Q: Suppose the proportion of female students in a university is $$p = 0.25. If a $$random sample of $$n = 100$$ students is taken, what is the standard deviation of the sample proportion?

$$\sqrt{\frac{0.25 \times 0.75}{100}}$$

Q: A poll finds that 42% of adults support a new policy, based on a sample of 1000 adults. What is the margin of error for a 95% confidence interval?

$$1.96 \times \sqrt{\frac{0.42 \times 0.58}{1000}}$$

Q: What is the empirical rule for the margin of error at a 95% confidence level?

The margin of error is $$2 \times$$ the standard error.

Q: What is the formula for the margin of error (ME) for a population proportion?

$$z^* \times \sqrt{\frac{p(1-p)}{n}}$$

Q: If a poll of 500 adults finds that 60% support a new law, what is the 99% confidence interval for the population proportion?

$$0.60 \pm 2.58 \times \sqrt{\frac{0.60 \times 0.40}{500}}$$

Question 1

Which of the following conditions must be met to apply the Central Limit Theorem (CLT) for sample proportions?

Accepted Answer

The sample must be randomly selected and independent, and both $$np$$ and $$n(1-p)$$ must be at least 10.

Question 2

Suppose the proportion of female students in a university is $$p = 0.25. If a $$random sample of $$n = 100$$ students is taken, what is the standard deviation of the sample proportion?

Accepted Answer

$$\sqrt{\frac{0.25 	imes 0.75}{100}}$$

Question 3

A poll finds that 42% of adults support a new policy, based on a sample of 1000 adults. What is the margin of error for a 95% confidence interval?

Accepted Answer

$$1.96 	imes \sqrt{\frac{0.42 	imes 0.58}{1000}}$$

Question 4

If a confidence interval for a population proportion is calculated as (0.38, 0.46), what does this mean?

Accepted Answer

We are 95% confident that the true population proportion is between 0.38 and 0.46.

Question 5

Which formula is used to calculate the standard error (SE) for a sample proportion?

Accepted Answer

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

Question 6

In a sample of 876 letters, the proportion of 'E's is found to be 0.147. If the true proportion is $$p = 0.12$$, what is the mean of the sampling distribution for the sample proportion?

Accepted Answer

$$0.12$$

Question 7

What is the empirical rule for the margin of error at a 95% confidence level?

Accepted Answer

The margin of error is $$2 	imes$$ the standard error.

Question 8

A researcher wants to estimate the proportion of people who prefer tea over coffee. Which of the following best describes a confidence interval?

Accepted Answer

An interval that is likely to contain the true population proportion.

Question 9

If $$p = 0.18$$ and $$n = 1000$$, what is the probability that the sample proportion is greater than $$0.25$$ according to the CLT?

Accepted Answer

Approximately 18%

Question 10

Which of the following statements about confidence intervals is FALSE?

Accepted Answer

A 90% confidence interval is wider than a 95% confidence interval.

Question 11

What is the formula for the margin of error (ME) for a population proportion?

Accepted Answer

$$z^* 	imes \sqrt{\frac{p(1-p)}{n}}$$

Question 12

If a poll of 500 adults finds that 60% support a new law, what is the 99% confidence interval for the population proportion?

Accepted Answer

$$0.60 \pm 2.58 	imes \sqrt{\frac{0.60 	imes 0.40}{500}}$$

Question 13

Why is it important that $$np \geq 10$$ and $$n(1-p) \geq 10$$ when applying the CLT to sample proportions?

Accepted Answer

To ensure the sampling distribution is approximately normal.

Question 14

A confidence interval for a population proportion is calculated as $$\hat{p} \pm z^* 	imes SE. $$What does $$z^*$$ represent?

Accepted Answer

The critical value corresponding to the desired confidence level.

Question 15

If the sample size increases, what happens to the margin of error for a confidence interval?

Accepted Answer

The margin of error decreases.

Question 16

Which scenario best illustrates the use of the CLT for sample proportions?

Accepted Answer

Estimating the proportion of defective items in a factory by taking a random sample of 200 items.

Central Limit Theorem and Confidence Intervals for Population Proportions

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