Q: A poll of 900 registered voters found that 540 support a new policy. Construct a 95% confidence interval for the population proportion $$p of $$voters who support the policy. Use $$z_{\alpha/2} = 1.96.$$

$$ 0.6 \pm 1.96 \sqrt{\frac{0.6 \times 0.4}{900}} $$

Q: A researcher wants to estimate the proportion of adults who exercise regularly. She wants a margin of error of 0.04 with 95% confidence but has no preliminary estimate for $$p. $$What is the minimum sample size she should use? ($$z_{\alpha/2} = 1.96)$$

$$ n = \left( \frac{1.96}{0.04} \$$right)^2$$ \times 0.25 $$

Q: In a survey of 800 people, 480 said they use public transportation. Construct a 99% confidence interval for the population proportion $$p of $$people who use public transportation. Use $$z_{\alpha/2} = 2.576.$$

$$ 0.6 \pm 2.576 \sqrt{\frac{0.6 \times 0.4}{800}} $$

Q: A news organization wants to estimate the proportion of viewers who trust their reporting. They want a 99% confidence interval with a margin of error of 0.03. A preliminary estimate suggests $$\hat{p} = 0.7. $$What is the minimum sample size required? ($$z_{\alpha/2} = 2.576)$$

$$ n = \left( \frac{2.576}{0.03} \$$right)^2$$ \times 0.7 \times 0.3 $$

Q: A sample of 1500 adults found that 1200 support renewable energy initiatives. What is the 90% confidence interval for the population proportion? ($$z_{\alpha/2} = 1.645)$$

$$ 0.8 \pm 1.645 \sqrt{\frac{0.8 \times 0.2}{1500}} $$

Q: A company wants to estimate the proportion of defective products in a large shipment. They want a 95% confidence interval with a margin of error of 0.02. No preliminary estimate is available. What is the minimum sample size required? ($$z_{\alpha/2} = 1.96)$$

$$ n = \left( \frac{1.96}{0.02} \$$right)^2$$ \times 0.25 $$

Question 1

Suppose a survey of 1200 college students found that 780 students prefer online classes. What is the point estimate for the population proportion of college students who prefer online classes?

Accepted Answer

$$ \frac{780}{1200} = 0.65 $$

Question 2

A poll of 900 registered voters found that 540 support a new policy. Construct a 95% confidence interval for the population proportion $$p of $$voters who support the policy. Use $$z_{\alpha/2} = 1.96.$$

Accepted Answer

$$ 0.6 \pm 1.96 \sqrt{\frac{0.6 	imes 0.4}{900}} $$

Question 3

Which of the following best describes the meaning of a 95% confidence interval for a population proportion?

Accepted Answer

If we repeat the sampling process many times, about 95% of the confidence intervals constructed will contain the true population proportion.

Question 4

A researcher wants to estimate the proportion of adults who exercise regularly. She wants a margin of error of 0.04 with 95% confidence but has no preliminary estimate for $$p. $$What is the minimum sample size she should use? ($$z_{\alpha/2} = 1.96)$$

Accepted Answer

$$ n = \left( \frac{1.96}{0.04} \$$right)^2$$ 	imes 0.25 $$

Question 5

In a survey of 800 people, 480 said they use public transportation. Construct a 99% confidence interval for the population proportion $$p of $$people who use public transportation. Use $$z_{\alpha/2} = 2.576.$$

Accepted Answer

$$ 0.6 \pm 2.576 \sqrt{\frac{0.6 	imes 0.4}{800}} $$

Question 6

Why do we use $$\hat{p} = 0.5$$ when no preliminary estimate for the population proportion is available in minimum sample size calculations?

Accepted Answer

Because $$\hat{p} = 0.5$$ maximizes the product $$\hat{p}\hat{q}$$, resulting in the largest required sample size.

Question 7

A news organization wants to estimate the proportion of viewers who trust their reporting. They want a 99% confidence interval with a margin of error of 0.03. A preliminary estimate suggests $$\hat{p} = 0.7. $$What is the minimum sample size required? ($$z_{\alpha/2} = 2.576)$$

Accepted Answer

$$ n = \left( \frac{2.576}{0.03} \$$right)^2$$ 	imes 0.7 	imes 0.3 $$

Question 8

Which of the following conditions must be met to use the normal approximation for constructing a confidence interval for a population proportion?

Accepted Answer

$$ n\hat{p} \geq 5 $$ and $$ n\hat{q} \geq 5 $$

Question 9

A sample of 1500 adults found that 1200 support renewable energy initiatives. What is the 90% confidence interval for the population proportion? ($$z_{\alpha/2} = 1.645)$$

Accepted Answer

$$ 0.8 \pm 1.645 \sqrt{\frac{0.8 	imes 0.2}{1500}} $$

Question 10

If a 95% confidence interval for a population proportion is (0.62, 0.70), which of the following statements is correct?

Accepted Answer

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

Question 11

A company wants to estimate the proportion of defective products in a large shipment. They want a 95% confidence interval with a margin of error of 0.02. No preliminary estimate is available. What is the minimum sample size required? ($$z_{\alpha/2} = 1.96)$$

Accepted Answer

$$ n = \left( \frac{1.96}{0.02} \$$right)^2$$ 	imes 0.25 $$

Question 12

Which of the following is the correct formula for the margin of error $$E$$ when constructing a confidence interval for a population proportion?

Accepted Answer

$$ E = z_{\alpha/2} \sqrt{\frac{\hat{p}\hat{q}}{n}} $$

Question 13

A survey of 500 people found that 350 prefer brand A over brand B. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$ \frac{350}{500} = 0.7 $$

Question 14

If the sample size increases while keeping the confidence level and sample proportion the same, what happens to the margin of error?

Accepted Answer

The margin of error decreases.

Question 15

A 95% confidence interval for a population proportion is wider than a 90% confidence interval for the same data. Why?

Accepted Answer

A higher confidence level requires a larger critical value, increasing the margin of error.

Confidence Intervals for Population Proportions (Chapter 6.3) – Study Notes

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