# 6.3E: Using the Normal Distribution (Exercises)

- Page ID
- 22214

Exercise \(\PageIndex{7}\)

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

**Answer**

\(P(x < 1)\)

Exercise \(\PageIndex{8}\)

Is \(P(x < 1)\) equal to \(P(x \leq 1)\)? Why?

**Answer**

Yes, because they are the same in a continuous distribution: \(P(x = 1) = 0\)

Exercise \(\PageIndex{9}\)

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

Exercise \(\PageIndex{10}\)

What is the area to the right of three?

**Answer**

\(1 – P(x < 3)\) or \(P(x > 3)\)

Exercise \(\PageIndex{11}\)

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

Exercise \(\PageIndex{12}\)

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

**Answer**

\(1 - 0.543 = 0.457\)

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

\(X \sim N(54, 8)\)

Exercise \(\PageIndex{13}\)

Find the probability that \(x > 56\).

Exercise \(\PageIndex{14}\)

Find the probability that \(x < 30\).

**Answer**

0.0013

Exercise \(\PageIndex{15}\)

Find the 80^{th} percentile.

Exercise \(\PageIndex{16}\)

Find the 60^{th} percentile.

**Answer**

56.03

Exercise \(\PageIndex{17}\)

\(X \sim N(6, 2)\)

Find the probability that \(x\) is between three and nine.

Exercise \(\PageIndex{18}\)

\(X \sim N(–3, 4)\)

Find the probability that \(x\) is between one and four.

**Answer**

0.1186

Exercise \(\PageIndex{19}\)

\(X \sim N(4, 5)\)

Find the maximum of \(x\) in the bottom quartile.

Exercise \(\PageIndex{20}\)

*Use the following information to answer the next three exercise:* The life of Sunshine CD players is normally distributed with a 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. Find the probability that a CD player 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 \(x\).)*Figure \(\PageIndex{12}\).*

**Answer**

- Check student’s solution.
- 3, 0.1979

Exercise \(\PageIndex{21}\)

Find the probability that a CD player will last between 2.8 and six years.

- Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.

\(P(\)__________ \(< x <\) __________\()\) = __________*Figure \(\PageIndex{13}\).*

Exercise \(\PageIndex{22}\)

Find the 70^{th} percentile of the distribution for the time a CD player lasts.

- Sketch the situation. Label and scale the axes. Shade the region corresponding to the lower 70%.

\(P(x < k) =\) __________ Therefore, \(k =\) _________*Figure \(\PageIndex{14}\).*

**Answer**

- Check student’s solution.
- 0.70, 4.78 years

## Contributors and Attributions

Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/30189442-699...b91b9de@18.114.