Proportion Given only Standard Deviation and Mean

• Nov 14th 2012, 08:22 PM
lpg0005
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. • Nov 15th 2012, 01:10 AM chiro 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). • Nov 15th 2012, 01:17 AM a tutor 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

• Nov 15th 2012, 08:33 PM
lpg0005
Re: Proportion Given only Standard Deviation and Mean
Awesome. Thank you so much. You were a ton of help with this problem!
• Nov 15th 2012, 08:54 PM
lpg0005
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?
• Nov 15th 2012, 10:25 PM
chiro
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.