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6.3E: Using the Normal Distribution (Exercises)

  • Page ID
    22214
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    Exercise \(\PageIndex{7}\)

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

    alt
    Figure \(\PageIndex{8}\).

    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?

    alt
    Figure \(\PageIndex{10}\).

    Exercise \(\PageIndex{10}\)

    What is the area to the right of three?

    alt
    Figure \(\PageIndex{11}\).

    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 80th percentile.

    Exercise \(\PageIndex{16}\)

    Find the 60th 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.

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

      alt

      Figure \(\PageIndex{12}\).

      \(P(0 < x <\) ____________\() =\) ___________ (Use zero for the minimum value of \(x\).)

    Answer

    1. Check student’s solution.
    2. 3, 0.1979

    Exercise \(\PageIndex{21}\)

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

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

      alt

      Figure \(\PageIndex{13}\).

      \(P(\)__________ \(< x <\) __________\()\) = __________

    Exercise \(\PageIndex{22}\)

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

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

      alt

      Figure \(\PageIndex{14}\).

      \(P(x < k) =\) __________ Therefore, \(k =\) _________

    Answer

    1. Check student’s solution.
    2. 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.


    This page titled 6.3E: Using the Normal Distribution (Exercises) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.