Question 1

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

Accepted Answer

The random variable must be categorical.

Question 2

A random variable $$x is $$called discrete if:

Accepted Answer

It has a finite or countable number of possible values.

Question 3

Given the probability distribution for the number of heads when tossing two coins: $$P(0) = 0.25$$, $$P(1) = 0.5$$, $$P(2) = 0.25$$, what is the expected value (mean) of the number of heads?

Accepted Answer

$$\sum x \cdot P(x) = 1.0$$

Question 4

Which of the following scenarios describes a continuous random variable?

Accepted Answer

The body temperature of a randomly selected patient.

Question 5

A probability distribution table for a random variable $$x$$ has probabilities $$0.2, 0.3, 0.4, 0.1. $$What is the sum of the probabilities, and does this table satisfy the requirements for a probability distribution?

Accepted Answer

The sum is $$1.0$$; yes, it satisfies the requirements.

Question 6

Which of the following is a key property of a binomial probability distribution?

Accepted Answer

The trials are independent and there is a fixed number of trials.

Question 7

Suppose you are sampling without replacement from a population of 10,000 and your sample size is 300. According to the 5% guideline, can you treat the selections as independent?

Accepted Answer

Yes, because $$\frac{300}{10000} = 0.03 is $$less than 0.05.

Question 8

Given a binomial experiment with $$n = 8$$, $$p = 0.3$$, what is the probability of exactly 2 successes? Use the binomial probability formula.

Accepted Answer

$$P(2) = \displaystyle \frac{8!}{2!6!} (0.3)^2$ (0.7)^6$

Question 9

A survey finds that 85% of adults know what Twitter is. If 5 adults are randomly selected, what is the probability that exactly 3 know what Twitter is? (Use $$n=5$$, $$x=3$$, $$p=0.85$$, $$q=0.15)$$

Accepted Answer

$$P(3) = \displaystyle \frac{5!}{3!2!} (0.85)^3$ (0.15)^2$

Question 10

Which of the following best describes the rare event rule for inferential statistics?

Accepted Answer

If an outcome is very unlikely under a given assumption and it occurs, the assumption is probably not correct.

Question 11

For a binomial distribution with $$n = 460$$ and $$p = 0.5$$, what is the mean and standard deviation?

Accepted Answer

$$\mu = np = 230$$, $$\sigma = \sqrt{npq} = 10.7$$

Question 12

Using the range rule of thumb, which values are considered significantly high for a binomial distribution with $$\mu = 230$$ and $$\sigma = 10.7$$?

Accepted Answer

Values $$\geq 251.4$$

Question 13

Which of the following is the correct formula for the variance of a discrete probability distribution?

Accepted Answer

$$\$$sigma^2$$ = \sum [(x - \mu)^2$ \cdot P(x)]$$

Question 14

A probability distribution has $$P(x) = 0.2$$ for $$x=1$$, $$P(x) = 0.5$$ for $$x=2$$, and $$P(x) = 0.3$$ for $$x=3. $$What is the expected value?

Accepted Answer

$$E = 1 \cdot 0.2 + 2 \cdot 0.5 + 3 \cdot 0.3 = 2.1$$

Question 15

In a medical study, the probability of a patient responding to a new drug is 0.4. If 6 patients are treated, what is the probability that exactly 4 respond? (Use the binomial formula.)

Accepted Answer

$$P(4) = \displaystyle \frac{6!}{4!2!} (0.4)^4$ (0.6)^2$

Question 16

A probability distribution is given by $$P(x) = 0.3$$ for $$x=0$$, $$P(x) = 0.4$$ for $$x=1$$, $$P(x) = 0.3$$ for $$x=2. $$What is the variance if the mean is $$\mu = 1.0$$?

Accepted Answer

$$\$$sigma^2$$ = (0-1)^2$ \cdot 0.3 + (1-1)^2$ \cdot 0.4 + (2-1)^2$ \cdot 0.3 = 0.6$$

Probability Distributions and Binomial Probability Distributions: Study Notes for Statistics Students

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