Q: You want to estimate the mean IQ of statistics students. The population standard deviation is known to be $$15. $$What is the minimum sample size needed to estimate the mean within $$3$$ points with 95% confidence?

$$n = \left( \dfrac{z_{0.025} \times 15}{3} \$$right)^2$$

Question 1

A Pew Research Center survey of 1007 adults found that 85% know what Twitter is. Construct a 95% confidence interval for the population proportion $$p. $$Which of the following is the correct formula for the margin of error $$E in $$this context?

Accepted Answer

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

Question 2

Suppose you want to estimate the proportion of adults who buy clothing online. You have no prior estimate for $$\hat{p}. $$What value should you use for $$\hat{p}$$ when determining the minimum sample size required for a given margin of error?

Accepted Answer

$$\hat{p} = 0.5$$

Question 3

A clinical trial using the XSORT method resulted in 574 babies, 525 of whom were girls. Construct a 95% confidence interval for the proportion of girls. Which of the following is the correct point estimate for $$p$$?

Accepted Answer

$$\hat{p} = \dfrac{525}{574}$$

Question 4

Given a confidence interval for a population proportion $$0.278 \u003c p \u003c 0.338$$, what is the margin of error $$E$$?

Accepted Answer

$$E = \dfrac{0.338 - 0.278}{2} = 0.03$$

Question 5

A manager wants to estimate the percentage of U.S. adults who use the internet. If a previous poll found 73% use the internet, what is the formula for the minimum sample size $$n$$ required to estimate the proportion within a margin of error $$E at $$confidence level $$1-\alpha$$?

Accepted Answer

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

Question 6

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

Accepted Answer

The population size must be at least 10 times the sample size.

Question 7

A sample of 49 subjects treated with garlic had a mean LDL cholesterol change of $$0.4 mg/dL $$and a standard deviation of $$21.0 mg/dL. $$Which distribution should be used to construct a 95% confidence interval for the mean change?

Accepted Answer

The $$t-$$distribution, because $$\sigma is $$unknown.

Question 8

Given a confidence interval for a mean $$43 \u003c \mu \u003c 57$$, what is the point estimate for $$\mu$$?

Accepted Answer

$$\mu = \dfrac{43 + 57}{2} = 50$$

Question 9

A sample of 40 men has a mean weight of $$172.55 lb. If $$the population standard deviation is $$\sigma = 26 lb$$, what is the 95% confidence interval for the population mean $$\mu$$?

Accepted Answer

$$172.55 \pm z_{0.025} \dfrac{26}{\sqrt{40}}$$

Question 10

You want to estimate the mean IQ of statistics students. The population standard deviation is known to be $$15. $$What is the minimum sample size needed to estimate the mean within $$3$$ points with 95% confidence?

Accepted Answer

$$n = \left( \dfrac{z_{0.025} 	imes 15}{3} \$$right)^2$$

Question 11

A 99% confidence interval for the mean number of calories in a type of cookie is calculated from a sample of 15 cookies. Which distribution should be used?

Accepted Answer

The $$t-$$distribution, because $$\sigma is $$unknown and $$n \u003c 30.$$

Question 12

A poll reports a 95% confidence interval for a proportion as $$(0.62, 0.70). $$What is the best point estimate for the population proportion?

Accepted Answer

$$\hat{p} = \dfrac{0.62 + 0.70}{2} = 0.66$$

Question 13

Which of the following statements about sample size and poll reliability is correct?

Accepted Answer

The reliability of a poll depends mainly on the sample size and sampling method, not the population size.

Question 14

A 98% confidence interval for the mean mercury content in tuna sushi is calculated. If the interval is entirely above 1 part per million, what does this suggest?

Accepted Answer

The mean mercury content likely exceeds the FDA guideline.

Question 15

Which formula is used to estimate the population mean $$\mu$$ when the population standard deviation $$\sigma is $$unknown?

Accepted Answer

$$\bar{x} \pm t_{\alpha/2, n-1} \dfrac{s}{\sqrt{n}}$$

Question 16

If you are given a list of data values and need to estimate the population standard deviation $$\sigma$$ using the range rule of thumb, which formula should you use?

Accepted Answer

$$\sigma \approx \dfrac{	ext{range}}{4}$$

Question 17

A 95% confidence interval for a population mean is $$(100, 120). $$What is the margin of error $$E$$?

Accepted Answer

$$E = \dfrac{120 - 100}{2} = 10$$

Question 18

Which of the following is a correct interpretation of a 95% confidence interval for a population proportion?

Accepted Answer

We are 95% confident that the interval contains the true population proportion.

Estimating Population Proportions and Means: Confidence Intervals and Sample Size Determination

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