Q: Given a sample mean $$\bar{x} = 28$$, population standard deviation $$\sigma = 5$$, and sample size $$n = 40$$, what is the margin of error for a 99% confidence interval for the population mean?

$$2.58 \times \dfrac{5}{\sqrt{40}} = 2.04$$

Q: For the newspaper survey above, what is the margin of error for a 95% confidence interval for the true proportion? (Use $$z_{0.025} = 1.96)$$

$$1.96 \times \sqrt{\dfrac{0.41 \times 0.59}{1600}} \approx 0.024$$

Question 1

Which of the following best describes a confidence interval estimate for a population mean?

Accepted Answer

It is a range of values, calculated from the sample data, that is likely to contain the population mean with a specified level of confidence.

Question 2

Suppose you want to estimate the average in-state tuition fee at a university. You collect a random sample and calculate a sample mean of $2,300. Which type of estimate is this?

Accepted Answer

Point estimate

Question 3

Given a sample mean $$\bar{x} = 28$$, population standard deviation $$\sigma = 5$$, and sample size $$n = 40$$, what is the margin of error for a 99% confidence interval for the population mean?

Accepted Answer

$$2.58 	imes \dfrac{5}{\sqrt{40}} = 2.04$$

Question 4

Which formula is used to calculate a 95% confidence interval for the population mean when the population standard deviation $$\sigma is $$known?

Accepted Answer

$$\bar{x} \pm Z_{\alpha/2} \dfrac{\sigma}{\sqrt{n}}$$

Question 5

When is it appropriate to use the t-distribution instead of the z-distribution for constructing a confidence interval for the mean?

Accepted Answer

When the population standard deviation $$\sigma is $$unknown and the sample size is small ($$n \u003c 30)$$

Question 6

A random sample of 6 students' GPAs is: 3.20, 3.75, 3.10, 2.95, 3.40, 3.00. The sample mean is $$3.20$$ and sample standard deviation is $$0.34. $$What is the 95% confidence interval for the population mean GPA? (Use $$t_{0.025,5} = 2.571)$$

Accepted Answer

$$3.20 \pm 2.571 \dfrac{0.34}{\sqrt{6}} = [2.84, 3.56]$$

Question 7

What does a 95% confidence interval of [2.84, 3.56] for the mean GPA mean?

Accepted Answer

We are 95% confident that the true mean GPA is between 2.84 and 3.56.

Question 8

Which property distinguishes the t-distribution from the normal (z) distribution?

Accepted Answer

The t-distribution has heavier tails than the normal distribution.

Question 9

A metropolitan newspaper surveys 1,600 subscribers and finds that 656 want more local news coverage. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\dfrac{656}{1600} = 0.41$$

Question 10

For the newspaper survey above, what is the margin of error for a 95% confidence interval for the true proportion? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$1.96 	imes \sqrt{\dfrac{0.41 	imes 0.59}{1600}} \approx 0.024$$

Question 11

Which condition must be satisfied to use the normal approximation for a confidence interval for a proportion?

Accepted Answer

$$np \u003e 5$$ and $$n(1-p) \u003e 5$$

Question 12

If you want to estimate a population mean with a margin of error no greater than $$E$$, and you know the population standard deviation $$\sigma$$, which formula gives the required sample size $$n$$?

Accepted Answer

$$n = \left( \dfrac{Z_{\alpha/2} \sigma}{E} \$$right)^2$$

Question 13

You want to estimate the average weekly internet usage of households with a margin of error of 2 minutes at 90% confidence. If $$\sigma = 15$$ minutes and $$Z_{0.05} = 1.65$$, what sample size is required?

Accepted Answer

$$n = \left( \dfrac{1.65 	imes 15}{2} \$$right)^2$$ = 153$$

Question 14

To estimate a population proportion $$p$$ with a margin of error $$E at a $$confidence level $$(1-\alpha)$$, which formula gives the required sample size $$n$$?

Accepted Answer

$$n = \dfrac{Z_{\alpha/2}^2 p(1-p)}{E^2$}$$

Question 15

If the desired margin of error decreases, what happens to the required sample size for a given confidence level and population standard deviation?

Accepted Answer

The required sample size increases.

Question 16

A 95% confidence interval for a population proportion is calculated as $$[0.39, 0.43]. $$What is the correct interpretation?

Accepted Answer

We are 95% confident that the true proportion is between 39% and 43%.

Question 17

Which of the following statements about degrees of freedom (df) in the t-distribution is correct?

Accepted Answer

df is the number of independent observations used to calculate the statistic, typically $$n-1$$ for a sample mean.

Confidence Interval Estimation: Concepts, Calculations, and Applications

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