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

$$P(x \u003c 80) = \frac{80 - 70}{120 - 70} = \frac{10}{50} = 0.20$$

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 $$z_{30} = -0.86$$, $$z_{70} = 1.30$$, $$P(z \u003c 1.30) = 0.9032$$, $$P(z \u003c -0.86) = 0.1949)$$

Accepted Answer

$$P(30 \u003c X \u003c 70) = 0.9032 - 0.1949 = 0.7083$$

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

$$np \u003e 5$$ and $$nq \u003e 5$$

Question 11

The exponential cumulative distribution function is given by $$P(x \u003c a) = 1 - e$$^{-\lambda a}$$. If $$\lambda = 0.25$$ and a = 2$, what is P(x$$ \u003c 2)$$?

Accepted Answer

$$P(x \u003c 2) = 1 - e$$^{-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 $$X \u003c 27$$? (Given $$z = -0.70$$, $$P(z \u003c -0.70) = 0.2420)$$

Accepted Answer

$$P(X \u003c 27) = 0.2420$$

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 \u003c x \u003c b$

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

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