Q: If a normal distribution has $$\mu = 12$$ and $$\sigma = 3$$, what value $$x$$ corresponds to the 95th percentile? (Use $$z = 1.645)$$

$$x = 12 + (1.645 \times 3) = 16.94$$

Question 1

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

Accepted Answer

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

Question 2

Which formula represents the probability density function for 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 $$x = 14$$?

Accepted Answer

$$z = \frac{14-12}{3} = 0.67$$

Question 4

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

Accepted Answer

Right-skewed and not symmetrical

Question 5

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

Accepted Answer

$$0.9032 - 0.1949 = 0.7083$$

Question 6

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

Accepted Answer

The distribution is right-skewed.

Question 7

The time between customer arrivals at a deli counter follows an exponential distribution with a mean of 4 minutes. What is the probability that the next customer will arrive within 2 minutes? (Use $$\lambda = 1/4$$ and the cumulative distribution function $$P(x \leq a) = 1 - e$$^{-\lambda a}$$)

Accepted Answer

$$1 - e$$^{-0.5}$$ \approx 0.3935$$

Question 8

Which formula gives the z-score for a value $$x in a $$normal distribution?

Accepted Answer

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

Question 9

A uniform distribution has minimum $$a = 70$$ and maximum $$b = 120. $$What is the probability that a randomly selected value is less than 80? (Use $$P(x \leq x_2$) = \frac{x_2$ - a}{b - a})$$

Accepted Answer

$$\frac{80-70}{120-70} = \frac{10}{50} = 0.2$$

Question 10

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

Accepted Answer

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

Question 11

According to the empirical rule, what percentage of values fall within two standard deviations of the mean in a normal distribution?

Accepted Answer

95%

Question 12

If a normal distribution has $$\mu = 12$$ and $$\sigma = 3$$, what value $$x$$ corresponds to the 95th percentile? (Use $$z = 1.645)$$

Accepted Answer

$$x = 12 + (1.645 	imes 3) = 16.94$$

Question 13

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 14

A binomial distribution can be approximated by a normal distribution when which condition is met?

Accepted Answer

$$np \geq 5$$ and $$nq \geq 5$$

Question 15

For a continuous uniform distribution with $$a = 10$$ and $$b = 30$$, what is the mean?

Accepted Answer

$$\frac{a+b}{2} = \frac{10+30}{2} = 20$$

Question 16

Which formula gives the standard deviation of a continuous uniform distribution with endpoints $$a$$ and $$b$$?

Accepted Answer

$$\sqrt{\frac{(b-a)^2$}{12}}$$

Question 17

If the average time between failures in a business process is 10 hours, what is the value of $$\lambda$$ for the exponential distribution?

Accepted Answer

$$\lambda = \frac{1}{10}$$

Question 18

Which of the following is TRUE about the probability density function for a continuous uniform distribution?

Accepted Answer

It is constant for all values between $$a$$ and $$b.$$

Question 19

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

Accepted Answer

0.2420

Continuous Probability Distributions: Normal, Exponential, and Uniform Distributions

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