6.3E: Using the Normal Distribution (Exercises)

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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.

Figure \(\PageIndex{12}\).
\(P(0 < x <\) ____________\() =\) ___________ (Use zero for the minimum value of \(x\).)

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.

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

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%.

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

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.