# 6.E: The Standard Normal Distribution (Optional Exercises)

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Exercise \(\PageIndex{7}\)

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 \(\PageIndex{8}\)

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

Exercise \(\PageIndex{9}\)

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

\(\sigma =\) _______

**Answer**

2

Exercise \(\PageIndex{10}\)

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 \(\PageIndex{11}\)

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

What is the median?

**Answer**

–4

Exercise \(\PageIndex{12}\)

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

\(\sigma =\) _______

Exercise \(\PageIndex{13}\)

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

\(\mu =\) _______

**Answer**

–2

Exercise \(\PageIndex{14}\)

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

Exercise \(\PageIndex{15}\)

What does standardizing a normal distribution do to the mean?

**Answer**

The mean becomes zero.

Exercise \(\PageIndex{16}\)

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

Exercise \(\PageIndex{17}\)

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

**Answer**

\(z = 2\)

Exercise \(\PageIndex{18}\)

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

Exercise \(\PageIndex{19}\)

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 \(\PageIndex{20}\)

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

Exercise \(\PageIndex{21}\)

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

**Answer**

\(x = 20\)

Exercise \(\PageIndex{22}\)

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

Exercise \(\PageIndex{23}\)

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

**Answer**

\(x = 6.5\)

Exercise \(\PageIndex{24}\)

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

Exercise \(\PageIndex{25}\)

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 \(\PageIndex{26}\)

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

Exercise \(\PageIndex{27}\)

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 \(\PageIndex{28}\)

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

Exercise \(\PageIndex{29}\)

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

**Answer**

\(z = –1.67\)

Exercise \(\PageIndex{30}\)

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

Exercise \(\PageIndex{31}\)

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 \(\PageIndex{32}\)

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 \(\PageIndex{33}\)

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 \(\PageIndex{34}\)

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 \(\PageIndex{35}\)

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 \(\PageIndex{36}\)

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 \(\PageIndex{37}\)

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 \(\PageIndex{38}\)

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 \(\PageIndex{39}\)

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

**Answer**

about 4%

Exercise \(\PageIndex{40}\)

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 \(\PageIndex{41}\)

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 \(\PageIndex{42}\)

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

Exercise \(\PageIndex{43}\)

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

**Answer**

about 50%

Exercise \(\PageIndex{44}\)

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

Exercise \(\PageIndex{45}\)

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

**Answer**

about 27%

Exercise \(\PageIndex{46}\)

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 \(\PageIndex{47}\)

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

**Answer**

The lifetime of a Sunshine CD player measured in years.

Exercise \(\PageIndex{48}\)

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