## 6.1 The Standard Normal Distribution

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* = ____________.

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

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* ~ *N*(3, 5)

*σ* = _______

What does a *z*-score measure?

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

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* = 7, if it is 0.133 standard deviations to the left of the mean?

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

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

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

Suppose *X* ~ *N*(–1, 2). What is the *z*-score of *x* = 2?

Suppose *X* ~ *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?

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.

In a normal distribution, *x* = –2 and *z* = 6. This tells you that *x* = –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.

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?

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)?

Suppose *X* ~ *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* ~ *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).

Suppose *X* ~ *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 one standard deviation?

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

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 wearable fitness devices is normally distributed with mean of 4.1 years and a standard deviation of 1.3 years. A wearable fitness device is guaranteed for three years. We are interested in the length of time a wearable fitness device lasts.

*X* ~ _____(_____,_____)

## 6.2 Using the Normal Distribution

What is the area to the right of one?

How would you represent the area to the left of three in a probability statement?

If the area to the left of *x* in a normal distribution is 0.123, what is the area to the right of *x*?

If the area to the right of *x* in a normal distribution is 0.543, what is the area to the left of *x*?

*Use the following information to answer the next four exercises:*

*X* ~ *N*(54, 8)

Find the probability that *x* > 56.

Find the 80^{th} percentile.

*X* ~ *N*(6, 2)

Find the probability that *x* is between three and nine.

*X* ~ *N*(4, 5)

Find the maximum of *x* in the bottom quartile.

*Use the following information to answer the next three exercise:* The life of wearable fitness devices is normally distributed with a mean of 4.1 years and a standard deviation of 1.3 years. A wearable fitness device is guaranteed for three years. We are interested in the length of time a wearable fitness device lasts. Find the probability that a wearable fitness device will break down during the guarantee period.

- Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.
*P*(0 <*x*< ____________) = ___________ (Use zero for the minimum value of*x*.)

Find the probability that a wearable fitness device will last between 2.8 and six years.

- Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.
*P*(__________ <*x*< __________) = __________

Find the 70^{th} percentile of the distribution for the time a wearable fitness device lasts.

- Sketch the situation. Label and scale the axes. Shade the region corresponding to the lower 70%.
*P*(*x*<*k*) = __________ Therefore,*k*= _________