Compute the mean of a portion of a normal distribution

This is a tricky one for an old guy like me. Can you teach an old guy new tricks? It's been 20+ years since my last stats class.

I want to efficiently cut cheese. Lets say that I have a customer that wants to buy exact weight pieces of cheese, they will pay me $2 for each piece; but only if it weighs 8oz or more. They will still buy pieces that weigh less than 8oz but they will only pay me $1/lb for those pieces....and gosh, I pay $2/lb for the cheese so I loose money on those pieces.

When I cut cheese, it is normally distributed. I certainly don't want to cut the pieces too big because I'd be giving away cheese, but it I cut to small I loose money on those pieces. So how do I determine what my target mean should be in order to cut the greatest amount of pieces over 8oz? For example sake, lets say the standard deviation is 0.3oz.

I can figure out the area under the curve, so I know the percentage of pieces above 8oz and the percentage below, I also know the average piece weight, but I think i need to know the average piece weight of the pieces above 8oz and the average piece weight of the pieces below 8oz. in order to calculate costs and I can't figure out how to do that part.

Your help would be much appriciated.

I would approach it some thing like this:

Let M be the total number of pieces with average weight m (in oz). Let C be your total cost at $2/# so that
C = 2 (m/16) M
or
M = (8/m) C

Let n be the number of pieces over 8 oz. On average n depends on how far 8 is from m, so we will assume that it is 'a' standard deviations, i.e.
n/M = P(x > m + a s)
or
n = M P(x > m + a s)
= 8 P(x > m + a s) / m
where P is the probability distribution function of the weight of the pieces and s is the standard deviation. We also note that
m + a s = 8
so that
a = (8 - m) / s

Your income I is then given by
I = 2 n + (M - n) / 2 = 1.5 n + M / 2 = 4 C (3 P + 1) / m
where I have changed your $1/# to $0.50/piece with the assumption that you would be honest enough to try to make those pieces 'close to' 8 oz. so that you would be getting about $1/#.

Your relative profit p is then
p = (I - C) / C = 4 (3 P + 1) / m - 1


As a practical problem, assuming P is a known distribution, i.e. Normal, it would allow for building a table and finding how to maximize your profit. Or, if you had rather, go ahead and take the derivative and work it from that end.


BTW: Playing around I got you should shoot for about 8.6 oz. but that's certainly no guarantee.