Question 1

Which of the following best describes the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

It describes all possible values of $$\hat{p}$$ from repeated random samples of size $$n.$$

Question 2

Given a population proportion $$p = 0.15$$ and a sample size $$n = 120$$, what is the mean of the sampling distribution of $$\hat{p}$$?

Accepted Answer

$$\mu_{\hat{p}} = 0.15$$

Question 3

What is the formula for the standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

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

Question 4

In the context of the image, what does the shaded area represent?

Accepted Answer

The probability that the sample proportion $$\hat{p} is $$less than or equal to 0.12.

Question 5

If $$p = 0.15$$ and $$n = 120$$, what is the value of $$np$$ and $$n(1-p)$$, and why are these important?

Accepted Answer

$$np = 18$$, $$n(1-p) = 102$$; these values ensure the sampling distribution is approximately normal.

Question 6

A random sample of 120 Americans finds 26 with hearing trouble. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{26}{120} \approx 0.217$$

Question 7

What is the z-score formula for a sample proportion $$\hat{p}$$?

Accepted Answer

$$z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}}$$

Question 8

If the z-score for $$\hat{p} = 0.12 is$$ $$z = -0.92$$, what does this value indicate?

Accepted Answer

The sample proportion is 0.92 standard deviations below the mean.

Question 9

If $$P(z \leq -0.92) \approx 0.1788$$, how would you interpret this probability in context?

Accepted Answer

There is about a 17.88% chance that a random sample of 120 Americans will have at most 12% with hearing trouble.

Question 10

If a sample proportion $$\hat{p} = 0.217$$ yields a z-score of $$z = 2.06$$, what is the probability of obtaining a sample proportion of 0.217 or higher if $$p = 0.15$$?

Accepted Answer

$$P(\hat{p} \geq 0.217) = 1 - P(z \leq 2.06) \approx 0.0197$$

Question 11

Why might a sample proportion of $$\hat{p} = 0.217 be $$considered unusual if the true population proportion is $$p = 0.15$$?

Accepted Answer

Because the probability of obtaining such a sample proportion or higher is very low (about 2%).

Question 12

Which condition must be satisfied for the sampling distribution of $$\hat{p} to be $$approximately normal?

Accepted Answer

Both $$np \geq 15$$ and $$n(1-p) \geq 15$$

Question 13

If you want to estimate the proportion of students who live off-campus, which statistic would you use from your sample?

Accepted Answer

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

Question 14

Suppose you survey 100 students and find that 60 live off-campus. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{60}{100} = 0.6$$

Question 15

Which of the following is NOT a plausible conclusion if a sample proportion is much higher than the population proportion?

Accepted Answer

The sample size was too large.

Question 16

If the sampling distribution of $$\hat{p} is $$approximately normal, which real-world scenario would justify using the normal approximation?

Accepted Answer

Surveying 120 Americans about hearing trouble when $$p = 0.15$$

Sampling Distributions of Sample Proportions

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