Q: If a normal distribution has mean $$\mu = 100,000$$ gallons and standard deviation $$\sigma = 10,000$$ gallons, what production level corresponds to the 90th percentile?

$$100,000 + 1.28 \times 10,000 = 112,800$$ gallons

Question 1

Which of the following is a characteristic of a discrete random variable?

Accepted Answer

It can only assume countable, distinct values.

Question 2

Suppose the number of customers arriving at a bank in an hour is modeled as a random variable. Which type of random variable is this, and why?

Accepted Answer

Discrete, because the number of customers is countable.

Question 3

Which of the following is NOT a requirement for a binomial experiment?

Accepted Answer

The probability of success changes from trial to trial.

Question 4

A company sells a $$10,000 insurance policy for 290. $$The probability of a claim is 0.001. What is the expected gain for the company per policy? Use the following outcomes: gain $$290 with probability 0.999, loss 9$$,710 with probability 0.001.

Accepted Answer

$280.29

Question 5

Which formula gives the probability of exactly $$k$$ successes in $$n$$ independent binomial trials with probability $$p of $$success?

Accepted Answer

$$P(X = k) = \frac{n!}{k!(n-k)!} p^k (1-p$$)^{n-k}$$

Question 6

A machine produces 10% defective items. If 5 items are selected at random, what is the probability that exactly 3 are defective? Use the binomial formula.

Accepted Answer

$$P(X=3) = \frac{5!}{3!2!} (0.1)^3$ (0.9)^2$

Question 7

Which of the following statements about the normal distribution is FALSE?

Accepted Answer

It is only defined for discrete random variables.

Question 8

If $$X is $$normally distributed with mean $$\mu = 10$$ and standard deviation $$\sigma = 1.5$$, what is the $$z-$$score for $$X = 8$$?

Accepted Answer

$$z = -1.33$$

Question 9

Given a standard normal distribution, what is the probability that $$z is $$between 0 and 1.23?

Accepted Answer

Approximately 0.3907

Question 10

Suppose the time between charges for a cell phone is normally distributed with mean 10 hours and standard deviation 1.5 hours. What is the probability that a phone lasts between 8 and 12 hours between charges?

Accepted Answer

Approximately 0.8176

Question 11

A new car model has in-city mileage that is normally distributed with mean 27 mpg and standard deviation 3 mpg. What is the probability that a randomly selected car gets less than 20 mpg?

Accepted Answer

Approximately 0.01

Question 12

Which of the following is a property of a probability mass function (PMF) for a discrete random variable?

Accepted Answer

The sum of all probabilities is 1.

Question 13

If $$X is a $$binomial random variable with $$n = 10$$ and $$p = 0.3$$, what is the expected value $$E[X]$$?

Accepted Answer

$$E[X] = 3$$

Question 14

Which of the following best describes the difference between a discrete and a continuous random variable?

Accepted Answer

Discrete variables have countable values; continuous variables have infinite possible values within an interval.

Question 15

A hotel guest satisfaction study finds that 26% of guests had better-than-expected sleep, and 83% of those would return. What is the probability that a randomly selected guest both had better-than-expected sleep and would return?

Accepted Answer

$$0.22$$

Question 16

For a continuous random variable $$X$$ with probability density function $$f(x)$$, which statement is true?

Accepted Answer

$$P(X = a) = 0$$ for any specific value $$a.$$

Question 17

If a normal distribution has mean $$\mu = 100,000$$ gallons and standard deviation $$\sigma = 10,000$$ gallons, what production level corresponds to the 90th percentile?

Accepted Answer

$$100,000 + 1.28 	imes 10,000 = 112,800$$ gallons

Question 18

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

Accepted Answer

It is only used for discrete data.

Question 19

If $$X is a $$continuous random variable with PDF $$f(x)$$, how do you find the probability that $$X$$ falls between $$a$$ and $$b$$?

Accepted Answer

$$P(a \u003c X \u003c b) = \int_a^b f(x) dx$$

Random Variables and Probability Distributions: Study Notes

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