# 6.2E: The Standard Normal Distribution (Exercises)

- Page ID
- 4683

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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

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

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

\(\sigma =\) _______

**Answer**

2

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. \(X =\) ______________.

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

What is the median?

**Answer**

–4

\(X \sim N(3, 5)\)

\(\sigma =\) _______

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

\(\mu =\) _______

**Answer**

–2

What does a \(z\)-score measure?

What does standardizing a normal distribution do to the mean?

**Answer**

The mean becomes zero.

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

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

**Answer**

\(z = 2\)

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

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

**Answer**

\(z = 2.78\)

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

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

**Answer**

\(x = 20\)

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

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

**Answer**

\(x = 6.5\)

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

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

**Answer**

\(x = 1\)

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

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

**Answer**

\(x = 1.97\)

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

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

**Answer**

\(z = –1.67\)

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

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**

\(z \approx –0.33\)

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.

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

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.

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

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.

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%

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?

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

**Answer**

about 4%

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).

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

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

About what percent of \(x\) values lie between the mean and three standard deviations?

**Answer**

about 50%

About what percent of \(x\) values lie between the mean and one standard deviation?

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

**Answer**

about 27%

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.

Define the random variable \(X\) in words. \(X =\) _______________.

**Answer**

The lifetime of a Sunshine CD player measured in years.

\(X \sim\) _____(_____,_____)