Q: Which of the following statements about mutually exclusive events is correct?

If events A and B are mutually exclusive, then $$P(A \text{ and } B) = 0.$$

Q: A hospital committee of 5 is to be chosen from 8 administrators, 10 physicians, and 15 nurses. What is the probability that the committee consists of 1 administrator, 1 physician, and 3 nurses?

$$\dfrac{8 \times 10 \times \binom{15}{3}}{\binom{33}{5}}$$

Q: Suppose $$X is a $$discrete random variable with the following probability distribution: $$P(X=1)=0.2$$, $$P(X=2)=0.5$$, $$P(X=3)=0.3. $$What is the mean $$\mu of$$ $$X$$?

$$\mu = 1 \times 0.2 + 2 \times 0.5 + 3 \times 0.3 = 2.1$$

Q: If $$P(A) = 0.4$$, what is $$P(\text{not } A)$$?

$$P(\text{not } A) = 1 - 0.4 = 0.6$$

Q: A binomial random variable has $$n=10$$ and $$p=0.3. $$What is its standard deviation?

$$\sigma = \sqrt{10 \times 0.3 \times 0.7}$$

Question 1

Which of the following statements about mutually exclusive events is correct?

Accepted Answer

If events A and B are mutually exclusive, then $$P(A 	ext{ and } B) = 0.$$

Question 2

A hospital committee of 5 is to be chosen from 8 administrators, 10 physicians, and 15 nurses. What is the probability that the committee consists of 1 administrator, 1 physician, and 3 nurses?

Accepted Answer

$$\dfrac{8 	imes 10 	imes \binom{15}{3}}{\binom{33}{5}}$$

Question 3

Suppose $$X is a $$discrete random variable with the following probability distribution: $$P(X=1)=0.2$$, $$P(X=2)=0.5$$, $$P(X=3)=0.3. $$What is the mean $$\mu of$$ $$X$$?

Accepted Answer

$$\mu = 1 	imes 0.2 + 2 	imes 0.5 + 3 	imes 0.3 = 2.1$$

Question 4

A multiple-choice test has 8 questions, each with 4 possible answers. If a student guesses on every question, what is the probability that they get exactly 5 questions correct?

Accepted Answer

$$P(X=5) = \binom{8}{5} (0.25)^5$ (0.75)^3$

Question 5

Which formula gives the standard deviation of the sampling distribution of the sample mean when the population standard deviation $$\sigma is $$known and the sample size is $$n$$?

Accepted Answer

$$\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}}$$

Question 6

A code word is constructed from the letters in DEXTROUSLY. If repetition is allowed, how many 5-letter code words are possible?

Accepted Answer

$$14^5$$

Question 7

Which of the following is the correct formula for the margin of error $$E$$ when estimating a population mean with known standard deviation $$\sigma$$ and sample size $$n at $$confidence level $$1-\alpha$$?

Accepted Answer

$$E = z_{\alpha/2} \cdot \dfrac{\sigma}{\sqrt{n}}$$

Question 8

A population is normally distributed with mean $$\mu = 28000$$ and standard deviation $$\sigma = 4200. $$What is the probability that a randomly selected value is at least $$26000$$?

Accepted Answer

$$P(X \geq 26000) = 0.6830$$

Question 9

Which statement best describes the Central Limit Theorem?

Accepted Answer

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

Question 10

If $$P(A) = 0.4$$, what is $$P(	ext{not } A)$$?

Accepted Answer

$$P(	ext{not } A) = 1 - 0.4 = 0.6$$

Question 11

A binomial random variable has $$n=10$$ and $$p=0.3. $$What is its standard deviation?

Accepted Answer

$$\sigma = \sqrt{10 	imes 0.3 	imes 0.7}$$

Question 12

Which formula is used to calculate the number of ways to arrange 5 distinct letters from 14 available, with no repetition and order matters?

Accepted Answer

$$14P5 = \dfrac{14!}{(14-5)!}$$

Question 13

A sample of 10 people who shovel snow manually has a mean max heart rate of 175 and a sample standard deviation of 15. What is the 95% confidence interval for the population mean?

Accepted Answer

$$175 \pm t_{0.025,9} \cdot \dfrac{15}{\sqrt{10}}$$

Question 14

If events A and B are independent, which formula correctly gives $$P(A 	ext{ and } B)$$?

Accepted Answer

$$P(A 	ext{ and } B) = P(A) 	imes P(B)$$

Question 15

Which formula is used to determine the required sample size $$n to $$achieve a specified margin of error $$E$$ for estimating a population mean with known $$\sigma$$?

Accepted Answer

$$n = \left( \dfrac{z_{\alpha/2} \cdot \sigma}{E} \$$right)^2$$

Question 16

A dartboard is divided into 6 equal regions labeled A, B, C, D, E, F. What is the probability of hitting either region A or region C with a single throw?

Accepted Answer

$$P(A 	ext{ or } C) = \dfrac{2}{6} = \dfrac{1}{3}$$

Question 17

Which of the following is a point estimate for the population mean?

Accepted Answer

The sample mean $$\bar{x}$$

Question 18

If a probability distribution for a discrete random variable $$X is $$given by $$P(X=3)=0.15$$, $$P(X=4)=0.20$$, $$P(X=5)=0.25$$, $$P(X=6)=0.20$$, $$P(X=7)=0.20$$, what is the probability that $$X is at $$least 5?

Accepted Answer

$$P(X \geq 5) = 0.25 + 0.20 + 0.20 = 0.65$$

STAT 2 Review: Probability, Distributions, and Estimation

Study Guide - Practice Questions

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