Question 1

Which of the following correctly represents the formula for the number of permutations of $$n$$ objects taken $$r at a $$time?

Accepted Answer

$$P(n, r) = \frac{n!}{(n - r)!}$$

Question 2

A committee of 4 is to be chosen from 10 people. In how many ways can this be done if the order does not matter?

Accepted Answer

$$\frac{10!}{4!6!}$$

Question 3

Given a discrete random variable $$X$$ with possible values $$x_i$$ and probabilities $$p(x_i)$$, which formula gives the mean (expected value) of $$X$$?

Accepted Answer

$$\mu = \sum x_i p(x_i)$$

Question 4

Which of the following is the correct formula for the standard deviation of a discrete random variable $$X$$?

Accepted Answer

$$\sigma = \sqrt{\sum (x_i - \mu)^2$ p(x_i)}$$

Question 5

A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads? (Use the binomial formula.)

Accepted Answer

$$\binom{5}{3} (0.5)^3$ (0.5)^2$

Question 6

If $$X is a $$binomial random variable with parameters $$n$$ and $$p$$, what is the formula for its mean $$\mu$$?

Accepted Answer

$$\mu = np$$

Question 7

For a binomial random variable $$X$$ with parameters $$n$$ and $$p$$, what is the formula for its standard deviation $$\sigma$$?

Accepted Answer

$$\sigma = \sqrt{np(1-p)}$$

Question 8

A normal random variable $$X$$ has mean $$\mu$$ and standard deviation $$\sigma. $$What is the $$z-$$score for a value $$x$$?

Accepted Answer

$$z = \frac{x - \mu}{\sigma}$$

Question 9

Suppose the heights of adult men are normally distributed with mean 70 inches and standard deviation 3 inches. What is the $$z-$$score for a man who is 76 inches tall?

Accepted Answer

$$z = 2$$

Question 10

If a sample of size $$n is $$drawn from a population with mean $$\mu$$ and standard deviation $$\sigma$$, what is the standard deviation of the sample mean $$\bar{x}$$?

Accepted Answer

$$\frac{\sigma}{\sqrt{n}}$$

Question 11

A population has mean $$\mu = 50$$ and standard deviation $$\sigma = 10. $$What is the standard deviation of the sample mean for samples of size $$n = 25$$?

Accepted Answer

$$2$$

Question 12

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

Accepted Answer

Its standard deviation is $$\frac{\sigma}{\sqrt{n}}.$$

Question 13

A medical researcher wants to know the probability that, in a random sample of 100 people, exactly 60 have blood type O, given that the probability of blood type O in the population is 0.6. Which distribution and formula should be used?

Accepted Answer

Binomial distribution: $$P(X = 60) = \binom{100}{60} (0.6$$)^{60}$$ (0.4$$)^{40}$$

Question 14

If $$X is a $$binomial random variable with $$n = 20$$ and $$p = 0.3$$, what is the expected value (mean) of $$X$$?

Accepted Answer

$$6$$

Question 15

A quality control engineer samples 36 light bulbs from a factory where the mean lifetime is 1200 hours and the standard deviation is 120 hours. What is the standard deviation of the sample mean?

Accepted Answer

$$20$$

Question 16

Which of the following best describes the Central Limit Theorem as it relates to the sampling distribution of the sample mean?

Accepted Answer

As the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the population's distribution.

Question 17

A student calculates the $$z-$$score for a test score of 85, given a class mean of 75 and a standard deviation of 5. What is the $$z-$$score?

Accepted Answer

$$2$$

Question 18

Which of the following is NOT a property of the binomial distribution?

Accepted Answer

The probability of success can change from trial to trial.

Question 19

A researcher wants to know the probability that, in a sample of 50 people, at least 40 prefer a new product, given that the probability of preference is 0.8. Which formula should be used?

Accepted Answer

$$P(X \geq 40) = \sum_{x=40}^{50} \binom{50}{x} (0.8)^x (0.2$$)^{50-x}$$

Key Concepts in Probability, Discrete Distributions, and Sampling

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