6.1E: The Standard Normal Distribution (Exercises)

Last updated

Nov 10, 2019
Save as PDF
- 6.2: The Standard Normal Distribution
- 6.3: Using the Normal Distribution

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

Exercise 6.2.7

A bottle of water contains 12.05 fluid ounces with a standard deviation of 0.01 ounces. Define the random variable $X$ in words. $X =$ ____________.

Answer

ounces of water in a bottle

Exercise 6.2.8

A normal distribution has a mean of 61 and a standard deviation of 15. What is the median?

Exercise 6.2.9

$X \sim N(1, 2)$

$\sigma =$ _______

Answer

Exercise 6.2.10

A company manufactures rubber balls. The mean diameter of a ball is 12 cm with a standard deviation of 0.2 cm. Define the random variable $X$ in words. $X =$ ______________.

Exercise 6.2.11

$X \sim N(-4, 1)$

What is the median?

Answer

–4

Exercise 6.2.12

$X \sim N(3, 5)$

$\sigma =$ _______

Exercise 6.2.13

$X \sim N(-2, 1)$

$\mu =$ _______

Answer

–2

Exercise 6.2.14

What does a $z$ -score measure?

Exercise 6.2.15

What does standardizing a normal distribution do to the mean?

Answer

The mean becomes zero.

Exercise 6.2.16

Is $X \sim N(0, 1)$ a standardized normal distribution? Why or why not?

Exercise 6.2.17

What is the $z$ -score of $x = 12$ , if it is two standard deviations to the right of the mean?

Answer

$z = 2$

Exercise 6.2.18

What is the $z$ -score of $x = 9$ , if it is 1.5 standard deviations to the left of the mean?

Exercise 6.2.19

What is the $z$ -score of $x = -2$ , if it is 2.78 standard deviations to the right of the mean?

Answer

$z = 2.78$

Exercise 6.2.20

What is the $z$ -score of $x = 7$ , if it is 0.133 standard deviations to the left of the mean?

Exercise 6.2.21

Suppose $X \sim N(2, 6)$ . What value of x has a z-score of three?

Answer

$x = 20$

Exercise 6.2.22

Suppose $X \sim N(8, 1)$ . What value of $x$ has a $z$ -score of –2.25?

Exercise 6.2.23

Suppose $X \sim N(9, 5)$ . What value of $x$ has a $z$ -score of –0.5?

Answer

$x = 6.5$

Exercise 6.2.24

Suppose $X \sim N(2, 3)$ . What value of $x$ has a $z$ -score of –0.67?

Exercise 6.2.25

Suppose $X \sim N(4, 2)$ . What value of $x$ is 1.5 standard deviations to the left of the mean?

Answer

$x = 1$

Exercise 6.2.26

Suppose $X \sim N(4, 2)$ . What value of $x$ is two standard deviations to the right of the mean?

Exercise 6.2.27

Suppose $X \sim N(8, 9)$ . What value of $x$ is 0.67 standard deviations to the left of the mean?

Answer

$x = 1.97$

Exercise 6.2.28

Suppose $X \sim N(-1, 12)$ . What is the $z$ -score of $x = 2$ ?

Exercise 6.2.29

Suppose $X \sim N(12, 6)$ . What is the $z$ -score of $x = 2$ ?

Answer

$z = –1.67$

Exercise 6.2.30

Suppose $X \sim N(9, 3)$ . What is the $z$ -score of $x = 9$ ?

Exercise 6.2.31

Suppose a normal distribution has a mean of six and a standard deviation of 1.5. What is the $z$ -score of $x = 5.5$ ?

Answer

$z \approx –0.33$

Exercise 6.2.32

In a normal distribution, $x = 5$ and $z = –1.25$ . This tells you that $x = 5$ is ____ standard deviations to the ____ (right or left) of the mean.

Exercise 6.2.33

In a normal distribution, $x = 3$ and $z = 0.67$ . This tells you that $x = 3$ is ____ standard deviations to the ____ (right or left) of the mean.

Answer

0.67, right

Exercise 6.2.34

In a normal distribution, $x = –2$ and $z = 6$ . This tells you that $z = –2$ is ____ standard deviations to the ____ (right or left) of the mean.

Exercise 6.2.35

In a normal distribution, $x = –5$ and $z = –3.14$ . This tells you that $x = –5$ is ____ standard deviations to the ____ (right or left) of the mean.

Answer

3.14, left

Exercise 6.2.36

In a normal distribution, $x = 6$ and $z = –1.7$ . This tells you that $x = 6$ is ____ standard deviations to the ____ (right or left) of the mean.

Exercise 6.2.37

About what percent of $x$ values from a normal distribution lie within one standard deviation (left and right) of the mean of that distribution?

Answer

about 68%

Exercise 6.2.38

About what percent of the $x$ values from a normal distribution lie within two standard deviations (left and right) of the mean of that distribution?

Exercise 6.2.39

About what percent of $x$ values lie between the second and third standard deviations (both sides)?

Answer

about 4%

Exercise 6.2.40

Suppose $X \sim N(15, 3)$ . Between what $x$ values does 68.27% of the data lie? The range of $x$ values is centered at the mean of the distribution (i.e., 15).

Exercise 6.2.41

Suppose $X \sim N(-3, 1)$ . Between what $x$ values does 95.45% of the data lie? The range of $x$ values is centered at the mean of the distribution (i.e., –3).

Answer

between –5 and –1

Exercise 6.2.42

Suppose $X \sim N(-3, 1)$ . Between what $x$ values does 34.14% of the data lie?

Exercise 6.2.43

About what percent of $x$ values lie between the mean and three standard deviations?

Answer

about 50%

Exercise 6.2.44

About what percent of $x$ values lie between the mean and one standard deviation?

Exercise 6.2.45

About what percent of $x$ values lie between the first and second standard deviations from the mean (both sides)?

Answer

about 27%

Exercise 6.2.46

About what percent of $x$ values lie between the first and third standard deviations(both sides)?

Use the following information to answer the next two exercises: The life of Sunshine CD players is normally distributed with mean of 4.1 years and a standard deviation of 1.3 years. A CD player is guaranteed for three years. We are interested in the length of time a CD player lasts.

Exercise 6.2.47

Define the random variable $X$ in words. $X =$ _______________.

Answer

The lifetime of a Sunshine CD player measured in years.

Exercise 6.2.48

$X \sim$ _____(_____,_____)

Search

Text Color

Text Size

Margin Size

Font Type

Support Center

How can we help?