Q: If the population proportion $$p = 0.13$$ and the sample size $$n = 120$$, what is the standard deviation of the sampling distribution of $$\hat{p}$$?

$$\sigma_{\hat{p}} = \sqrt{\dfrac{0.13 \times 0.87}{120}}$$

Question 1

Which of the following best defines the sample proportion $$\hat{p} in a $$statistical study?

Accepted Answer

The proportion of a sample with a certain characteristic.

Question 2

Suppose a survey of 1200 adults finds that 352 plan to vote in an upcoming election. What is the sample proportion $$\hat{p} of $$adults planning to vote?

Accepted Answer

$$\hat{p} = \dfrac{352}{1200}$$

Question 3

As the sample size $$n$$ increases, what happens to the standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

It decreases.

Question 4

What is the mean of the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\mu_{\hat{p}} = p$$

Question 5

Which formula gives the standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\sigma_{\hat{p}} = \sqrt{\dfrac{p(1-p)}{n}}$$

Question 6

Under which condition is the sampling distribution of $$\hat{p}$$ approximately normal?

Accepted Answer

When both $$n p \geq 10$$ and $$n(1-p) \geq 10$$

Question 7

A random sample of 120 Americans is taken. If 13% of Americans have hearing trouble, what is the expected number of people in the sample with hearing trouble?

Accepted Answer

$$120 	imes 0.13 = 15.6$$

Question 8

If the sample size $$n is $$more than 5% of the population size $$N$$, what is a potential issue when using the formulas for the sampling distribution of $$\hat{p}$$?

Accepted Answer

The formulas may underestimate the standard deviation.

Question 9

Suppose in a survey, 24 out of 120 Americans who regularly listen to music using headphones have hearing trouble. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \dfrac{24}{120} = 0.2$$

Question 10

If the population proportion $$p = 0.13$$ and the sample size $$n = 120$$, what is the standard deviation of the sampling distribution of $$\hat{p}$$?

Accepted Answer

$$\sigma_{\hat{p}} = \sqrt{\dfrac{0.13 	imes 0.87}{120}}$$

Question 11

A Gallup poll finds that 60% of Americans believe the state of sexual values is declining. If a random sample of 100 Americans is taken, what is the mean of the sampling distribution of $$\hat{p}$$?

Accepted Answer

$$\mu_{\hat{p}} = 0.60$$

Question 12

Which of the following best describes the effect of increasing sample size $$n on $$the shape of the sampling distribution of $$\hat{p}$$?

Accepted Answer

It becomes more approximately normal.

Question 13

If $$p = 0.13$$, $$n = 120$$, and you want to find the probability that at most 12% of the sample has hearing trouble, which distribution should you use?

Accepted Answer

The normal approximation to the binomial distribution.

Question 14

Which of the following is NOT a requirement for using the normal approximation for the sampling distribution of $$\hat{p}$$?

Accepted Answer

Population must be at least 20 times the sample size.

Question 15

A researcher wants to estimate the proportion of students who own a laptop. She samples 200 students and finds 150 own a laptop. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \dfrac{150}{200} = 0.75$$

Question 16

If the population proportion $$p is $$unknown, which statistic is used to estimate it from a sample?

Accepted Answer

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

Question 17

Which of the following real-world scenarios best illustrates the concept of a sampling distribution of a sample proportion?

Accepted Answer

Repeatedly surveying different groups of 100 voters to estimate the proportion supporting a candidate.

Question 18

If $$p = 0.6$$, $$n = 100$$, and you want to compute the probability that at least 65% of the sample supports a policy, which steps should you follow?

Accepted Answer

Find the mean and standard deviation of $$\hat{p}$$, standardize 0.65, and use the normal distribution.

Question 19

Why is the sample proportion $$\hat{p}$$ considered a random variable?

Accepted Answer

Because its value varies from sample to sample.

Section 8.2: Distribution of the Sample Proportion

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