Q: A life insurance policy pays out $$120,000 if $$the insured dies within a year. The probability of survival is $$0.9987$$, and the premium is $$175. $$What is the expected value for the insured?

$$(0.9987 \times -175) + (0.0013 \times (120,000 - 175))$$

Question 1

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

Accepted Answer

The random variable can be categorical.

Question 2

A probability histogram differs from a relative frequency histogram in which way?

Accepted Answer

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

Question 3

Given a discrete random variable $$x$$ with probability distribution $$P(x)$$, which formula correctly calculates the mean (expected value) of $$x$$?

Accepted Answer

$$\displaystyle \mu = \sum x \cdot P(x)$$

Question 4

Suppose you toss two coins. Let $$x be $$the number of heads. What is the probability that $$x = 1$$?

Accepted Answer

$$0.50$$

Question 5

Which scenario describes a continuous random variable?

Accepted Answer

The body temperature of a randomly selected adult.

Question 6

A probability distribution table for job interview mistakes lists categories such as 'inappropriate attire' and 'checking phone' with probabilities summing to $$1.57. $$Why does this NOT describe a valid probability distribution?

Accepted Answer

Because the random variable is not numerical and the probabilities do not sum to 1.

Question 7

For a discrete probability distribution, which formula gives the variance $$\$$sigma^2$$?

Accepted Answer

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

Question 8

Using the range rule of thumb, which value is considered significantly high?

Accepted Answer

$$\mu + 2\sigma$$

Question 9

A life insurance policy pays out $$120,000 if $$the insured dies within a year. The probability of survival is $$0.9987$$, and the premium is $$175. $$What is the expected value for the insured?

Accepted Answer

$$(0.9987 	imes -175) + (0.0013 	imes (120,000 - 175))$$

Question 10

Which of the following is a requirement for a binomial probability distribution?

Accepted Answer

The trials must be independent.

Question 11

In a binomial experiment with $$n = 10$$ trials and probability of success $$p = 0.05$$, what is the probability of exactly $$2$$ successes? Use the binomial formula.

Accepted Answer

$$\displaystyle P(2) = \binom{10}{2} (0.05)^2$ (0.95)^8$

Question 12

For a binomial distribution, what is the formula for the mean $$\mu$$?

Accepted Answer

$$\displaystyle \mu = np$$

Question 13

In the context of the central limit theorem, what is the standard deviation of the sampling distribution of the sample mean $$\bar{x}$$ for samples of size $$n$$?

Accepted Answer

$$\displaystyle \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$

Question 14

A sample of $$n = 106$$ body temperatures has a mean of $$98.2^	ext{o}F. If $$the population mean is $$98.6^	ext{o}F $$and standard deviation is $$0.62^	ext{o}F$$, what is the $$z-$$score for the sample mean?

Accepted Answer

$$\displaystyle z = \frac{98.2 - 98.6}{0.62 / \sqrt{106}}$$

Question 15

In the Boeing 737 seat width example, why is the probability of the mean hip width of 126 randomly selected males exceeding 16.0 in. much lower than the probability for a single male?

Accepted Answer

Because the standard deviation of the sample mean is much smaller than that of individual values.

Question 16

Which of the following statements about the central limit theorem is TRUE?

Accepted Answer

For any population, the sampling distribution of the sample mean approaches normality as $$n$$ increases.

Question 17

If the probability of observing $$x or $$more successes in $$n$$ trials is less than $$0.05$$, what can be concluded?

Accepted Answer

The result is considered significantly high.

Question 18

A binomial experiment has $$n = 460$$ trials and $$p = 0.5. $$What is the mean and standard deviation?

Accepted Answer

$$\mu = 230.0$$, $$\sigma = 10.7$$

Discrete Probability Distributions: Key Concepts and Applications

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