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# The Standard Normal Distribution (Exercises)


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

2

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

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

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