Q: A researcher wants to estimate the proportion of adults who make online purchases with a margin of error of $$0.03 at a 95$$% confidence level. If a prior estimate of $$\hat{p} = 0.8 is $$available, what is the minimum sample size required?

$$n = \left[ \dfrac{(1.96)^2$ \times 0.8 \times 0.2}{0.03^2$} \right]$$

Question 1

A Gallup poll surveyed 1487 adults and found that 43% have a Facebook page. What is the best point estimate of the population proportion $$p of $$all adults who have a Facebook page?

Accepted Answer

$$0.43$$

Question 2

Which of the following is the correct interpretation of a 95% confidence interval for a population proportion $$p$$ given by $$0.405 \u003c p \u003c 0.455$$?

Accepted Answer

We are 95% confident that the interval from $$0.405 to$$ $$0.455$$ actually does contain the true value of the population proportion $$p.$$

Question 3

Which formula is used to calculate the margin of error $$E$$ for estimating a population proportion $$p$$?

Accepted Answer

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

Question 4

If you want to estimate a population proportion with a margin of error no greater than $$0.03$$ and you have no prior estimate for $$\hat{p}$$, what value should you use for $$\hat{p}\hat{q} in $$the sample size formula?

Accepted Answer

$$0.25$$

Question 5

Which of the following statements about the Student $$t$$ distribution is FALSE?

Accepted Answer

The Student $$t$$ distribution has a standard deviation of $$1$$ for all sample sizes.

Question 6

A sample of 106 body temperatures has a mean of $$98.20^{\circ}F$ and a standard deviation of 0.62$$^{\circ}$$F. $$Which distribution should be used to construct a 95% confidence interval for the population mean body temperature?

Accepted Answer

Student $$t$$ distribution

Question 7

Which formula is used to determine the minimum sample size $$n$$ needed to estimate a population proportion $$p$$ with a specified margin of error $$E$$ when an estimate $$\hat{p} is $$available?

Accepted Answer

$$n = \left[ \dfrac{z_{\alpha/2}^2 \hat{p}\hat{q}}{E^2$} \right]$$

Question 8

A 95% confidence interval for a population mean $$\mu is $$calculated as $$[98.0, 98.4]. $$What is the point estimate of $$\mu$$?

Accepted Answer

$$98.2$$

Question 9

Which of the following is NOT a condition for constructing a confidence interval for a population proportion $$p$$?

Accepted Answer

The population standard deviation $$\sigma$$ must be known.

Question 10

Suppose you want to estimate the mean IQ score for statistics students with 95% confidence and a margin of error of 3 IQ points. If $$\sigma = 15$$, which formula should you use to determine the required sample size $$n$$?

Accepted Answer

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

Question 11

Which critical value $$z_{\alpha/2}$$ should be used for a 99% confidence interval?

Accepted Answer

$$2.575$$

Question 12

A researcher wants to estimate the proportion of adults who make online purchases with a margin of error of $$0.03 at a 95$$% confidence level. If a prior estimate of $$\hat{p} = 0.8 is $$available, what is the minimum sample size required?

Accepted Answer

$$n = \left[ \dfrac{(1.96)^2$ 	imes 0.8 	imes 0.2}{0.03^2$} ight]$$

Question 13

Which of the following best describes the relationship between the confidence level and $$\alpha$$?

Accepted Answer

Confidence level $$= 1 - \alpha$$

Question 14

When constructing a confidence interval for a population mean $$\mu$$ with $$\sigma$$ known and $$n \u003e 30$$, which distribution should be used?

Accepted Answer

Standard normal ($$z) $$distribution

Question 15

Which of the following is a correct formula for a confidence interval for a population mean $$\mu$$ when $$\sigma is $$not known?

Accepted Answer

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

Estimating Parameters and Determining Sample Sizes

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