample Size

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