Q: A survey of 2,000 people finds that 70% support a new law. What is the standard deviation of the sampling distribution of $$\hat{p}$$?

$$SD(\hat{p}) = \sqrt{\frac{0.7 \times 0.3}{2000}}$$

Question 1

Which formula should you use to determine the minimum sample size $$n$$ required to estimate a population proportion with a specified margin of error $$ME$$ and confidence level, given an estimated proportion $$\hat{p}$$?

Accepted Answer

$$n = \left( \frac{z^*}{ME} \$$right)^2$$ \hat{p}(1-\hat{p})$$

Question 2

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 less than 3%. She has no prior estimate for $$p. $$What value should she use for $$\hat{p} to $$ensure the largest (most conservative) sample size?

Accepted Answer

$$\hat{p} = 0.5$$

Question 3

If a pilot study finds that 0.5% of credit card offers are accepted, what is the minimum sample size needed to estimate the true acceptance rate within 0.1% margin of error at 95% confidence? Use $$z^* = 1.96.$$

Accepted Answer

$$n = \left( \frac{1.96}{0.001} \$$right)^2$$ (0.005)(0.995) \approx 19,112$$

Question 4

Which of the following statements about confidence intervals is TRUE?

Accepted Answer

A confidence interval is about the population parameter, not the sample statistic.

Question 5

What happens to the standard deviation of the sampling distribution of $$\hat{p} if $$the sample size $$n is $$quadrupled?

Accepted Answer

It is halved.

Question 6

A poll reports a margin of error of ±3% at a 95% confidence level. What is the minimum sample size used, assuming no prior estimate for $$p$$?

Accepted Answer

$$n = \left( \frac{1.96}{0.03} \$$right)^2$$ (0.5)(0.5) \approx 1068$$

Question 7

Which of the following is a potential problem when constructing confidence intervals?

Accepted Answer

Using a biased sample.

Question 8

If you want to reduce the margin of error from 3% to 1% in a survey, how does the required sample size change?

Accepted Answer

It increases by a factor of 9.

Question 9

The number of standard errors to move away from the mean of the sampling distribution to correspond to the specified level of confidence is called the ________.

Accepted Answer

critical value

Question 10

Which of the following is FALSE regarding confidence intervals?

Accepted Answer

The higher the level of confidence, the narrower the confidence interval.

Question 11

A company wants to estimate the proportion of customers satisfied with their service. They want a 99% confidence interval with a margin of error of 2%. If they have no prior estimate for $$p$$, what $$z^*$$ value should they use?

Accepted Answer

$$z^* = 2.576$$

Question 12

Which of the following best describes the effect of increasing the sample size on the margin of error, all else being equal?

Accepted Answer

The margin of error decreases.

Question 13

A survey of 2,000 people finds that 70% support a new law. What is the standard deviation of the sampling distribution of $$\hat{p}$$?

Accepted Answer

$$SD(\hat{p}) = \sqrt{\frac{0.7 	imes 0.3}{2000}}$$

Question 14

Why is it important to round up the calculated sample size when planning a survey?

Accepted Answer

To ensure the margin of error does not exceed the desired value.

Question 15

Which of the following is NOT a recommended practice when interpreting confidence intervals?

Accepted Answer

Assuming the Normal model always works regardless of sample size or conditions.

Sampling Distribution Models and Confidence Intervals for Proportions: Sample Size Determination

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