Question 1

A researcher surveys a random sample of 100 people and finds that 55 of them eat breakfast every morning. What is the value of $$\hat{p} ($$sample proportion) in this scenario?

Accepted Answer

$$\hat{p} = 0.55$$

Question 2

Given a sample proportion $$\hat{p} = 0.55$$ and $$n = 100$$, what is the value of $$q ($$where $$q = 1 - \hat{p}$$)?

Accepted Answer

$$q = 0.45$$

Question 3

What is the formula for the margin of error $$E$$ for a confidence interval for a proportion at a 95% confidence level?

Accepted Answer

$$E = 1.96 \sqrt{\dfrac{\hat{p}q}{n}}$$

Question 4

If a sample of 100 people yields $$\hat{p} = 0.55$$, $$q = 0.45$$, and you use a 95% confidence level, what is the margin of error $$E$$? (Use $$z = 1.96)$$

Accepted Answer

$$E = 0.098$$

Question 5

A survey of 100 people finds that 60 drink coffee every morning. Construct the 95% confidence interval for the proportion of coffee drinkers. (Use $$z = 1.96)$$

Accepted Answer

$$0.507 \u003c p \u003c 0.693$$

Question 6

What sample size is needed to estimate a population proportion within 4 percentage points (0.04) at a 95% confidence level, assuming no prior estimate for $$\hat{p}$$?

Accepted Answer

$$n = 601$$

Question 7

If you want to estimate a proportion within 5 percentage points (0.05) at a 95% confidence level and have no prior estimate for $$\hat{p}$$, what sample size should you use?

Accepted Answer

$$n = 385$$

Question 8

Which formula is used to construct a confidence interval for a population mean when the population standard deviation is unknown?

Accepted Answer

$$\bar{x} \pm t_{\alpha/2} \dfrac{s}{\sqrt{n}}$$

Question 9

A sample of 49 has a mean of 20 and a sample standard deviation of 3. What is the 95% confidence interval for the population mean? (Use $$t_{0.025, 48} \approx 2.01)$$

Accepted Answer

$$19.14 \u003c \mu \u003c 20.86$$

Question 10

Which of the following 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 calculated intervals would contain the true population proportion.

Question 11

A sample of 40 has a mean of 30 and a sample standard deviation of 4. What is the margin of error for a 95% confidence interval? (Use $$t_{0.025, 39} = 2.021)$$

Accepted Answer

$$E = 1.278$$

Question 12

If a 95% confidence interval for a mean is $$28.722 \u003c \mu \u003c 31.278$$, what is the sample mean?

Accepted Answer

$$\bar{x} = 30$$

Question 13

Which of the following is NOT an assumption for constructing a confidence interval for a population mean using the $$t-$$distribution?

Accepted Answer

The population standard deviation is known.

Question 14

A medical researcher wants to estimate the proportion of adults who exercise regularly. She wants the margin of error to be no more than 3% at a 95% confidence level. What is the minimum sample size required (assume no prior estimate for $$\hat{p}$$)?

Accepted Answer

$$n = 1068$$

Question 15

If the sample size increases while keeping the confidence level the same, what happens to the margin of error?

Accepted Answer

The margin of error decreases.

Question 16

A 95% confidence interval for a proportion is calculated as $$0.452 \u003c p \u003c 0.648. $$What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = 0.55$$

Question 17

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

Accepted Answer

Increasing the confidence level from 95% to 99%.

Question 18

A sample of 100 adults is surveyed, and 40 report drinking coffee daily. What is the 95% confidence interval for the proportion of coffee drinkers? (Use $$z = 1.96)$$

Accepted Answer

$$0.304 \u003c p \u003c 0.496$$

Question 19

Which of the following statements is TRUE regarding confidence intervals?

Accepted Answer

A 99% confidence interval is wider than a 95% confidence interval for the same data.

Confidence Intervals and Sample Size in Statistics

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