Q: Suppose 24% of M&M’s are blue. If you take a random sample of 100 M&M’s, what is the standard deviation of the sampling distribution of the sample proportion of blue M&M’s?

$$\sqrt{0.24 \times 0.76 / 100} = 0.0427$$

Q: A mechanic tests 128 new cars and finds that 14% of new cars have electrical problems. What is the standard deviation of the sampling distribution of the sample proportion?

$$\sqrt{0.14 \times 0.86 / 128} = 0.031$$

Question 1

Which of the following is NOT an assumption or condition required for using the Normal model for the sampling distribution of sample proportions?

Accepted Answer

The population must be normally distributed

Question 2

Suppose 24% of M&M’s are blue. If you take a random sample of 100 M&M’s, what is the standard deviation of the sampling distribution of the sample proportion of blue M&M’s?

Accepted Answer

$$\sqrt{0.24 	imes 0.76 / 100} = 0.0427$$

Question 3

A college reports that 22% of its students have a BMI over 30. In a random sample of 200 students, 31 have a BMI over 30. What is the sample proportion?

Accepted Answer

$$\frac{31}{200} = 0.155$$

Question 4

Which of the following best explains why larger samples have less variability in their sampling distributions?

Accepted Answer

Larger samples better reflect the population, reducing the standard error.

Question 5

If $$p is $$very close to 0, what is the shape of the sampling distribution of the sample proportion?

Accepted Answer

Skewed right

Question 6

A mechanic tests 128 new cars and finds that 14% of new cars have electrical problems. What is the standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

$$\sqrt{0.14 	imes 0.86 / 128} = 0.031$$

Question 7

In a class of 90 students, 13% are left-handed. What is the expected number of left-handed students?

Accepted Answer

$$90 	imes 0.13 = 11.7$$

Question 8

Which of the following is a correct interpretation of the 68-95-99.7 rule for the sampling distribution of sample proportions?

Accepted Answer

About 95% of sample proportions will fall within 2 standard deviations of the mean.

Question 9

Which scenario would violate the independence assumption for using the Normal model for sample proportions?

Accepted Answer

Sampling students from the same family for a survey on study habits.

Question 10

If you increase the sample size from 100 to 400, what happens to the standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

It decreases by a factor of 2.

Question 11

Which of the following is TRUE about the mean of the sampling distribution of the sample proportion?

Accepted Answer

It is equal to the population proportion $$p.$$

Question 12

A sample proportion is calculated as $$\hat{p} = 0.18$$ from a sample of $$n = 150. If $$the population proportion is $$p = 0.22$$, what is the $$z-$$score for this sample proportion? (Use $$\sigma_{\hat{p}} = \sqrt{0.22 	imes 0.78 / 150} = 0.034)$$

Accepted Answer

$$z = \frac{0.18 - 0.22}{0.034} = -1.18$$

Question 13

Which of the following is a potential consequence of confusing the sampling distribution with the distribution of the sample?

Accepted Answer

Misinterpreting the variability expected in repeated samples.

Question 14

Why might the Normal model not be appropriate for the sampling distribution of the sample median?

Accepted Answer

The Central Limit Theorem does not guarantee normality for medians.

Question 15

A survey finds that 15 out of 90 students are left-handed. If the population proportion is 0.13, what is the probability that there are more left-handed students than seats (15 seats) in a random sample of 90? (Use $$\sigma_{\hat{p}} = 0.035$$ and $$p = 0.13)$$

Accepted Answer

About 14.5%

Sampling Distribution Models and Confidence Intervals for Proportions

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