Q: If $$p = 0.3$$, $$n = 100$$, what is the standard deviation of the sampling distribution of $$\hat{p}$$?

$$\sqrt{\dfrac{0.3 \times 0.7}{100}} = 0.0458$$

Question 1

Which of the following best describes the concept of a sampling distribution?

Accepted Answer

A distribution of summary statistics (such as sample proportions) from all possible samples of the same size from the same population.

Question 2

Suppose you want to estimate the proportion of green M&M's in a large bag. You take a random sample of 100 M&M's and find 18 are green. What is your best guess for the true proportion of green M&M's in the population?

Accepted Answer

$$0.18$$

Question 3

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

Accepted Answer

$$\sqrt{\dfrac{pq}{n}}$$

Question 4

If the true proportion of orange M&M's in a population is $$p = 0.20$$ and you take a random sample of $$n = 50$$, what is the expected mean of the sampling distribution of $$\hat{p}$$?

Accepted Answer

$$0.20$$

Question 5

Which of the following statements about sampling distributions is TRUE?

Accepted Answer

The sampling distribution of $$\hat{p} is $$approximately normal if the sample size is large and $$p is $$not too close to 0 or 1.

Question 6

A researcher collects 1,000 random samples of size $$n = 30$$ from a population with $$p = 0.4. $$What will the histogram of the sample proportions $$\hat{p}$$ most likely look like?

Accepted Answer

Symmetric and unimodal, centered around $$0.4.$$

Question 7

Why do we use the sampling distribution when making statistical inferences about a population proportion?

Accepted Answer

To quantify how much sample statistics are expected to vary from sample to sample.

Question 8

If $$p = 0.3$$, $$n = 100$$, what is the standard deviation of the sampling distribution of $$\hat{p}$$?

Accepted Answer

$$\sqrt{\dfrac{0.3 	imes 0.7}{100}} = 0.0458$$

Question 9

Which of the following is NOT a property of the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

It is always identical to the population distribution.

Question 10

A student claims that the sample proportion $$\hat{p}$$ from a single sample is always equal to the population proportion $$p. $$Which concept best explains why this is incorrect?

Accepted Answer

Sampling variability (sampling error)

Question 11

Suppose the true proportion of red M&M's is $$p = 0.13. If $$you take a random sample of $$n = 200$$, what is the probability that your sample proportion $$\hat{p}$$ falls within one standard deviation of the mean, according to the 68–95–99.7 rule?

Accepted Answer

Approximately 68%

Question 12

Which of the following best describes the difference between a sample distribution and a sampling distribution?

Accepted Answer

A sample distribution is the distribution of data from one sample, while a sampling distribution is the distribution of a statistic from many samples.

Question 13

If you increase the sample size $$n$$ when estimating a population proportion, what happens to the standard deviation of the sampling distribution of $$\hat{p}$$?

Accepted Answer

It decreases.

Question 14

A company wants to estimate the proportion of defective products in a large shipment. They take several random samples of size $$n = 50$$ and calculate the sample proportion of defectives in each. What is the purpose of examining the sampling distribution of these sample proportions?

Accepted Answer

To understand how much the sample proportions are expected to vary from sample to sample.

Question 15

Which of the following is a correct interpretation of the standard deviation of the sampling distribution of $$\hat{p}$$?

Accepted Answer

It measures the typical distance between a sample proportion and the true population proportion.

Sampling Distribution Models and Confidence Intervals for Proportions

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