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

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. $$X =$$ ____________.

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 $$\PageIndex{9}$$

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

$$\sigma =$$ _______

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?

–4

Exercise $$\PageIndex{12}$$

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

$$\sigma =$$ _______

Exercise $$\PageIndex{13}$$

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

$$\mu =$$ _______

–2

Exercise $$\PageIndex{14}$$

What does a $$z$$-score measure?

Exercise $$\PageIndex{15}$$

What does standardizing a normal distribution do to the mean?

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?

$$z = 2$$

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?

$$z = 2.78$$

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?

$$x = 20$$

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?

$$x = 6.5$$

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?

$$x = 1$$

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?

$$x = 1.97$$

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 $$x = 2$$?

$$z = –1.67$$

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 $$x = 5.5$$?

$$z \approx –0.33$$

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.

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.

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?

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)?

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).

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?

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)?

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. $$X =$$ _______________.

Exercise $$\PageIndex{48}$$
$$X \sim$$ _____(_____,_____)