Q: Given a population proportion $$p = 0.116$$ and a sample size $$n = 100$$, what is the standard deviation of the sampling distribution of the sample proportion?

$$\sqrt{\dfrac{0.116 \times 0.884}{100}}$$

Q: A sample of size $$n = 50 is $$taken from a population with $$p = 0.75. $$What is the standard deviation of the sample proportion?

$$\sqrt{\dfrac{0.75 \times 0.25}{50}}$$

Question 1

Which of the following best describes the Central Limit Theorem (CLT) for sample means?

Accepted Answer

The sampling distribution of the sample mean is approximately normal for large enough sample sizes, regardless of the population's distribution.

Question 2

A manufacturer knows that 0.2% of its AA batteries are defective. If a university orders 25,000 batteries, what is the expected number of defective batteries?

Accepted Answer

$$25,000 	imes 0.002 = 50$$

Question 3

Which condition must be checked to ensure the independence of observations in a sample?

Accepted Answer

Randomization Condition

Question 4

Given a population proportion $$p = 0.116$$ and a sample size $$n = 100$$, what is the standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

$$\sqrt{\dfrac{0.116 	imes 0.884}{100}}$$

Question 5

A poll finds that 13.4% of a sample of 925 U.S. births were premature. The true population proportion is 11.6%. What is the z-score for this sample proportion if the standard deviation is 0.011?

Accepted Answer

$$z = \dfrac{0.134 - 0.116}{0.011} = 1.71$$

Question 6

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

Accepted Answer

Equal Variance Condition

Question 7

If a population has mean $$\mu = 176$$ and standard deviation $$\sigma = 61$$, what is the standard deviation of the sampling distribution of the mean for samples of size $$n = 100$$?

Accepted Answer

$$\dfrac{61}{\sqrt{100}} = 6.1$$

Question 8

A survey of 300 voters finds that 52% plan to vote 'Yes' on a budget. Which condition must be checked to use the Normal model for the sample proportion?

Accepted Answer

Both $$np \geq 10$$ and $$nq \geq 10$$

Question 9

Why do we use sampling distributions in statistics?

Accepted Answer

To understand the variability of sample statistics and make inferences about the population.

Question 10

A balcony is designed to hold 100 adults. The mean adult weight is 176 lbs with a standard deviation of 61 lbs. What is the probability that the average weight of 100 adults exceeds 190 lbs?

Accepted Answer

About 1.1%

Question 11

Which of the following best describes the Law of Large Numbers?

Accepted Answer

As the sample size increases, the sample mean gets closer to the population mean.

Question 12

If the population distribution is highly skewed, what should you do before using the Normal model for the sampling distribution of the mean?

Accepted Answer

Increase the sample size to at least 100.

Question 13

Which formula gives the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\dfrac{\sigma}{\sqrt{n}}$$

Question 14

A sample of size $$n = 50 is $$taken from a population with $$p = 0.75. $$What is the standard deviation of the sample proportion?

Accepted Answer

$$\sqrt{\dfrac{0.75 	imes 0.25}{50}}$$

Question 15

Which of the following is a correct interpretation of a z-score of 2.295 in the context of the balcony problem?

Accepted Answer

The observed average weight is 2.295 standard deviations above the mean.

Sampling Distributions and the Central Limit Theorem: Study Notes for Introductory Statistics

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