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.

Question 2

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

Accepted Answer

The probabilities must be based on categorical variables.

Question 3

Given the probability distribution for the number of heads in two coin tosses: $$x = 0, 1, 2$$ with $$P(x) = 0.25, 0.5, 0.25$$ respectively, what is the mean ($$\mu) of $$the distribution?

Accepted Answer

$$\mu = 1.0$$

Question 4

A table lists the proportion of unlicensed software in four countries, with values summing to 2.09. Why does this table NOT represent a probability distribution?

Accepted Answer

The sum of the probabilities is not equal to 1.

Question 5

Which formula is used to calculate the variance ($$\$$sigma^2$$) of a discrete probability distribution?

Accepted Answer

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

Question 6

If the mean number of females in two births is $$\mu = 1.0$$ and the standard deviation is $$\sigma = 0.7$$, what is the range of values considered 'not significant' according to the range rule of thumb?

Accepted Answer

$$-0.4 to$$ $$2.4$$

Question 7

Which of the following statements about expected value is TRUE?

Accepted Answer

The expected value is the same as the mean of the probability distribution.

Question 8

A probability distribution has $$P(x) = 0.2, 0.5, 0.3$$ for $$x = 1, 2, 3. $$What is the standard deviation ($$\sigma) if $$the mean ($$\mu) is$$ $$2.1$$?

Accepted Answer

$$\sigma = 0.9$$

Question 9

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

Accepted Answer

Rejecting an assumption if an observed outcome has a very low probability under that assumption.

Question 10

If $$P(x 	ext{ or more}) \leq 0.05$$, what can be concluded about the value $$x$$?

Accepted Answer

$$x is a $$significantly high value.

Question 11

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

Accepted Answer

The body temperature of a randomly selected person.

Question 12

Given the probability distribution for the number of females in two births: $$x = 0, 1, 2$$ with $$P(x) = 0.25, 0.5, 0.25$$, what is the variance ($$\$$sigma^2$$)?

Accepted Answer

$$\$$sigma^2$$ = 0.5$$

Question 13

A probability histogram differs from a relative frequency histogram in that:

Accepted Answer

It uses probabilities on the vertical axis instead of relative frequencies.

Question 14

If the probability of an event is 0.05, which of the following statements is most accurate?

Accepted Answer

The event is unlikely but possible.

Question 15

Which of the following is a correct interpretation of the mean of a probability distribution?

Accepted Answer

It is the long-run average value of the random variable.

Question 16

Suppose you observe 20 girls in 100 births, and the probability of this outcome under the assumption of equal likelihood is very low. What should you conclude?

Accepted Answer

The assumption of equal likelihood is probably not correct.

Question 17

Which of the following is NOT a property of a probability distribution?

Accepted Answer

Probabilities can be negative.

Question 18

If $$\mu = 1.0$$ and $$\sigma = 0.7$$, is 2 females in 2 births a significantly high value according to the range rule of thumb?

Accepted Answer

No, because 2 is less than $$\mu + 2\sigma.$$

Question 19

Which formula would you use to find the expected value $$E of a $$discrete random variable $$x$$?

Accepted Answer

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

5.1 Probability Distributions: Discrete Random Variables and Their Properties

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