Q: A probability distribution is valid if which of the following conditions are met?

$$\sum P(x) = 1$$ \,\text{and}\, 0 \leq P(x) \leq 1$$ \text{for all} \; x$$

Q: For a binomial distribution with $$n = 10$$ and $$p = 0.1$$, what is the standard deviation?

$$\sigma = \sqrt{n p q} = \sqrt{10 \times 0.1 \times 0.9}$$

Q: A random variable $$X$$ has the following probability distribution: $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3. $$What is the mean $$\mu$$?

$$\mu = 0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 1.1$$

Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable that has a finite or countable number of possible values, typically resulting from a counting process.

Question 2

Given the probability distribution for rolling a fair six-sided die, what is the probability of rolling a number greater than 4?

Accepted Answer

$$\frac{2}{6}$$

Question 3

Which formula correctly calculates the mean $$\mu of a $$discrete probability distribution?

Accepted Answer

$$\mu = \sum [x \cdot P(x)]$$

Question 4

A probability distribution is valid if which of the following conditions are met?

Accepted Answer

$$\sum P(x) = 1$$ \,	ext{and}\, 0 \leq P(x) \leq 1$$ 	ext{for all} \; x$$

Question 5

Suppose the mean number of callers to a radio station during a commercial is $$\mu = 1.6$$ and the standard deviation is $$\sigma = 1.1. $$According to the range rule of thumb, what is the maximum usual number of callers?

Accepted Answer

$$1.6 + 2 	imes 1.1 = 3.8$$

Question 6

Which scenario best illustrates the rare event rule for inferential statistics?

Accepted Answer

A fair coin is tossed 1000 times and 992 heads are observed.

Question 7

If $$P(x 	ext{ or more}) \u003c 0.05 in a $$binomial experiment, what can be concluded about the observed value of $$x$$?

Accepted Answer

The value of $$x is $$unusually high.

Question 8

A binomial experiment has $$n = 5$$ trials and probability of success $$p = 0.3. $$What is the mean number of successes?

Accepted Answer

$$\mu = n \cdot p = 5 \cdot 0.3 = 1.5$$

Question 9

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

Accepted Answer

The trials must be dependent on each other.

Question 10

A survey finds that 30% of teenagers have part-time jobs. If 5 teenagers are selected at random, what is the probability that exactly 3 have part-time jobs? Use the binomial formula.

Accepted Answer

$$P(3) = \binom{5}{3} (0.3)^3$ (0.7)^2$

Question 11

For a binomial distribution with $$n = 10$$ and $$p = 0.1$$, what is the standard deviation?

Accepted Answer

$$\sigma = \sqrt{n p q} = \sqrt{10 	imes 0.1 	imes 0.9}$$

Question 12

A random variable $$X$$ has the following probability distribution: $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3. $$What is the mean $$\mu$$?

Accepted Answer

$$\mu = 0 	imes 0.2 + 1 	imes 0.5 + 2 	imes 0.3 = 1.1$$

Question 13

Which of the following statements about continuous random variables is correct?

Accepted Answer

They can take on infinitely many values associated with measurements on a continuous scale.

Question 14

If the mean $$\mu of a $$probability distribution is 4 and the standard deviation $$\sigma is 1.5$$, what is the minimum usual value according to the range rule of thumb?

Accepted Answer

$$4 - 2 	imes 1.5 = 1$$

Question 15

A student guesses on all 13 questions of a multiple-choice test, each with 4 options. What is the expected number of correct answers?

Accepted Answer

$$\mu = n \cdot p = 13 \cdot 0.25 = 3.25$$

Question 16

Which of the following is a real-world example of a binomial experiment?

Accepted Answer

Testing whether each of 20 light bulbs works or does not work.

Question 17

If a probability distribution has $$x$$ values that are all integers, how should the mean and standard deviation be rounded?

Accepted Answer

To one decimal place.

Question 18

Which formula gives the variance $$\$$sigma^2$$ of a discrete probability distribution using the shortcut method?

Accepted Answer

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

Probability Distributions and Random Variables: Key Concepts and Applications

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