Q: Suppose you want to estimate the mean IQ score for statistics students with a margin of error of $$3$$ points and $$\sigma = 15. $$What is the minimum sample size required for a 95% confidence level? (Use $$z_{0.025} = 1.96.)$$

$$n = \left( \dfrac{1.96 \times 15}{3} \$$right)^2$$ = 96.04 \approx 97$$

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}$$F. $$Construct a 95% confidence interval for the population mean $$\mu. $$Which distribution should you use?

Accepted Answer

Student $$t$$ distribution, because $$\sigma is $$not known.

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

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

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 requirement 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 a margin of error of $$3$$ points and $$\sigma = 15. $$What is the minimum sample size required for a 95% confidence level? (Use $$z_{0.025} = 1.96.)$$

Accepted Answer

$$n = \left( \dfrac{1.96 	imes 15}{3} \$$right)^2$$ = 96.04 \approx 97$$

Question 11

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

Accepted Answer

Confidence level $$= 1 - \alpha$$

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 $$\hat{p} = 0.80 is $$available, what is the minimum sample size required? (Use $$z_{0.025} = 1.96.)$$

Accepted Answer

$$n = \left( \dfrac{1.96}{0.03} \$$right)^2$$ (0.80)(0.20) \approx 683$$

Question 13

Which of the following is TRUE about the critical value $$z_{\alpha/2}$$ for a 95% confidence level?

Accepted Answer

$$z_{\alpha/2} = 1.96$$

Question 14

When estimating a population mean $$\mu$$ and the population standard deviation $$\sigma is $$known, which distribution should be used to construct the confidence interval?

Accepted Answer

Standard normal ($$z) $$distribution

Question 15

A 99% confidence interval for a population proportion $$p is $$wider than a 90% confidence interval for the same data. Why?

Accepted Answer

A higher confidence level requires a larger critical value, increasing the margin of error.

Question 16

Which of the following is a requirement for using the binomial distribution to construct a confidence interval for a population proportion?

Accepted Answer

There are at least 5 successes and 5 failures in the sample.

Question 17

If a confidence interval for $$p$$ does NOT contain $$0.50$$, what can you conclude about the population proportion?

Accepted Answer

There is sufficient evidence at the given confidence level that $$p is $$not $$0.50.$$

Estimating Parameters and Determining Sample Sizes

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