Q: Suppose a population of college students has a proportion of female students $$p = 0.60. If a $$random sample of $$n = 100$$ students is taken, what is the standard deviation of the sample proportion $$\hat{p}$$?

$$\sqrt{\dfrac{0.60 \times 0.40}{100}}$$

Q: A manufacturer claims that the proportion of defective pieces is $$p = 0.10. In a $$random sample of $$n = 75$$, what is the probability that the sample contains a proportion of $$\hat{p} \leq 0.05$$ defective pieces? Assume normal approximation is appropriate.

Calculate $$z = \dfrac{0.05 - 0.10}{\sqrt{\dfrac{0.10 \times 0.90}{75}}}$$ and find $$P(z)$$ from the standard normal table.

Q: A sample of $$n = 100$$ students shows a sample proportion $$\hat{p} = 0.58$$ for females. If the population proportion is $$p = 0.60$$, what is the $$z-$$score for this sample proportion?

$$z = \dfrac{0.58 - 0.60}{\sqrt{\dfrac{0.60 \times 0.40}{100}}}$$

Q: Two independent samples are taken: one with $$n_1 = 1000$ and proportion $$\hat{p}_1 = 0.55$$, and another with n_2$ = 900$$ and proportion $$\hat{p}_2 = 0.60. $$What is the standard error for the difference in sample proportions $$\hat{p}_1 - \hat{p}_2$$?

$$\sqrt{\dfrac{0.55 \times 0.45}{1000} + \dfrac{0.60 \times 0.40}{900}}$$

Q: A sample of $$n = 100$$ students shows a sample proportion $$\hat{p} = 0.52$$ for a certain trait. What is the 95% confidence interval for the proportion? (Use $$z^* = 1.96)$$

$$0.52 \pm 1.96 \sqrt{\dfrac{0.52 \times 0.48}{100}}$$

Question 1

Suppose a population of college students has a proportion of female students $$p = 0.60. If a $$random sample of $$n = 100$$ students is taken, what is the standard deviation of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\sqrt{\dfrac{0.60 	imes 0.40}{100}}$$

Question 2

A manufacturer claims that the proportion of defective pieces is $$p = 0.10. In a $$random sample of $$n = 75$$, what is the probability that the sample contains a proportion of $$\hat{p} \leq 0.05$$ defective pieces? Assume normal approximation is appropriate.

Accepted Answer

Calculate $$z = \dfrac{0.05 - 0.10}{\sqrt{\dfrac{0.10 	imes 0.90}{75}}}$$ and find $$P(z)$$ from the standard normal table.

Question 3

Which formula is used to construct a confidence interval for a sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} \pm z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$$

Question 4

If the sample size $$n$$ increases while keeping the population proportion $$p$$ constant, what happens to the standard deviation of the sampling distribution of $$\hat{p}$$?

Accepted Answer

It decreases.

Question 5

A simple random sample of 900 residents is picked from a country, and 627 of them are right-handed. What is the sample proportion $$\hat{p} of $$right-handed residents?

Accepted Answer

$$\dfrac{627}{900}$$

Question 6

Which of the following best describes the effect of increasing the confidence level on the width of a confidence interval for a proportion?

Accepted Answer

The interval becomes wider.

Question 7

In a survey, 95% confidence interval for a proportion $$p is $$calculated as $$(0.52, 0.62). $$Which of the following is the correct interpretation?

Accepted Answer

We are 95% confident that the true proportion $$p is $$between 0.52 and 0.62.

Question 8

A sample of $$n = 100$$ students shows a sample proportion $$\hat{p} = 0.58$$ for females. If the population proportion is $$p = 0.60$$, what is the $$z-$$score for this sample proportion?

Accepted Answer

$$z = \dfrac{0.58 - 0.60}{\sqrt{\dfrac{0.60 	imes 0.40}{100}}}$$

Question 9

Which condition must be satisfied to use the normal model for the sampling distribution of sample proportions?

Accepted Answer

$$np \geq 10$$ and $$n(1-p) \geq 10$$

Question 10

Two independent samples are taken: one with $$n_1 = 1000$ and proportion $$\hat{p}_1 = 0.55$$, and another with n_2$ = 900$$ and proportion $$\hat{p}_2 = 0.60. $$What is the standard error for the difference in sample proportions $$\hat{p}_1 - \hat{p}_2$$?

Accepted Answer

$$\sqrt{\dfrac{0.55 	imes 0.45}{1000} + \dfrac{0.60 	imes 0.40}{900}}$$

Question 11

A sample of $$n = 100$$ students shows a sample proportion $$\hat{p} = 0.52$$ for a certain trait. What is the 95% confidence interval for the proportion? (Use $$z^* = 1.96)$$

Accepted Answer

$$0.52 \pm 1.96 \sqrt{\dfrac{0.52 	imes 0.48}{100}}$$

Question 12

If the sample proportion $$\hat{p} is $$close to 0 or 1, which of the following is true about the normal approximation for the sampling distribution?

Accepted Answer

The normal approximation may not be appropriate.

Question 13

A researcher wants to estimate the proportion of residents who are right-handed. Which of the following would increase the precision of the confidence interval?

Accepted Answer

Increasing the sample size.

Question 14

Which of the following statements is true regarding the difference between bulk proportions in two populations?

Accepted Answer

The confidence interval for the difference uses the standard error of $$\hat{p}_1 - \hat{p}_2.$$

Question 15

A sample of $$n = 100$$ students shows a sample proportion $$\hat{p} = 0.52. If $$the population proportion is $$p = 0.50$$, what is the probability that the sample proportion is at least $$0.52$$? (Assume normal approximation is appropriate.)

Accepted Answer

Calculate $$z = \dfrac{0.52 - 0.50}{\sqrt{\dfrac{0.50 	imes 0.50}{100}}}$$ and find $$P(z)$$ from the standard normal table.

Question 16

Which of the following is a correct statement about the meaning of a 99% confidence interval for a population proportion?

Accepted Answer

If we repeated the sampling process many times, about 99% of the intervals would contain the true proportion.

Sampling Distributions and Confidence Intervals for Proportions

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