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jeremymaestro
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Tyro
Asked: 2021-04-06T03:37:23+00:00 In: ,

solve the problem. ( statistics )

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A homework consists of 40 independent problems is given. On the average, it takes 5 minutes to solve a problem, with a standard deviation of 2 minutes. Find the probability that the homework will be completed in less than 3 hours.

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  2. acedstud

    Let Y be a random variable representing the amount of time it takes to solve a problem, then E(Y)=5 minutes and \sigma_Y=2 minutes.

    Let X be a random variable representing the amount of time it takes to solve 40 problems, the X=40Y.

    Using the laws of expected value and variance, E(cX)=cE(X) and Var(cX)=c^2Var(X), where c is a constant and Var(X)=\sigma_X^2, we have that E(X)=E(40Y)=40E(Y)=40\times5=200 minutes and \sigma_X=\sqrt{Var(X)}=\sqrt{Var(40Y)}=\sqrt{40^2\sigma_Y^2}=\sqrt{40^2\times2^2}=40\times2=80 minutes.

    The probability that a  normally distributed variable, X, with a mean, \mu, and standard deviation, \sigma, is less than a value, x is given by P(X<x)=P\left(z<\dfrac{x-\mu}{\sigma}\right), where z is the area under the standard normal curve.

    Given: \mu=E(X)=200 minutes, \sigma=80 minutes, 3 hours \equiv 180 minutes.

    Thus, P(X<180)=P\left(z<\dfrac{180-200}{80}\right)

    =P\left(z<\dfrac{-20}{80}\right)=P(z<-0.25)=1-P(z<0.25)

    Using the table of the area under the standard normal curve or a calculator, we have that P(z<0.25)=0.5987.

    Therefore, the probability that the home work will be completed in less than 3 hours is 1 – 0.5987 = 0.4013.

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