Q: In a medical study, the population proportion of patients responding to a new drug is $$p = 0.7. If a $$random sample of $$n = 200$$ patients is taken, what is the standard deviation of the sampling distribution of the sample proportion?

$$\sqrt{\frac{0.7 \times 0.3}{200}}$$

Question 1

In the context of the 2010 California gubernatorial election, polling organizations used exit polls to predict the winner. What statistical concept allows us to estimate how close the sample proportion from the exit poll is to the true population proportion?

Accepted Answer

Sampling Distribution

Question 2

Suppose you take a random sample of size $$n$$ from a population with proportion $$p of $$outcomes in a particular category. What is the mean of the sampling distribution of the sample proportion?

Accepted Answer

$$p$$

Question 3

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

Accepted Answer

Population distribution gives probabilities for individual outcomes; sampling distribution gives probabilities for sample statistics.

Question 4

If the sample size $$n is $$sufficiently large such that $$np \geq 10$$ and $$n(1-p) \geq 10$$, what can be said about the shape of the sampling distribution of the sample proportion?

Accepted Answer

It is approximately normal.

Question 5

A polling organization samples 3889 voters and finds that 46.9% voted for Brown. The official population proportion was 53.8%. What does this difference illustrate?

Accepted Answer

Sampling error due to random variation.

Question 6

Which formula gives the standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$ for a random sample of size $$n$$ from a population with proportion $$p$$?

Accepted Answer

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

Question 7

Why is it important for the sample size to be large when using the normal approximation for the sampling distribution of the sample proportion?

Accepted Answer

Because the normal approximation is only valid when both $$np$$ and $$n(1-p)$$ are sufficiently large.

Question 8

A researcher wants to estimate the proportion of students who prefer online classes. She samples 100 students and finds 60 prefer online classes. If the true population proportion is 0.55, what is the mean of the sampling distribution of the sample proportion?

Accepted Answer

$$0.55$$

Question 9

Which of the following statements about sampling distributions is FALSE?

Accepted Answer

Sampling distributions are only relevant for proportions.

Question 10

In a medical study, the population proportion of patients responding to a new drug is $$p = 0.7. If a $$random sample of $$n = 200$$ patients is taken, what is the standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

$$\sqrt{\frac{0.7 	imes 0.3}{200}}$$

Question 11

Which scenario best illustrates the concept of a sampling distribution?

Accepted Answer

Plotting the means from many repeated samples of the same size.

Question 12

If the population proportion $$p is 0.4 $$and the sample size $$n is 100$$, what is the expected value (mean) of the sample proportion?

Accepted Answer

$$0.4$$

Question 13

A sample statistic is considered a random variable because:

Accepted Answer

Its value varies from sample to sample.

Question 14

Which of the following is NOT a type of sampling distribution?

Accepted Answer

Sampling distribution of the population parameter

Question 15

In the context of exit polls, why might the sample proportion differ from the population proportion?

Accepted Answer

Because of sampling variability.

Question 16

A university wants to estimate the proportion of students who commute. If the sample size is increased, what happens to the standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

It decreases.

Question 17

Which condition must be met for the sampling distribution of the sample proportion to be approximately normal?

Accepted Answer

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

Question 18

If a sample proportion is used to estimate a population proportion, what does the sampling distribution tell us?

Accepted Answer

The probabilities of all possible values the sample proportion can take.

Question 19

A company surveys 500 customers and finds that 80% are satisfied. If the true population proportion is 0.78, what is the standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

$$\sqrt{\frac{0.78 	imes 0.22}{500}}$$

Question 20

Which of the following is a real-world example of using a sampling distribution?

Accepted Answer

Estimating the proportion of defective products in a factory by repeatedly sampling boxes.

(Lecture 16) Sampling Distributions: Understanding Sample Proportions and Their Variability

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