Question 1

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

Accepted Answer

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

Question 2

Which formula correctly represents the probability density function (PDF) of the normal distribution?

Accepted Answer

$$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e$$^{-(1/2)((x-\mu)/\sigma)^2}$$

Question 3

A normal distribution has a mean $$\mu = 12$$ and standard deviation $$\sigma = 3. $$What is the $$z-$$score for $$x = 8.5$$?

Accepted Answer

$$z = \frac{8.5 - 12}{3} = -1.17$$

Question 4

Which of the following best describes the exponential probability distribution?

Accepted Answer

It is right-skewed and describes the time between events.

Question 5

Suppose the time between customer arrivals at a deli counter follows an exponential distribution with a mean of 4 minutes. What is the value of $$\lambda$$ for this distribution?

Accepted Answer

$$\lambda = 0.25$$

Question 6

Given a uniform distribution on the interval $$[a, b]$$, what is the formula for the mean?

Accepted Answer

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

Question 7

A statistics final exam time is uniformly distributed between 70 and 120 minutes. What is the probability that a randomly selected student finishes in less than 80 minutes?

Accepted Answer

$$P(x \u003c 80) = \frac{80 - 70}{120 - 70} = 0.2$$

Question 8

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

Accepted Answer

The distribution is skewed to the right.

Question 9

If $$X is a $$continuous random variable with a uniform distribution on $$[2, 10]$$, what is the probability density function $$f(x)$$?

Accepted Answer

$$f(x) = \frac{1}{10 - 2} = 0.125$$

Question 10

A company experiences a 6% error rate in its drive-through orders. For a sample of 120 orders, can the normal distribution be used to approximate the binomial distribution for the number of errors? Why?

Accepted Answer

Yes, because $$np = 7.2$$ and $$nq = 112.8$$, both greater than 5.

Question 11

Which Excel function would you use to find the cumulative probability that a normal random variable is less than or equal to $$x$$?

Accepted Answer

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

Question 12

If the annual snowfall in Minneapolis is normally distributed with $$\mu = 46$$ inches and $$\sigma = 18.5$$ inches, what is the probability that between 30 and 70 inches will fall next year? (Use $$z_{30} = -0.86$$, $$z_{70} = 1.30$$, $$P(z \u003c -0.86) = 0.1949$$, $$P(z \u003c 1.30) = 0.9032)$$

Accepted Answer

$$0.9032 - 0.1949 = 0.7083$$

Question 13

Which of the following is the formula for the cumulative distribution function (CDF) of the exponential distribution?

Accepted Answer

$$F(a) = 1 - e$$^{-\lambda a}$$

Question 14

A normal distribution has $$\mu = 34$$ and $$\sigma = 10. $$What is the probability that a randomly selected value is less than 27? (Given $$z = -0.70$$, $$P(z \u003c -0.70) = 0.2420)$$

Accepted Answer

0.2420

Question 15

Which of the following best describes the effect of increasing the standard deviation $$\sigma in a $$normal distribution, while keeping the mean $$\mu$$ constant?

Accepted Answer

The distribution becomes wider and flatter.

Question 16

If $$X is $$exponentially distributed with mean $$\mu = 5$$, what is the standard deviation of $$X$$?

Accepted Answer

$$\sigma = 5$$

Question 17

Which of the following is a key difference between the normal and exponential distributions?

Accepted Answer

The normal distribution is symmetrical, while the exponential distribution is right-skewed.

Question 18

A uniform distribution is defined on the interval $$[3, 9]. $$What is the standard deviation?

Accepted Answer

$$\sigma = \frac{9 - 3}{\sqrt{12}} = 1.732$$

Continuous Probability Distributions: Normal, Exponential, and Uniform Distributions

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