Q: A researcher wants a 95% confidence interval for a population proportion with a margin of error of 1%. If the estimated proportion is $$0.5$$, what sample size is required? (Use $$z^* = 1.96$$ for 95% confidence.)

$$n = \left( \frac{1.96^2$ \times 0.5 \times 0.5}{0.01^2$} \right) = 9,604$$

Question 1

Suppose you want to estimate the proportion of STLCC students who are 'traditional aged' (18-24 years old). Out of 15,649 students, 7,804 are in this age group. What is the point estimate for the population proportion $$p$$?

Accepted Answer

$$\frac{7,804}{15,649} \approx 0.499$$

Question 2

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

Accepted Answer

We are 95% confident that the interval contains the true value of the population proportion $$p.$$

Question 3

Given a confidence interval for $$p of$$ $$(0.491, 0.507)$$, what is the margin of error $$E$$?

Accepted Answer

$$E = \frac{0.507 - 0.491}{2} = 0.008$$

Question 4

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

Accepted Answer

The population mean must be known.

Question 5

If you want to decrease the width of your confidence interval for a population proportion, which of the following should you do?

Accepted Answer

Increase the sample size.

Question 6

A researcher wants a 95% confidence interval for a population proportion with a margin of error of 1%. If the estimated proportion is $$0.5$$, what sample size is required? (Use $$z^* = 1.96$$ for 95% confidence.)

Accepted Answer

$$n = \left( \frac{1.96^2$ 	imes 0.5 	imes 0.5}{0.01^2$} ight) = 9,604$$

Question 7

Which of the following statements about the margin of error $$E is $$correct?

Accepted Answer

It is the maximum likely amount by which the sample statistic misses the population parameter.

Question 8

If a sample of 25 students is used to estimate the proportion of 'traditional aged' students at STLCC, which of the following is a likely issue with the resulting confidence interval?

Accepted Answer

The sample size may be too small to satisfy the requirement of at least 5 successes and 5 failures.

Question 9

Which formula correctly expresses a confidence interval for a population proportion $$p$$?

Accepted Answer

$$\hat{p} - E \u003c p \u003c \hat{p} + E$$

Question 10

A 95% confidence interval for the proportion of traditional aged students at STLCC is $$(0.491, 0.507). $$What does this mean in the context of the population?

Accepted Answer

We are 95% confident that the true proportion of traditional aged students at STLCC is between 0.491 and 0.507.

Question 11

If the confidence level is increased from 95% to 99%, what happens to the width of the confidence interval for a population proportion, all else being equal?

Accepted Answer

The width increases.

Question 12

Which of the following is the best point estimate for the population proportion $$p$$?

Accepted Answer

The sample proportion $$\hat{p}.$$

Question 13

A sample of 1,000 people is taken to estimate the proportion of voters who support a new policy. 540 support the policy. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{540}{1000} = 0.54$$

Question 14

Which of the following is a correct way to express a confidence interval for a population proportion?

Accepted Answer

$$(\hat{p} - E, \hat{p} + E)$$

Question 15

Which of the following is a common misconception about confidence intervals?

Accepted Answer

There is a 95% chance that the true parameter is in the interval.

Question 16

If you want to estimate a population proportion with a smaller margin of error, but keep the confidence level the same, what must you do?

Accepted Answer

Increase the sample size.

Question 17

Which of the following scenarios would violate the requirements for constructing a confidence interval for a population proportion?

Accepted Answer

A sample with only 3 successes and 22 failures.

Question 18

A confidence interval for a population proportion is $$(0.40, 0.48). $$What is the point estimate $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{0.40 + 0.48}{2} = 0.44$$

Question 19

In a real-world scenario, why might the proportion of 'traditional aged' students at STLCC change from year to year?

Accepted Answer

Because the population of students changes each year.

Estimating Population Proportions and Determining Sample Sizes (Chapter 7.1 Study Notes)

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