Certain transportation company has a fleet of 210 vehicles. The average age of the vehicles is 4.25 years, with a standard deviation of 18 m

Question

Certain transportation company has a fleet of 210 vehicles. The average age of the vehicles is 4.25 years, with a standard deviation of 18 months. In a random sample of 40 vehicles, what is the probability that the average age of vehicles in the sample will be less than 4 years

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Hailey 3 months 2021-10-13T16:58:05+00:00 1 Answer 0 views 0

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    2021-10-13T16:59:56+00:00

    Answer:

    z = \frac{4-4.25}{\frac{1.5}{\sqrt{40}}}= -1.054

    And we can find the following probability:

     P(z<-1.054) = 0.146

    And the last probability can be founded using  the normal standard distribution or excel.

    Step-by-step explanation:

    For this case we define the random variable X as the ages of vehicles. We know the following info for this variable:

    \bar X = 4.25 represent the mean

    \sigma =18/12=1.5 represent the deviation in years

    They select a sample size of n=40>30. And they want to find this probability:

     P(\bar X<40)

    Since the sample size is large enough we can use the central limit theorem and the distribution for the sample mean would be:

    \bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})

    We can use the z score formula given by:

     z = \frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}

    And if we find the z score for 4 we got:

    z = \frac{4-4.25}{\frac{1.5}{\sqrt{40}}}= -1.054

    And we can find the following probability:

     P(z<-1.054) = 0.146

    And the last probability can be founded using  the normal standard distribution or excel.

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