Q: Given a sample mean $$\bar{x} = 74$$, sample standard deviation $$s = 23$$, and $$n = 32$$, what is the 95% confidence interval for the population mean if $$t_{\alpha/2} = 2.04$$?

$$(74 - 2.04 \times \frac{23}{\sqrt{32}},\ 74 + 2.04 \times \frac{23}{\sqrt{32}})$$

Q: A researcher wants to estimate the mean increase in SAT Math scores with a margin of error of 5 points and a standard deviation of 50 points at a 95% confidence level. What is the minimum sample size required?

$$n = \left[ \frac{z_{\alpha/2} \times 50}{5} \right]^2$$

Question 1

Which of the following best describes a parameter in statistics?

Accepted Answer

A numerical summary of a population.

Question 2

A researcher wants to estimate the proportion of college students who own a car. She surveys 200 students and finds that 90 own a car. What is the point estimate for the population proportion?

Accepted Answer

$$\frac{90}{200} = 0.45$$

Question 3

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

Accepted Answer

$$E = z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 4

A 95% confidence interval for a population proportion is (0.25, 0.35). What is the margin of error?

Accepted Answer

$$E = \frac{0.35 - 0.25}{2} = 0.05$$

Question 5

If you increase the confidence level from 95% to 99% while keeping the sample size the same, what happens to the width of the confidence interval?

Accepted Answer

The confidence interval becomes wider.

Question 6

Which of the following is a requirement for constructing a confidence interval for a population proportion?

Accepted Answer

$$np \geq 5$$ and $$nq \geq 5$$

Question 7

A 90% confidence interval for the mean processing time of insurance policies is (44.4, 55.6) days. What is the point estimate of the mean?

Accepted Answer

$$\frac{44.4 + 55.6}{2} = 50.0$$

Question 8

Which distribution should be used to construct a confidence interval for a population mean when the population standard deviation $$\sigma is $$unknown and the sample size is small?

Accepted Answer

Student's $$t-$$distribution

Question 9

Given a sample mean $$\bar{x} = 74$$, sample standard deviation $$s = 23$$, and $$n = 32$$, what is the 95% confidence interval for the population mean if $$t_{\alpha/2} = 2.04$$?

Accepted Answer

$$(74 - 2.04 	imes \frac{23}{\sqrt{32}},\ 74 + 2.04 	imes \frac{23}{\sqrt{32}})$$

Question 10

A researcher wants to estimate the mean increase in SAT Math scores with a margin of error of 5 points and a standard deviation of 50 points at a 95% confidence level. What is the minimum sample size required?

Accepted Answer

$$n = \left[ \frac{z_{\alpha/2} 	imes 50}{5} ight]^2$$

Question 11

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

Accepted Answer

As the sample size increases, the $$t-$$distribution approaches the standard normal distribution.

Question 12

A 99% confidence interval for the mean box office receipts is (5.22, 31.62) million dollars. What is the margin of error?

Accepted Answer

$$E = \frac{31.62 - 5.22}{2} = 13.2$$

Question 13

If a confidence interval for a population proportion is (0.113, 0.287), what is the point estimate $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{0.113 + 0.287}{2} = 0.20$$

Question 14

Which of the following is NOT a requirement for using the formula for a confidence interval for a population mean when $$\sigma is $$unknown?

Accepted Answer

The population must be binomially distributed.

Question 15

A survey finds that 28 out of 140 workers aspire to have their boss’s job. What is the 95% confidence interval for the population proportion? ($$z_{\alpha/2} = 1.96)$$

Accepted Answer

$$(0.2 - 1.96 	imes \sqrt{\frac{0.2 	imes 0.8}{140}},\ 0.2 + 1.96 	imes \sqrt{\frac{0.2 	imes 0.8}{140}})$$

Question 16

Which of the following best explains why a 95% confidence interval does not always contain the true population parameter?

Accepted Answer

Because 5% of all such intervals constructed from repeated samples will not contain the true parameter.

Question 17

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

Accepted Answer

$$\hat{p} = 0.5$$

Question 18

A 95% confidence interval for the mean selling price of used cars is desired with a margin of error of $$100 and a population standard deviation of 2000. $$What is the minimum sample size required?

Accepted Answer

$$n = \left[ \frac{z_{\alpha/2} 	imes 2000}{100} ight]^2$$

Estimating Parameters and Determining Sample Sizes: Confidence Intervals for Proportions and Means

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