Calculating a Normal Distribution With a Set Mean

Hello,

I came across a problem that I was hoping to get some help on.

I have ~$3,260 to spend on 6 different items priced at:

$5 each
$5.50 each
$7.70 each
$10 each
$11 each
$15 each

I also need the mean across all items to be equal to ~$8.36.

It doesn't really matter if I'm slightly above, or slightly below the average amount/total amount, but the count of each item has to be a whole number (I can't buy 1/2 a $5 item).

Does anyone know of a formula out there that would allow me to calculate how many of each I require if I assume a normal distribution?

Moreover, something that would be reusable for if I were to have ~$469 to buy the same 6 items where the mean across all items are equal to ~$7.10?

Thank you in advance!
Rob

Rob636 said:

Hello,

I came across a problem that I was hoping to get some help on.

I have ~$3,260 to spend on 6 different items priced at:

$5 each
$5.50 each
$7.70 each
$10 each
$11 each
$15 each

I also need the mean across all items to be equal to ~$8.36.

It doesn't really matter if I'm slightly above, or slightly below the average amount/total amount, but the count of each item has to be a whole number (I can't buy 1/2 a $5 item).

Does anyone know of a formula out there that would allow me to calculate how many of each I require if I assume a normal distribution?

Moreover, something that would be reusable for if I were to have ~$469 to buy the same 6 items where the mean across all items are equal to ~$7.10?

Thank you in advance!
Rob

Click to expand...

The total number of items should be $3260/$8.36 = 390.
"One item" represents 1/390 = 0.256% of the area of the normal distribution.
If you want less than "one item" to be outside the tail on the low side, the z-value corresponding to $5 would be about 2.8.
Try sigma = ($8.36 - $5.00)/2.8 = $1.20
Look up the six different z-values and find the corresponding f(z).
Can you normalize those so the sum is 390? [Round to integers]
What is the Expectation value? Is it close enough to $8.36?
Can you diddle the numbers for $7.70 and $10.00 to get close enough?
Otherwise, adjust sigma and try again.