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Math Help - Proportion Given only Standard Deviation and Mean

  1. #1
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    Proportion Given only Standard Deviation and Mean

    Regular-grade gasoline prices for all the gas stations in Lee County have a normal distribution with a mean of $2.95 and
    standard deviation of $0.16. Use this information and the standard Normal distribution to answer the next two questions.

    What proportion of gas stations will charge more than $3.15 per gallon?

    A manager of a discount gas station wants to set the price for gasoline at his station where only 10% of the stations in
    Lee County will charge less than his price? What price should he charge for gasoline at his station?



    I have been staring at this question for way too long. I cannot figure out the correct answer to this because I only know how to do it when I am given a population.
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  2. #2
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    Re: Proportion Given only Standard Deviation and Mean

    Hey lpg0005.

    To start you off if X is Normal with mean mu and variance sigma^2 then Z = [X - mu]/sigma where Z is Normal(0,1) and probability tables can be used to get probabilities for P(Z < z) for various values of z.

    So if you want to find P(X < x) then remember that this is the same as finding P(Z < (x-mu)/sigma) where Z is Normal (0,1).
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  3. #3
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    Re: Proportion Given only Standard Deviation and Mean

    How many standard deviations above the mean is $3.15?

    \frac{315-295}{16}=1.25

    You can now use the standard normal distribution.

    Z~N(0,1)

    You need P(Z>1.25). This is equal to 1 - P(Z<1.25). This can be found in a suitable table or maybe on your calculator.

    P(Z<1.25)=0.8944

    So your answer is 1 - 0.8944 = 0.1056.
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  4. #4
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    Re: Proportion Given only Standard Deviation and Mean

    Awesome. Thank you so much. You were a ton of help with this problem!
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  5. #5
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    Re: Proportion Given only Standard Deviation and Mean

    But can you help me with the second part of the question?

    A manager of a discount gas station wants to set the price for gasoline at his station where only 10% of the stations in
    Lee County will charge less than his price? What price should he charge for gasoline at his station?
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  6. #6
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    Re: Proportion Given only Standard Deviation and Mean

    If you Have P(X < x) = 0.1 and you know the distribution of X, then you can solve for x using the cumulative distribution.

    If X is normal then convert to a standard normal and look up the appropriate value for x* = (x-mu)/sigma and re-arrange to solve for x.
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