Q: The time between charges for a cell phone battery is normally distributed with $$\sigma = 1.5$$ hours. If only 5% of batteries should require charging in 13 hours or less, what must the mean time between charges be?

Use $$z = -1.645$$, $$13 = \mu + (-1.645) \times 1.5 \implies \mu = 15.47$$ hours.

Question 1

Which of the following statements best describes the properties of a normal distribution?

Accepted Answer

It is symmetrical, unimodal, and the mean, median, and mode are all equal.

Question 2

Given a normal distribution with mean $$\mu = 50$$ and standard deviation $$\sigma = 10$$, what is the z-value for $$x = 65$$?

Accepted Answer

$$z = \frac{65 - 50}{10} = 1.5$$

Question 3

A restaurant’s tips are normally distributed with $$\mu = 12$$ and $$\sigma = 3. $$What is the probability that a tip is less than $$8$$ dollars? (Use the standard normal table for calculation.)

Accepted Answer

First, calculate $$z = \frac{8 - 12}{3} = -1.33. $$The probability is approximately $$0.0918.$$

Question 4

If the time to complete a patio set is normally distributed with $$\mu = 22.14$$ minutes and $$\sigma = 6.09$$ minutes, what proportion of sets are completed in 30 minutes or less?

Accepted Answer

Calculate $$z = \frac{30 - 22.14}{6.09} \approx 1.29. $$The proportion is about $$0.9015.$$

Question 5

Which formula is used to convert a value $$x$$ from a normal distribution with mean $$\mu$$ and standard deviation $$\sigma to a $$standard normal $$z-$$value?

Accepted Answer

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

Question 6

A graduate program admits only the top 10% of applicants based on a normally distributed entrance exam with $$\mu = 800$$ and $$\sigma = 50. $$What is the minimum score required for admission?

Accepted Answer

Find $$z$$ for the top 10%: $$z = 1.28. $$Then $$x = 800 + 1.28 	imes 50 = 864.$$

Question 7

The time between charges for a cell phone battery is normally distributed with $$\sigma = 1.5$$ hours. If only 5% of batteries should require charging in 13 hours or less, what must the mean time between charges be?

Accepted Answer

Use $$z = -1.645$$, $$13 = \mu + (-1.645) 	imes 1.5 \implies \mu = 15.47$$ hours.

Question 8

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

Accepted Answer

$$f(x) = \frac{1}{b - a}$$ for $$a \leq x \leq b$$

Question 9

A dentist’s exam time is uniformly distributed between 15 and 20 minutes. What is the probability that an exam takes 17 minutes or less?

Accepted Answer

Probability $$= \frac{17 - 15}{20 - 15} = 0.4$$

Question 10

For a uniform distribution on $$[a, b]$$, what are the mean and standard deviation?

Accepted Answer

Mean: $$\frac{a + b}{2}$$, Standard deviation: $$\frac{b - a}{\sqrt{12}}$$

Question 11

Which of the following scenarios is best modeled by an exponential distribution?

Accepted Answer

The time between arrivals of customers at a bank drive-in teller window.

Question 12

The exponential probability density function is given by which formula?

Accepted Answer

$$f(x) = \frac{1}{\mu} e$$^{-x/\mu}$$ for x$$ \geq 0$$

Question 13

A 911 call system receives calls with a mean time between calls of 8 minutes (exponential distribution). What is the probability that the time between two calls exceeds 14 minutes?

Accepted Answer

Probability $$= e$$^{-14/8}$$ \approx 0.174$$

Question 14

Which of the following best describes the effect of increasing the standard deviation $$\sigma in a $$normal distribution?

Accepted Answer

The distribution becomes more spread out and flatter.

Question 15

If $$P(z  1.33)$$?

Accepted Answer

$$0.0918$$

Question 16

Which Excel function would you use to find the cumulative probability for a normal distribution?

Accepted Answer

NORM.DIST

Question 17

A continuous random variable $$X is $$uniformly distributed between $$a = 10$$ and $$b = 20. $$What is $$P(12 \leq X \leq 15)$$?

Accepted Answer

$$P = \frac{15 - 12}{20 - 10} = 0.3$$

Question 18

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

Accepted Answer

The total area under the curve is less than 1.

Introduction to Continuous Probability Distributions

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