Q: A statistics final exam takes between 70 and 120 minutes to complete, following a uniform distribution. What is the probability that a randomly selected student finishes in less than 80 minutes?

$P(x < 80) = \frac{80 - 70}{120 - 70} = \frac{10}{50} = 0.20$

Q: If $X$ is normally distributed with $\mu = 46$ and $\sigma = 18.5$, what is the probability that $X$ is between 30 and 70? (Use $z_{30} = -0.86$, $z_{70} = 1.30$, $P(z < 1.30) = 0.9032$, $P(z < -0.86) = 0.1949$)

$P(30 < X < 70) = 0.9032 - 0.1949 = 0.7083$

Q: The exponential cumulative distribution function is given by $P(x < a) = 1 - e^{-\lambda a}$. If $\lambda = 0.25$ and $a = 2$, what is $P(x < 2)$?

$P(x < 2) = 1 - e^{-0.5} \approx 0.3935$

Q: Which Excel function would you use to find the cumulative probability for a normal distribution with mean $\mu$, standard deviation $\sigma$, and value $x$?

NORM.DIST(x, mean, standard_dev, TRUE)

Q: A fast-food chain has a 6% error rate in drive-through orders. For the next 120 customers, what is the mean and standard deviation for the number of errors?

Mean: $7.2$, Standard deviation: $2.60$

Q: Which of the following is the formula for the standard deviation of a continuous uniform distribution defined on $[a, b]$?

$\sigma = \frac{b - a}{\sqrt{12}}$

Q: Which of the following is TRUE about the probability density function for a continuous uniform distribution on $[a, b]$?

$f(x) = \frac{1}{b - a}$ \,\text{for}\, a < x < b$

Question 1

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

Accepted Answer

The probability of a continuous random variable being exactly a specific value is always zero.

Question 2

A company tracks the time (in minutes) customers spend on hold, which follows a normal distribution with mean $\mu = 12$ and standard deviation $\sigma = 3$. What is the $z$-score for a customer who spends 8 minutes on hold?

Accepted Answer

$z = \frac{8 - 12}{3} = -1.33$

Question 3

Which of the following best describes the shape of the exponential probability distribution?

Accepted Answer

Skewed to the right with a long right tail

Question 4

Suppose the time between customer arrivals at a store follows an exponential distribution with a mean of 5 minutes. What is the value of $\lambda$ (the rate parameter) for this distribution?

Accepted Answer

$\lambda = \frac{1}{5} = 0.2$

Question 5

A uniform distribution is defined on the interval $[a, b]$. What is the formula for the mean of this distribution?

Accepted Answer

$\mu = \frac{a + b}{2}$

Question 6

A statistics final exam takes between 70 and 120 minutes to complete, following a uniform distribution. What is the probability that a randomly selected student finishes in less than 80 minutes?

Accepted Answer

$P(x < 80) = \frac{80 - 70}{120 - 70} = \frac{10}{50} = 0.20$

Question 7

Which of the following is NOT a characteristic of the normal probability distribution?

Accepted Answer

It is skewed to the right.

Question 8

If $X$ is normally distributed with $\mu = 46$ and $\sigma = 18.5$, what is the probability that $X$ is between 30 and 70? (Use $z_{30} = -0.86$, $z_{70} = 1.30$, $P(z < 1.30) = 0.9032$, $P(z < -0.86) = 0.1949$)

Accepted Answer

$P(30 < X < 70) = 0.9032 - 0.1949 = 0.7083$

Question 9

Which formula is used to calculate the $z$-score for a value $x$ in a normal distribution?

Accepted Answer

$z = \frac{x - \mu}{\sigma}$

Question 10

A business analyst wants to use the normal distribution to approximate a binomial distribution. Which condition must be met for this approximation to be valid?

Accepted Answer

$np > 5$ and $nq > 5$

Question 11

The exponential cumulative distribution function is given by $P(x < a) = 1 - e^{-\lambda a}$. If $\lambda = 0.25$ and $a = 2$, what is $P(x < 2)$?

Accepted Answer

$P(x < 2) = 1 - e^{-0.5} \approx 0.3935$

Question 12

Which Excel function would you use to find the cumulative probability for a normal distribution with mean $\mu$, standard deviation $\sigma$, and value $x$?

Accepted Answer

NORM.DIST(x, mean, standard_dev, TRUE)

Question 13

For a standard normal distribution, what is the probability that a value falls within one standard deviation of the mean?

Accepted Answer

Approximately 68%

Question 14

A fast-food chain has a 6% error rate in drive-through orders. For the next 120 customers, what is the mean and standard deviation for the number of errors?

Accepted Answer

Mean: $7.2$, Standard deviation: $2.60$

Question 15

Which of the following is the formula for the standard deviation of a continuous uniform distribution defined on $[a, b]$?

Accepted Answer

$\sigma = \frac{b - a}{\sqrt{12}}$

Question 16

If a random variable $X$ follows a normal distribution with $\mu = 34$ and $\sigma = 10$, what is the probability that $X < 27$? (Given $z = -0.70$, $P(z < -0.70) = 0.2420$)

Accepted Answer

$P(X < 27) = 0.2420$

Question 17

Which of the following is TRUE about the probability density function for a continuous uniform distribution on $[a, b]$?

Accepted Answer

$f(x) = \frac{1}{b - a}$ \,	ext{for}\, a < x < b$

Continuous Probability Distributions (Chapter 6): Study Notes for Business Statistics

Study Guide - Practice Questions

Study Guide - Flashcards