The amount of time that a ten year old boy plays video games in a week is normal

Question

The amount of time that a ten year old boy plays video games in a week is normally distributed with a mean of 10 hours and a standard deviation of 4 hours. (a) Suppose 9 ten-year-old boys are chosen. What is the probability that the sample mean time for playing video games per week is between 8 and 12 hours? (b) Suppose a boy is considered addicted if he plays computer games for more than 16 hours a week. If 12 ten year old boys are to be chosen at random, find the probability that at least one of them are addicted to video games.

Explanation / Answer

THE MEAN = 10

STANDARD DEVIATION = 4

THE DISTRIBUTION ID NORMAL

THERFORE THE FORMULA TO BE USED = Z = (X-MEAN)/STANDARD DEVIATION

A) N = 9

For x = 8 , z = (8 - 10) / 4/SQRT(9) = - 1.5 and for x = 12, z = (12 - 10) / 4/SQRT(9) = 1.5

Hence P(8 < x < 10) = P(-1.5 < z < 1.5) = [area to the left of z = 1.5] - [area to the left of -1.5]

= 0.9332 - 0.0668 = 0.8664

B) N = 12

P(X>16) =

For x = 16, z = (16 - 10) / 4/SQRT(12) = 5.19

Hence P(x > 16) = P(z > 5.19) = [total area] - [area to the left of 5.19]

1 - [area to the left of 5.19]

now from the z table we will take the value of z score = 0.999

    = 1 - 0.999 = 0.0001

ATLEAST 1 = 1 - P(0)

P(0) = 0.999^12 = 0.9880

P(ATLEAST 1) = 1 -0.9880 = 0.0120

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