Question 1

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

Accepted Answer

The random variable x must be categorical.

Question 2

Suppose you are analyzing the number of heads in three coin tosses. Which type of random variable is this?

Accepted Answer

A discrete random variable

Question 3

Given the probability distribution for the number of females in two births: $$x = 0, 1, 2$$ with $$P(x) = 0.25, 0.50, 0.25$$ respectively, what is the mean number of females?

Accepted Answer

$$1.0$$

Question 4

Which formula correctly calculates the variance $$\$$sigma^2$$ of a discrete probability distribution?

Accepted Answer

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

Question 5

A table lists countries and the proportion of unlicensed software in each. Why does this table NOT describe a probability distribution?

Accepted Answer

The variable x is categorical, not numerical.

Question 6

If the mean $$\mu of a $$probability distribution is 2 and the standard deviation $$\sigma is 0.5$$, what is the range of values considered 'not significant' according to the range rule of thumb?

Accepted Answer

Between $$1.0$$ and $$3.0$$

Question 7

Which of the following best describes the expected value $$E of a $$discrete random variable $$x$$?

Accepted Answer

The mean value of the outcomes, calculated as $$E = \sum x \cdot P(x)$$

Question 8

A casino game offers a $$1/38$$ chance to win $$175$$ dollars and a $$37/38$$ chance to lose $$5. $$What is the expected value of a $$5$$ bet?

Accepted Answer

$$-0.26$$

Question 9

Which of the following statements about probability histograms is TRUE?

Accepted Answer

The vertical axis represents probabilities, not relative frequencies.

Question 10

If $$P(x 	ext{ or more}) \leq 0.05$$, what can you conclude about the value x in a probability distribution?

Accepted Answer

x is a significantly high value.

Question 11

Suppose the mean number of defective items in a shipment is $$\mu = 3$$ and the standard deviation is $$\sigma = 1. $$According to the range rule of thumb, which number of defective items would be considered significantly low?

Accepted Answer

$$1 or $$fewer

Question 12

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

Accepted Answer

The body temperature of a randomly selected patient

Question 13

A probability distribution is given by $$P(x) = \frac{1}{2} \frac{2-x!}{x!}$$ for $$x = 0, 1, 2. $$What is $$P(1)$$?

Accepted Answer

$$0.50$$

Question 14

If the probability of an event is very small and the event occurs, what does the rare event rule suggest?

Accepted Answer

The assumption under which the probability was calculated is probably not correct.

Question 15

Which of the following is the correct formula for the standard deviation $$\sigma of a $$discrete probability distribution?

Accepted Answer

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

Question 16

A probability distribution has $$\mu = 4$$ and $$\sigma = 1.5. $$According to the range rule of thumb, what is the minimum value considered significantly high?

Accepted Answer

$$7.0$$

Question 17

Which of the following best describes a discrete random variable?

Accepted Answer

A variable that can take on a countable number of values.

Question 18

If $$P(x 	ext{ or fewer}) \leq 0.05$$, what can you conclude about the value x in a probability distribution?

Accepted Answer

x is a significantly low value.

Question 19

A student claims that a probability distribution is valid because all probabilities are between 0 and 1. What else must be true for the distribution to be valid?

Accepted Answer

The sum of all probabilities must be 1 and the random variable must be numerical.

Probability Distributions: Concepts, Properties, and Applications

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