Q: Given a sample proportion $$\hat{p} = 0.60$$, sample size $$n = 100$$, and a 95% confidence level, what is the correct formula for the margin of error $$E$$?

$$E = 1.96 \sqrt{\frac{0.60 \times 0.40}{100}}$$

Q: A researcher wants to estimate the proportion of adults who drink coffee every day. She wants a margin of error of 5% at a 95% confidence level and has no prior estimate for $$\hat{p}. $$What is the minimum sample size she should use?

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

Q: A sample of 40 adults has a mean daily caffeine intake of 30 mg and a sample standard deviation of 4 mg. What is the margin of error for a 95% confidence interval? (Use $$t^* = 2.021)$$

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

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 scenario?

Accepted Answer

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

Question 2

Given a sample proportion $$\hat{p} = 0.60$$, sample size $$n = 100$$, and a 95% confidence level, what is the correct formula for the margin of error $$E$$?

Accepted Answer

$$E = 1.96 \sqrt{\frac{0.60 	imes 0.40}{100}}$$

Question 3

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

$$\hat{p} = 0.25$$

Question 4

A researcher wants to estimate the proportion of adults who drink coffee every day. She wants a margin of error of 5% at a 95% confidence level and has no prior estimate for $$\hat{p}. $$What is the minimum sample size she should use?

Accepted Answer

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

Question 5

You survey 100 people and find that 60 drink coffee every morning. What is the 95% confidence interval for the proportion of coffee drinkers?

Accepted Answer

$$0.507 \u003c p \u003c 0.693$$

Question 6

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 7

A sample of 49 students has a mean test score of 20 and a sample standard deviation of 3. What is the 95% confidence interval for the population mean?

Accepted Answer

$$19.16 \u003c \mu \u003c 20.84$$

Question 8

When constructing a confidence interval for a population mean with unknown population standard deviation and a small sample size, which distribution should you use?

Accepted Answer

The $$t-$$distribution

Question 9

A sample of 40 adults has a mean daily caffeine intake of 30 mg and a sample standard deviation of 4 mg. What is the margin of error for a 95% confidence interval? (Use $$t^* = 2.021)$$

Accepted Answer

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

Question 10

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

Accepted Answer

$$\bar{x} = 30.00$$

Question 11

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

Accepted Answer

Decreasing the sample size

Question 12

A 95% confidence interval for a proportion is $$0.452 \u003c p \u003c 0.648. $$What is the margin of error?

Accepted Answer

$$E = 0.098$$

Question 13

If you want to decrease the margin of error in your confidence interval for a proportion, which of the following should you do?

Accepted Answer

Increase the sample size

Question 14

Which formula is used to calculate the sample size needed for estimating a population proportion with a specified margin of error $$E at a 95$$% confidence level?

Accepted Answer

$$n = \frac{(1.96)^2$ \hat{p}(1-\hat{p})}{E^2$}$$

Question 15

A sample of 100 people is surveyed, and 45% report eating breakfast daily. What is the value of $$\hat{q}$$?

Accepted Answer

$$\hat{q} = 0.55$$

Question 16

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

Accepted Answer

The population standard deviation is known

Question 17

If the sample size increases while all other factors remain constant, what happens to the margin of error for a confidence interval?

Accepted Answer

It decreases

Question 18

A 95% confidence interval for a mean is calculated as $$\bar{x} \pm E. If$$ $$\bar{x} = 20$$ and $$E = 0.84$$, what is the interval?

Accepted Answer

$$19.16 \u003c \mu \u003c 20.84$$

Question 19

Which of the following real-world scenarios best illustrates the use of a confidence interval for a population mean?

Accepted Answer

Estimating the average blood pressure of adults in a city based on a sample

Confidence Intervals for Proportions and Means in Statistics

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