Q: If $$X is a $$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 (expected value) of $$X$$?

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

Q: A binomial experiment consists of 10 trials with probability of success $$p=0.4. $$What is the probability of exactly 3 successes?

$$\binom{10}{3} \times 0.4^3$ \times 0.6^7$

Q: A random variable $$X$$ has the following probability distribution: $$P(X=0)=0.1$$, $$P(X=1)=0.6$$, $$P(X=2)=0.3. $$What is the standard deviation of $$X$$?

$$\sqrt{0.1 \times (0-1.3)^2$ + 0.6 \times (1-1.3)^2$ + 0.3 \times (2-1.3)^2$} \approx 0.64$$

Q: A quality control manager inspects 20 items. If the probability that an item is defective is $$0.05$$, what is the probability that exactly 2 items are defective?

$$\binom{20}{2} \times 0.05^2$ \times 0.95$$^{18}$$

Question 1

A hospital records whether patients recover from a certain treatment. If the probability that a patient recovers is 0.8, what is the probability that two randomly selected patients both recover?

Accepted Answer

$$0.8 	imes 0.8 = 0.64$$

Question 2

Which of the following best describes a discrete random variable?

Accepted Answer

A variable that can take on only specific, countable values.

Question 3

A fair die is rolled. What is the probability of rolling a number less than 5?

Accepted Answer

$$\frac{4}{6} = \frac{2}{3}$$

Question 4

If $$X is a $$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 (expected value) of $$X$$?

Accepted Answer

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

Question 5

A binomial experiment consists of 10 trials with probability of success $$p=0.4. $$What is the probability of exactly 3 successes?

Accepted Answer

$$\binom{10}{3} 	imes 0.4^3$ 	imes 0.6^7$

Question 6

Which of the following is NOT a requirement for a binomial distribution?

Accepted Answer

Each trial must have more than two possible outcomes.

Question 7

A random variable $$X$$ has mean $$\mu = 5$$ and standard deviation $$\sigma = 2. If $$you take a random sample of size $$n=16$$, what is the standard deviation of the sample mean?

Accepted Answer

$$\frac{2}{\sqrt{16}} = 0.5$$

Question 8

If the probability of event $$A is$$ $$0.3$$ and the probability of event $$B is$$ $$0.5$$, and $$A$$ and $$B$$ are mutually exclusive, what is $$P(A \cup B)$$?

Accepted Answer

$$0.3 + 0.5 = 0.8$$

Question 9

A normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the $$z-$$score for a value of $$130$$?

Accepted Answer

$$\frac{130 - 100}{15} = 2$$

Question 10

Which statement best describes the sampling distribution of the sample mean?

Accepted Answer

It is the distribution of all possible sample means from samples of a given size drawn from a population.

Question 11

A pharmaceutical company tests a new drug on 100 patients. If the probability of a patient experiencing side effects is $$0.1$$, what is the expected number of patients with side effects?

Accepted Answer

$$100 	imes 0.1 = 10$$

Question 12

If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.2$$, what is $$P(A \cup B)$$?

Accepted Answer

$$0.4 + 0.5 - 0.2 = 0.7$$

Question 13

A random variable $$X$$ has the following probability distribution: $$P(X=0)=0.1$$, $$P(X=1)=0.6$$, $$P(X=2)=0.3. $$What is the standard deviation of $$X$$?

Accepted Answer

$$\sqrt{0.1 	imes (0-1.3)^2$ + 0.6 	imes (1-1.3)^2$ + 0.3 	imes (2-1.3)^2$} \approx 0.64$$

Question 14

A quality control manager inspects 20 items. If the probability that an item is defective is $$0.05$$, what is the probability that exactly 2 items are defective?

Accepted Answer

$$\binom{20}{2} 	imes 0.05^2$ 	imes 0.95$$^{18}$$

Question 15

If a variable is normally distributed with mean $$\mu = 50$$ and standard deviation $$\sigma = 10$$, what proportion of values are expected to fall within one standard deviation of the mean?

Accepted Answer

Approximately $$68\%$$

Question 16

A student scores 85 on a test where the mean is 75 and the standard deviation is 5. What is the student's $$z-$$score?

Accepted Answer

$$\frac{85 - 75}{5} = 2$$

Question 17

Which of the following is an example of a discrete random variable?

Accepted Answer

The number of cars passing through a toll booth in an hour.

Question 18

If the mean of a population is $$\mu = 60$$ and the standard deviation is $$\sigma = 12$$, what is the mean of the sampling distribution of the sample mean for samples of size $$n=36$$?

Accepted Answer

$$60$$

Question 19

A researcher wants to find the probability that a randomly selected person has blood type O or blood type A. If $$P(O) = 0.45$$, $$P(A) = 0.40$$, and $$P(O \cap A) = 0$$, what is $$P(O \cup A)$$?

Accepted Answer

$$0.45 + 0.40 = 0.85$$

Probability, Random Variables, and Sampling Distributions: Study Notes for Introductory Statistics

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