Fundamentals of Statistics

Michael Sullivan III

Chapter 7

The Normal Probability Distribution - all with Video Answers

Educators

Section 1

Properties of the Normal Distribution

Problem 1

State the two characteristics of the graph of a probability density function,

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Problem 2

To find the probabilities for continuous random variables, we do not use probability _______ functions, but instead we use probability ______ functions.

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Problem 3

Provide two interpretations of the area under the graph of a probability density function.

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Problem 4

Why do we standardize normal random variables to find the area under any normal curve?

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Problem 5

The points at $x=$ _____ and $x=$ ______ are the inflection points on the normal curve.

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Problem 6

As $\sigma$ increases, the normal density curve becomes more spread out. Knowing the area under the density curve must be $1,$ what effect does increasing $\sigma$ have on the height of the curve?

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Problem 7

Determine whether the graph can represent a normal density function. If it cannot, explain why.
(FIGURE CANNOT COPY)

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Problem 8

Determine whether the graph can represent a normal density function. If it cannot, explain why.
(FIGURE CANNOT COPY)

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Problem 9

Determine whether the graph can represent a normal density function. If it cannot, explain why.
(FIGURE CANNOT COPY)

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Problem 10

Determine whether the graph can represent a normal density function. If it cannot, explain why.
(FIGURE CANNOT COPY)

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Problem 11

Determine whether the graph can represent a normal density function. If it cannot, explain why.
(FIGURE CANNOT COPY)

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Problem 12

Determine whether the graph can represent a normal density function. If it cannot, explain why.
(FIGURE CANNOT COPY)

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Problem 13

Use the information presented in Examples 1 and 2 .
Find the probability that your friend is between 5 and 10 minutes late.

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Problem 14

Use the information presented in Examples 1 and 2 .
Find the probability that your friend is between 15 and 25 minutes late.

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Problem 15

Use the information presented in Examples 1 and 2 .
Find the probability that your friend is at least 20 minutes late.

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Problem 16

Use the information presented in Examples 1 and 2 .
Find the probability that your friend is no more than 5 minutes late.

Harsh Gadhiya

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Problem 17

The random-number generator on calculators randomly generates a number between 0 and $1 .$ The random variable $X,$ the number generated, follows a uniform probability distribution.
(a) Draw the graph of the uniform density function.
(b) What is the probability of generating a number between 0 and $0.2 ?$
(c) What is the probability of generating a number between 0.25 and $0.6 ?$
(d) What is the probability of generating a number greater than $0.95 ?$
(e) Use your calculator or statistical software to randomIy generate 200 numbers between 0 and $1 .$ What proportion of the numbers are between 0 and $0.2 ?$ Compare the result with part (b).

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Problem 18

Suppose the reaction time $X$ (in minutes) of a certain chemical process follows a uniform probability distribution with $5 \leq X \leq 10$
(a) Draw the graph of the density curve.
(b) What is the probability that the reaction time is between 6 and 8 minutes?
(c) What is the probability that the reaction time is between 5 and 8 minutes?
(d) What is the probability that the reaction time is less than 6 minutes?

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Problem 19

Determine whether or not the histogram indicates that a normal distribution could be used as a model for the variable.
The following relative frequency histogram represents the birth weights (in grams) of babies whose term was 36 weeks. (GRAPH CANNOT COPY)

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Problem 20

Determine whether or not the histogram indicates that a normal distribution could be used as a model for the variable.
The following relative frequency histogram represents the waiting time in line (in minutes) for the Demon Roller Coaster for 2000 randomly selected people on a Saturday afternoon in the summer. (GRAPH CANNOT COPY)

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Problem 21

Determine whether or not the histogram indicates that a normal distribution could be used as a model for the variable.
The following relative frequency histogram represents the length of phone calls on my wife's cell phone during the month of September. (GRAPH CANNOT COPY)

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Problem 22

Determine whether or not the histogram indicates that a normal distribution could be used as a model for the variable.
The following relative frequency histogram represents the incubation times of a random sample of Rhode Island Red Hens' eggs. (GRAPH CANNOT COPY)

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Problem 23

One graph in the following figure represents a normal distribution with mean $\mu=10$ and standard deviation $\sigma=3$ The other graph represents a normal distribution with mean $\mu=10$ and standard deviation $\sigma=2 .$ Determine which graph is which and explain how you know. (GRAPH CANNOT COPY)

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Problem 24

One graph in the following figure represents a normal distribution with mean $\mu=8$ and standard deviation $\sigma=2$ The other graph represents a normal distribution with mean $\mu=14$ and standard deviation $\sigma=2 .$ Determine which graph is which and explain how you know. (GRAPH CANNOT COPY)

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Problem 25

The graph of a normal curve is given. Use the graph to identify the value of $\mu$ and $\sigma .$ (GRAPH CANNOT COPY)

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Problem 26

The graph of a normal curve is given. Use the graph to identify the value of $\mu$ and $\sigma .$ (GRAPH CANNOT COPY)

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Problem 27

The graph of a normal curve is given. Use the graph to identify the value of $\mu$ and $\sigma .$ (GRAPH CANNOT COPY)

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Problem 28

The graph of a normal curve is given. Use the graph to identify the value of $\mu$ and $\sigma .$ (GRAPH CANNOT COPY)

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Problem 29

Suppose the monthly charge for cell phone plans in the United States is normally distributed with mean $\mu=\$ 62$ and standard deviation $\sigma=\$ 18 .$ (Source: Based on information obtained from Consumer Reports)
(a) Draw a normal curve with the parameters labeled.
(b) Shade the region that represents the proportion of plans that charge less than $\$ 44$
(c) Suppose the area under the normal curve to the left of $X=\$ 44$ is $0.1587 .$ Provide two interpretations of this result.

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Problem 30

Suppose the life of refrigerators is normally distributed with mean $\mu=14$ years and standard deviation $\sigma=2.5$ years. (Source: Based on information obtained from Consumer Reports)
(a) Draw a normal curve with the parameters labeled.
(b) Shade the region that represents the proportion of refrigerators that are kept for more than 17 years.
(c) Suppose the area under the normal curve to the right of $X=17$ is $0.1151 .$ Provide two interpretations of this result.

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Problem 31

The birth weights of full-term babies are normally distributed with mean $\mu=3400$ grams and $\sigma=505$ grams. (Source: Based on data obtained from the National Vital Statistics Report, Vol. 48, No.3)
(a) Draw a normal curve with the parameters labeled.
(b) Shade the region that represents the proportion of full-term babies who weigh more than 4410 grams.
(c) Suppose the area under the normal curve to the right of $X=4410$ is $0.0228 .$ Provide two interpretations of this result.

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Problem 32

The heights of 10 -year-old males are normally distributed with mean $\mu=55.9$ inches and $\sigma=5.7$ inches.
(a) Draw a normal curve with the parameters labeled.
(b) Shade the region that represents the proportion of $10-$ year-old males who are less than 46.5 inches tall.
(c) Suppose the area under the normal curve to the left of $X=46.5$ is $0.0496 .$ Provide two interpretations of this result.

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Problem 33

The lengths of human pregnancy are normally distributed with $\mu=266$ days and $\sigma=16$ days.
(a) The following figure represents the normal curve with $\mu=266$ days and $\sigma=16$ days. The area to the right of $X=280$ is $0.1908 .$ Provide two interpretations of this area. (GRAPH CANNOT COPY)(b) The following figure represents the normal curve with $\mu=266$ days and $\sigma=16$ days. The area between $X=230$ and $X=260$ is $0.3416 .$ Provide two interpretations of this area. (GRAPH CANNOT COPY)

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Problem 34

Elena conducts an experiment in which she fills up the gas tank on her Toyota Camry 40 times and records the miles per gallon for each fill-up. A histogram of the miles per gallon indicates that a normal distribution with mean of 24.6 miles per gallon and a standard deviation of 3.2 miles per gallon could be used to model the gas mileage for her car.
(a) The following figure represents the normal curve with $\mu=24.6$ miles per gallon and $\sigma=3.2$ miles per gallon. The area under the curve to the right of $X=26$ is $0.3309 .$ Provide two interpretations of this area. (GRAPH CANNOT COPY)
(b) The following figure represents the normal curve with $\mu=24.6$ miles per gallon and $\sigma=3.2$ miles per gallon. The area under the curve between $X=18$ and $X=21$ is 0.1107. Provide two interpretations of this area. (GRAPH CANNOT COPY)

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Problem 35

A random variable $X$ is normally distributed with $\mu=10$ and $\sigma=3$
(a) Compute $Z_{1}=\frac{X_{1}-\mu}{\sigma}$ for $X_{1}=8$
(b) Compute $Z_{2}=\frac{X_{2}-\mu}{\sigma}$ for $X_{2}=12$
(c) The area under the normal curve between $X_{1}=8$ and $X_{2}=12$ is $0.495 .$ What is the area between $Z_{1}$ and $Z_{2} ?$

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Problem 36

A random variable $X$ is normally distributed with $\mu=25$ and $\sigma=6$
(a) Compute $Z_{1}=\frac{X_{1}-\mu}{\sigma}$ for $X_{1}=18$
(b) Compute $Z_{2}=\frac{X_{2}-\mu}{\sigma}$ for $X_{2}=30$
(c) The area under the normal curve between $X_{1}=18$ and $X_{2}=30$ is $0.6760 .$ What is the area between$Z_{1}$and $Z_{2} ?$

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Problem 37

In the game of golf, distance control is just as important as how far a player hits the ball. Suppose Michael went to the driving range with his range finder and hit 75 golf balls with his pitching wedge and measured the distance each ball traveled (in yards). He obtained the following data: (TABLE CANNOT COPY)
(a) Use MINITAB or some other statistical software to construct a relative frequency histogram. Comment on the shape of the distribution.
(b) Use MINITAB or some other statistical software to draw the normal density function on the relative frequency histogram.
(c) Do you think the normal density function accurately describes the distance Michael hits a pitching wedge? Why?

Jon Southam

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Problem 38

The following frequency distribution represents the heights (in inches) of eighty randomly selected five-year-old females. (TABLE CANNOT COPY)
(a) Use MINITAB or some other statistical software to construct a relative frequency histogram. Comment on the shape of the distribution.
(b) Use MINITAB or some other statistical software to draw the normal density function on the relative frequency histogram.
(c) Do you think the normal density function accurately describes the heights of 5 -year-old females? Why?

Jon Southam

Numerade Educator