Essentials of Statistics

Mario F. Triola

Chapter 6

Normal Probability Distributions - all with Video Answers

Educators

Section 1

The Standard Normal Distribution

Problem 1

What's wrong with the following statement? "Because the digits 0,1 , $2, \ldots, 9$ are the normal results from lottery drawings, such randomly selected numbers have a normal distribution."

Demi Nelson

Demi Nelson

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Problem 2

A normal distribution is informally described as a probability distribution that is bell-shaped when graphed. Draw a rough sketch of a curve having the bell shape that is characteristic of a normal distribution.

Trinity Steen

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Problem 3

Identify the two requirements necessary for a normal distribution to be a standard normal distribution.

Trinity Steen

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Problem 4

What does the notation $z_{\alpha}$ indicate?

Trinity Steen

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Problem 5

Refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.
Greater than $3.00$ minutes

Trinity Steen

Numerade Educator

Problem 6

Refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.
Less than $4.00$ minutes

Trinity Steen

Numerade Educator

Problem 7

Refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.
Between 2 minutes and 3 minutes

Trinity Steen

Numerade Educator

Problem 8

Refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.
Between $2.5$ minutes and $4.5$ minutes

Trinity Steen

Numerade Educator

Problem 9

Find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation $1 .$
$$
z=0.44
$$

Carly Stoner

Numerade Educator

Problem 10

Find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation $1 .$
$$
z=-1.04
$$

Carly Stoner

Numerade Educator

Problem 11

Find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation $1 .$
$$
z=-0.84 \quad z=1.28
$$

Carly Stoner

Numerade Educator

Problem 12

Find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation $1 .$
$$
z=-1.07 \quad z=0.67
$$

Carly Stoner

Numerade Educator

Problem 13

Find the indicated $z$ score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation $1 .$
$$
0.8907
$$

Carly Stoner

Numerade Educator

Problem 14

Find the indicated $z$ score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation $1 .$
$$
0.3050
$$

Carly Stoner

Numerade Educator

Problem 15

Find the indicated $z$ score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation $1 .$
$$
0.9265
$$

Carly Stoner

Numerade Educator

Problem 16

Find the indicated $z$ score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation $1 .$
$$
0.2061
$$

Carly Stoner

Numerade Educator

Problem 17

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Less than }-1.23
$$

Sanchit Jain

Numerade Educator

Problem 18

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Less than }-1.96
$$

Trinity Steen

Numerade Educator

Problem 19

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Less than } 1.28
$$

Trinity Steen

Numerade Educator

Problem 20

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Less than } 2.56
$$

Trinity Steen

Numerade Educator

Problem 21

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Greater than } 0.25
$$

Trinity Steen

Numerade Educator

Problem 22

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Greater than } 0.18
$$

Trinity Steen

Numerade Educator

Problem 23

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Greater than }-2.00
$$

Trinity Steen

Numerade Educator

Problem 24

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Greater than }-3.05
$$

Trinity Steen

Numerade Educator

Problem 25

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Between } 2.00 \text { and } 3.00
$$

Trinity Steen

Numerade Educator

Problem 26

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Between } 1.50 \text { and } 2.50
$$

Trinity Steen

Numerade Educator

Problem 27

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Between and }-2.55 \text { and }-2.00
$$

Trinity Steen

Numerade Educator

Problem 28

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Between }-2.75 \text { and }-0.75
$$

Trinity Steen

Numerade Educator

Problem 29

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Between }-2.00 \text { and } 2.00
$$

Trinity Steen

Numerade Educator

Problem 30

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Between }-3.00 \text { and } 3.00
$$

Trinity Steen

Numerade Educator

Problem 31

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Between }-1.00 \text { and } 5.00
$$

Trinity Steen

Numerade Educator

Problem 32

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Between }-4.27 \text { and } 2.34
$$

Trinity Steen

Numerade Educator

Problem 33

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Less than } 4.55
$$

Trinity Steen

Numerade Educator

Problem 34

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Greater than }-3.75
$$

Trinity Steen

Numerade Educator

Problem 35

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Greater than } 0
$$

Trinity Steen

Numerade Educator

Problem 36

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.
$$
\text { Less than } 0 \text { . }
$$

Trinity Steen

Numerade Educator

Problem 37

Assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the bone density test score corresponding to the given information. Round results to two decimal places.
Find $P_{99}$, the 99 th percentile. This is the bone density score separating the bottom $99 \%$ from the top $1 \%$.

Trinity Steen

Numerade Educator

Problem 38

Assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the bone density test score corresponding to the given information. Round results to two decimal places.
Find $P_{10}$, the 10 th percentile. This is the bone density score separating the bottom $10 \%$ from the top $90 \%$.

Trinity Steen

Numerade Educator

Problem 39

Assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the bone density test score corresponding to the given information. Round results to two decimal places.
If bone density scores in the bottom $2 \%$ and the top $2 \%$ are used as cutoff points for levels that are too low or too high, find the two readings that are cutoff values.

Trinity Steen

Numerade Educator

Problem 40

Assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of $1 .$ In each case, draw a graph, then find the bone density test score corresponding to the given information. Round results to two decimal places.
Find the bone density scores that can be used as cutoff values separating the lowest $3 \%$ and highest $3 \%$.

Trinity Steen

Numerade Educator

Problem 41

Find the indicated critical value. Round results to two decimal places.
$$
z_{0.10}
$$

Trinity Steen

Numerade Educator

Problem 42

Find the indicated critical value. Round results to two decimal places.
$$
z_{0.02}
$$

Trinity Steen

Numerade Educator

Problem 43

Find the indicated critical value. Round results to two decimal places.
$$
z_{0.04}
$$

Trinity Steen

Numerade Educator

Problem 44

Find the indicated critical value. Round results to two decimal places.
$$
z_{0.15}
$$

Trinity Steen

Numerade Educator

Problem 45

Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2.
About __________$\%$ of the area is between $z=-1$ and $z=1$ (or within 1 standard deviation
of the mean).

Trinity Steen

Numerade Educator

Problem 46

Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2.
About ________ $\%$ of the area is between $z=-2$ and $z=2$ (or within 2 standard deviations of the mean).

Trinity Steen

Numerade Educator

Problem 47

Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2.
About _______ $\%$ of the area is between $z=-3$ and $z=3$ (or within 3 standard deviations of the mean).

Trinity Steen

Numerade Educator

Problem 48

Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2.
About $\quad \%$ of the area is between $z=-3.5$ and $z=3.5$ (or within $3.5$ standard deviations of the mean).

Trinity Steen

Numerade Educator

Problem 49

For bone density scores that are normally distributed with a mean of 0 and a standard deviation of 1 , find the percentage of scores that are
a. significantly high (or at least 2 standard deviations above the mean).
b. significantly low (or at least 2 standard deviations below the mean).
c. not significant (or less than 2 standard deviations away from the mean).

Carly Stoner

Numerade Educator

Problem 50

In a continuous uniform distribution,
$$
\mu=\frac{\text { minimum }+\text { maximum }}{2} \text { and } \sigma=\frac{\text { range }}{\sqrt{12}}
$$
a. Find the mean and standard deviation for the distribution of the waiting times represented in Figure $6-2$, which accompanies Exercises $5-8$.
b. For a continuous_uniform distribution with $\mu=0$ and $\sigma=1$, the minimum is $-\sqrt{3}$ and the maximum is $\sqrt{3}$. For this continuous uniform distribution, find the probability of randomly selecting a value between $-1$ and 1 , and compare it to the value that would be obtained by incorrectly treating the distribution as a standard normal distribution. Does the distribution affect the results very much?

KH

Khuzaima Hameed

Numerade Educator