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5.2 Using the normal distribution -- rrc math1020  (Page 2/24)

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The golf scores for a school team were normally distributed with a mean of 68 and a standard deviation of three.

Find the probability that a randomly selected golfer scored less than 65.

A personal computer is used for office work at home, research, communication, personal finances, education, entertainment, social networking, and a myriad of other things. Suppose that the average number of hours a household personal computer is used for entertainment is two hours per day. Assume the times for entertainment are normally distributed and the standard deviation for the times is half an hour.

a. Find the probability that a household personal computer is used for entertainment between 1.8 and 2.75 hours per day.

a. Let X = the amount of time (in hours) a household personal computer is used for entertainment. X ~ N (2, 0.5) where μ = 2 and σ = 0.5.

Find P (1.8< x <2.75).

The probability for which you are looking is the area between x = 1.8 and x = 2.75. P (1.8< x <2.75) = 0.5886

This is a normal distribution curve. The peak of the curve coincides with the point 2 on the horizontal axis. The values 1.8 and 2.75 are also labeled on the x-axis. Vertical lines extend from 1.8 and 2.75 to the curve. The area between the lines is shaded.

The probability that a household personal computer is used between 1.8 and 2.75 hours per day for entertainment is 0.5886.

b. Find the maximum number of hours per day that the bottom quartile of households uses a personal computer for entertainment.

b. To find the maximum number of hours per day that the bottom quartile of households uses a personal computer for entertainment, find the 25 th percentile, k , where P ( x < k ) = 0.25.

This is a normal distribution curve. The area under the left tail of the curve is shaded. The shaded area shows that the probability that x is less than k is 0.25. It follows that k = 1.67.

The maximum number of hours per day that the bottom quartile of households uses a personal computer for entertainment is 1.66 hours.

Try it

The golf scores for a school team were normally distributed with a mean of 68 and a standard deviation of three. Find the probability that a golfer scored between 66 and 70.

0.4950

There are approximately one billion smartphone users in the world today. In the United States the ages 13 to 55+ of smartphone users approximately follow a normal distribution with approximate mean and standard deviation of 36.9 years and 13.9 years, respectively.

a. Determine the probability that a random smartphone user in the age range 13 to 55+ is between 23 and 64.7 years old.

a. 0.8186

b. Determine the probability that a randomly selected smartphone user in the age range 13 to 55+ is at most 50.8 years old.

b. 0.8413

c. Find the 80 th percentile of this distribution, and interpret it in a complete sentence.

c.

  • The 80 th percentile is 48.6 years.
  • 80% of the smartphone users in the age range 13 – 55+ are 48.6 years old or less.

Try it

Use the information in [link] to answer the following questions.

  1. Find the 30 th percentile, and interpret it in a complete sentence.
  2. What is the probability that the age of a randomly selected smartphone user in the range 13 to 55+ is less than 27 years old.

Let X = a smart phone user whose age is 13 to 55+. X ~ N (36.9, 13.9)

  1. To find the 30 th percentile, find k such that P ( x < k ) = 0.30.
    29.6 years
    Thirty percent of smartphone users 13 to 55+ are at most 29.6 years and 70% are at least 29.6 years.
  2. Find P ( x <27)
    This is a normal distribution curve. The peak of the curve coincides with the point 36.9 on the horizontal axis. The point 27 is also labeled. A vertical line extends from 27 to the curve. The area under the curve to the left of 27 is shaded. The shaded area shows that P(x < 27) = 0.2342.

There are approximately one billion smartphone users in the world today. In the United States the ages 13 to 55+ of smartphone users approximately follow a normal distribution with approximate mean and standard deviation of 36.9 years and 13.9 years respectively. Using this information, answer the following questions (round answers to one decimal place).

a. Calculate the interquartile range ( IQR ).

a.

  • IQR = Q 3 – Q 1
  • Calculate Q 3 = 75 th percentile and Q 1 = 25 th percentile.
  • Q 3 = 46.2754
  • Q 1 = 27.5246
  • IQR = Q 3 – Q 1 = 18.7508

b. Forty percent of the ages that range from 13 to 55+ are at least what age?

b.

  • Find k where P ( x > k ) = 0.40 ("At least" translates to "greater than or equal to.")
  • 0.40 = the area to the right.
  • Area to the left = 1 – 0.40 = 0.60.
  • The area to the left of k = 0.60.
  • k = 40.42.
  • Forty percent of the ages that range from 13 to 55+ are at least 40.42 years.
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Source:  OpenStax, Statistics i - math1020 - red river college - version 2015 revision a - draft 2015-10-24. OpenStax CNX. Oct 24, 2015 Download for free at http://legacy.cnx.org/content/col11891/1.8
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