Q: If a sample proportion $$\hat{p} is 0.4 $$from a sample of size 400, what is the standard error of $$\hat{p}$$?

$$ \sqrt{\frac{0.4 \times 0.6}{400}} $$

Question 1

Which of the following best defines a point estimate for a population proportion?

Accepted Answer

A single value computed from a sample that serves as a best guess for the population proportion.

Question 2

A Gallup poll surveyed 1015 Americans aged 18 and older and found that 731 said they always wear a seat belt. What is the point estimate for the population proportion of Americans who always wear a seat belt?

Accepted Answer

$$ \frac{731}{1015} $$

Question 3

Which formula gives the standard error of the sample proportion $$ \hat{p} $$?

Accepted Answer

$$ \sqrt{\frac{p(1-p)}{n}} $$

Question 4

What does a 95% confidence interval for a population proportion mean?

Accepted Answer

If we took many samples, about 95% of the confidence intervals constructed would contain the true population proportion.

Question 5

Which of the following is the correct formula for a $$(1-\alpha) 	imes 100\%$$ confidence interval for a population proportion $$p$$?

Accepted Answer

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

Question 6

If the level of confidence increases from 90% to 99%, what happens to the margin of error for a confidence interval for a population proportion, assuming the sample size remains the same?

Accepted Answer

The margin of error increases.

Question 7

A researcher wants to estimate the proportion of teenagers who text while driving. She wants a margin of error of 2% with 95% confidence and has no prior estimate. What value should she use for $$p in $$the sample size formula?

Accepted Answer

$$ 0.50 $$

Question 8

Which of the following best describes the effect of increasing the sample size on the margin of error for a confidence interval for a population proportion?

Accepted Answer

The margin of error decreases.

Question 9

Suppose you construct a 95% confidence interval for a population proportion and obtain the interval (0.62, 0.70). Which of the following is a correct interpretation?

Accepted Answer

We are 95% confident that the true population proportion is between 0.62 and 0.70.

Question 10

Which formula gives the minimum sample size $$n$$ required to estimate a population proportion $$p$$ with margin of error $$E$$ and confidence level $$(1-\alpha) 	imes 100\%$$, when no prior estimate of $$p is $$available?

Accepted Answer

$$ n = \left( \frac{z_{\alpha/2}}{E} \$$right)^2$$ 0.25 $$

Question 11

What does the critical value $$z_{\alpha/2}$$ represent in the context of constructing a confidence interval for a population proportion?

Accepted Answer

The number of standard errors to move away from the sample proportion to capture the desired confidence level.

Question 12

A 99% confidence interval for a population proportion will be ______ a 90% confidence interval, assuming the same sample data.

Accepted Answer

Wider than

Question 13

Which of the following is NOT an assumption required for constructing a confidence interval for a population proportion?

Accepted Answer

The population is normally distributed.

Question 14

If a sample proportion $$\hat{p} is 0.4 $$from a sample of size 400, what is the standard error of $$\hat{p}$$?

Accepted Answer

$$ \sqrt{\frac{0.4 	imes 0.6}{400}} $$

Question 15

A researcher wants to estimate the proportion of voters who support a new policy. She wants a 95% confidence interval with a margin of error of 3%. She has no prior estimate for $$p. $$What is the minimum sample size she should use?

Accepted Answer

$$ n = \left( \frac{1.96}{0.03} \$$right)^2$$ 0.25 $$

Question 16

Which of the following statements is TRUE regarding the interpretation of a confidence interval?

Accepted Answer

A confidence interval gives a range of plausible values for the population parameter based on the sample data.

9.1 Estimating a Population Proportion: Confidence Intervals, Margin of Error, and Sample Size

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