Q: You survey 100 people and find that 60 drink coffee every morning. Construct a 95% confidence interval for the proportion. What is the margin of error $$E$$? (Use $$Z_{\alpha/2} = 1.96)$$

$$E = 1.96 \times \sqrt{\frac{0.6 \times 0.4}{100}} = 0.096$$

Q: What is the formula for the required sample size $$n to $$estimate a population proportion with margin of error $$E at 95$$% confidence, assuming no prior knowledge of $$\hat{p}$$?

$$n = \frac{(1.96)^2$ \times 0.25}{E^2$}$$

Q: A researcher wants to estimate the proportion of adults who drink coffee daily to within 5 percentage points (0.05) at 95% confidence, with no prior estimate for $$\hat{p}. $$What is the minimum sample size required?

$$n = \frac{(1.96)^2$ \times 0.25}{0.05^2$} = 384.16 ($$round up to 385)

Q: You have a sample mean of 30, sample standard deviation of 4, and sample size of 40. What is the margin of error $$E$$ for a 95% confidence interval for the mean? (Use $$t^* = 2.021)$$

$$E = 2.021 \times \frac{4}{\sqrt{40}} = 1.278$$

Q: Construct a 95% confidence interval for a mean of 20, sample size 49, sample standard deviation 3. What is the margin of error $$E$$? (Use $$t^* = 2.021$$ for 48 degrees of freedom.)

$$E = 2.021 \times \frac{3}{7} = 0.866$$

Question 1

A simple random sample of 100 people finds that 55 eat breakfast every morning. What is the value of $$\hat{p} ($$sample proportion) in this case?

Accepted Answer

$$\hat{p} = \frac{55}{100} = 0.55$$

Question 2

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

Accepted Answer

$$q = 0.45$$

Question 3

For a 95% confidence interval for a proportion, what is the critical z-score ($$Z_{\alpha/2}) $$you should use?

Accepted Answer

$$1.96$$

Question 4

You survey 100 people and find that 60 drink coffee every morning. Construct a 95% confidence interval for the proportion. What is the margin of error $$E$$? (Use $$Z_{\alpha/2} = 1.96)$$

Accepted Answer

$$E = 1.96 	imes \sqrt{\frac{0.6 	imes 0.4}{100}} = 0.096$$

Question 5

If you want to be within 4 percentage points (0.04) of the true proportion at a 95% confidence level, and you have no prior knowledge of the proportion, what value should you use for $$\hat{p} in $$the sample size formula?

Accepted Answer

$$\hat{p} = 0.5$$

Question 6

What is the formula for the required sample size $$n to $$estimate a population proportion with margin of error $$E at 95$$% confidence, assuming no prior knowledge of $$\hat{p}$$?

Accepted Answer

$$n = \frac{(1.96)^2$ 	imes 0.25}{E^2$}$$

Question 7

A researcher wants to estimate the proportion of adults who drink coffee daily to within 5 percentage points (0.05) at 95% confidence, with no prior estimate for $$\hat{p}. $$What is the minimum sample size required?

Accepted Answer

$$n = \frac{(1.96)^2$ 	imes 0.25}{0.05^2$} = 384.16 ($$round up to 385)

Question 8

You have a sample mean of 30, sample standard deviation of 4, and sample size of 40. What is the margin of error $$E$$ for a 95% confidence interval for the mean? (Use $$t^* = 2.021)$$

Accepted Answer

$$E = 2.021 	imes \frac{4}{\sqrt{40}} = 1.278$$

Question 9

Given a sample mean of 30, sample standard deviation of 4, sample size of 40, and $$t^* = 2.021$$, what is the 95% confidence interval for the population mean?

Accepted Answer

$$28.722 \u003c \mu \u003c 31.278$$

Question 10

Construct a 95% confidence interval for a mean of 20, sample size 49, sample standard deviation 3. What is the margin of error $$E$$? (Use $$t^* = 2.021$$ for 48 degrees of freedom.)

Accepted Answer

$$E = 2.021 	imes \frac{3}{7} = 0.866$$

Question 11

A sample of 49 students has a mean test score of 20 and a standard deviation of 3. What is the 95% confidence interval for the population mean? (Use $$t^* = 2.021)$$

Accepted Answer

$$19.134 \u003c \mu \u003c 20.866$$

Question 12

Which of the following best describes the interpretation of a 95% confidence interval for a population proportion?

Accepted Answer

We are 95% confident that the true population proportion lies within the calculated interval.

Question 13

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

Accepted Answer

The margin of error decreases.

Question 14

A hospital administrator wants to estimate the average length of stay for patients. She samples 36 patients, finds a mean of 5 days, and a standard deviation of 2 days. What is the 95% confidence interval for the mean? (Use $$t^* = 2.03$$ for 35 degrees of freedom.)

Accepted Answer

$$4.32 \u003c \mu \u003c 5.68$$

Question 15

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 16

A pollster wants to estimate the proportion of voters who support a new policy. She wants to be 95% confident and within 3 percentage points (0.03) of the true value. What is the minimum sample size required, assuming no prior estimate for $$\hat{p}$$?

Accepted Answer

$$n = \frac{(1.96)^2$ 	imes 0.25}{0.03^2$} = 1067$$

Question 17

If a 95% confidence interval for a proportion is $$0.452 \u003c p \u003c 0.648$$, what can you conclude?

Accepted Answer

We are 95% confident that the true proportion is between 45.2% and 64.8%.

Confidence Intervals for Proportions and Means

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