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noti
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Tyro
Asked: 2021-04-11T14:46:35+00:00 In: ,

sampling distribution

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the average age of private vehicles in Marilao Bulacan is 6 years. Assume that the standard deviation is 2 years. A random sample of 10 private vehicles is chosen, find the probability that the mean of private vehicles ages is

 

a. between 3 to 9 years

b. over 6 years

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1 Answer

  2. acedstud
    This answer was edited.

    Here, we assume that the average age of private vehicles in Marilao Bulacan is normally distributed.

    Let \bar{X} be a random variable representing the average age of private vehicles in Marilao Bulacan from a sample of size n, then the probability that \bar{X} is less than a value \bar{x} is given by P(\bar{X}<\bar{x})=P\left(z<\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}\right), where \mu is the population mean and \sigma is the population standard deviation.

    Here, \mu=6, \sigma=2, and n=10.

    a.) Thus, the probability that the mean of private vehicles ages is between 3 to 9 years is given by P(3<\bar{X}<9)=P(\bar{X}<9)-P(\bar{X}<3)=P\left(z<\dfrac{9-6}{2/\sqrt{10}}\right)-P\left(z<\dfrac{3-6}{2/\sqrt{10}}\right)

    =P(z<4.74)-P(z<-4.74)=P(z<4.74-[1-P(z<4.74)]=2P(z<4.74)-1

    Using the table of areas under the standard normal curve or a calculator, we have =2(1)-1=1.

    Therefore, the probability that the mean of private vehicles ages is between 3 to 9 years is 1.

     

    b.) The probability that the mean of private vehicles ages is over 6 years is given by P(\bar{X}>1)=1-P(\bar{X}<6)=1-P\left(z<\dfrac{6-6}{2/\sqrt{10}}\right)=1-P(z<0)

    Using the table of areas under the standard normal curve or a calculator, we have =1-0.5=0.5.

    Therefore, the probability that the mean of private vehicles ages is over 6 years is 0.5.

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