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Math Help - Statistics Problem

  1. #1
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    Statistics Problem

    A car fleet of a delivery company consists of 145 cars.

    Data recorded during the last month shows that the mean fuel consumption of cars is 6 liters per 100 km with standard deviation 1.4 liters and exhibit Uniform Distribution.

    Assume this characteristic to be the population characteristics and they
    do not change in time, and the fuel consumption of cars is stable. If 49 cars have been randomly selected for delivery today.....

    a) what is the distribution of the average fuel consumption of the selected cars?

    b) what is the probability that the average fuel consumption of the selected cars is less than 5.8 liters/100km ?

    c) which value of average fuel consumption is such that in only 5% cases we get average fuel consumption of selected cars higher than this value?

    d) At the end of day you have observed the average fuel consumption of the selected cars to be 7 liters,
    what you can claim about the characteristics (mean and standard deviation) of your car fleet?

    Any help you can provide with this problem would be greatly appreciated!
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  2. #2
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    Quote Originally Posted by tikimasala View Post
    A car fleet of a delivery company consists of 145 cars.

    Data recorded during the last month shows that the mean fuel consumption of cars is 6 liters per 100 km with standard deviation 1.4 liters and exhibit Uniform Distribution.

    Assume this characteristic to be the population characteristics and they
    do not change in time, and the fuel consumption of cars is stable. If 49 cars have been randomly selected for delivery today.....

    a) what is the distribution of the average fuel consumption of the selected cars?

    b) what is the probability that the average fuel consumption of the selected cars is less than 5.8 liters/100km ?

    c) which value of average fuel consumption is such that in only 5% cases we get average fuel consumption of selected cars higher than this value?

    d) At the end of day you have observed the average fuel consumption of the selected cars to be 7 liters,
    what you can claim about the characteristics (mean and standard deviation) of your car fleet?

    Any help you can provide with this problem would be greatly appreciated!
    X ~ U(a, b) where

    \frac{a + b}{2} = 6 \Rightarrow a + b = 12 .... (1)

    \frac{(b - a)^2}{12} = 1.4 \Rightarrow (b - a)^2 = 16.8 .... (2)

    Solve equations (1) and (2) for a and b to get the parameters of the distribution.
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  3. #3
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    Quote Originally Posted by tikimasala View Post
    A car fleet of a delivery company consists of 145 cars.

    Data recorded during the last month shows that the mean fuel consumption of cars is 6 liters per 100 km with standard deviation 1.4 liters and exhibit Uniform Distribution.

    Assume this characteristic to be the population characteristics and they
    do not change in time, and the fuel consumption of cars is stable. If 49 cars have been randomly selected for delivery today.....

    a) what is the distribution of the average fuel consumption of the selected cars?

    b) what is the probability that the average fuel consumption of the selected cars is less than 5.8 liters/100km ?

    c) which value of average fuel consumption is such that in only 5% cases we get average fuel consumption of selected cars higher than this value?

    d) At the end of day you have observed the average fuel consumption of the selected cars to be 7 liters,
    what you can claim about the characteristics (mean and standard deviation) of your car fleet?

    Any help you can provide with this problem would be greatly appreciated!
    a) Assuming the fuel consumptions of the cars are independent, you know the average fuel consumption (per 100 km) of the selected cars is 6 liters and the standard deviation of the average is 1.4 / \sqrt{49}.

    By the Central Limit Theorem, the average fuel consumption has an approximately Normal distribution with these parameters. You can use this fact to answer the rest of the questions.

    Notice that we did not need to use the fact that the individual cars' fuel consumptions have a Uniform distribution.
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    Quote Originally Posted by awkward View Post
    a) Assuming the fuel consumptions of the cars are independent, you know the average fuel consumption (per 100 km) of the selected cars is 6 liters and the standard deviation of the average is 1.4 / \sqrt{49}.

    By the Central Limit Theorem, the average fuel consumption has an approximately Normal distribution with these parameters. You can use this fact to answer the rest of the questions.

    Notice that we did not need to use the fact that the individual cars' fuel consumptions have a Uniform distribution.
    It's quite incredible in hindsight. I interpreted "what is the distribution of the average fuel consumption of the selected cars" as getting the distribution of the average for a car in the sample, rather than the distribution of the average of the sample .... Go figure!
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  5. #5
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    so how do i find it? thanks a lot
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  6. #6
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    Quote Originally Posted by tikimasala View Post
    so how do i find it? thanks a lot
    Read post #2 and #3 and decide which interpretation is correct (probably #3, in hindsight). If you need more help, please say what you need clarified and where you're still stuck.
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