Question 1162169
the critical z-score is -2.326347877.
the area to the left of this z-score will be .01, which is 1%.


you want less than or equal to 1% of all the packets made to be equal to 800.


the z-score formula is z = (x - m) / x
z is the z-score
x is the raw score
m is the raw mean
s is the standard deviation.


in this formula, z = -2.326347877, s = 5, and x = 800.
you want to solve for m.


formula becomes -2.326347877 = (800 - m) / 5.
multiply both sides of the equation by 5 to get:
-2.326347877 * 5 = 800 - m
add m to both sides of the equation and subtract -2.326347877 * 5 from both sides of the equation to get:
m = 800 - (-2.326347877 * 5) = 811.6317394.


that's what the mean needs to be so that less than or equal to 1% of the packet weighs less than 800 when the standard deviation is 5.


if the standard deviation is equal to 2, then the formula to find m becomes:
-2.326347877 = (800 - m) / 2.
solve for m as before to get:
m = 800 - (-2.326347877 * 2) = 804.6526958


that's what the mean needs to be so that less than or equal to 1% of the packet weighs less than 800 when the standard deviation is 2.


an average of 2,400,000 packets are produced each week.


when the mean is 811.6317394, the cost is 2,400,000 * (6 + .5 * 811.6317394) = 988,358,087.3 dollars.

when the mean is 804.6526958, the cost is 2,400,000 * (6 + .5 * 804.6526958) = 979,983,234.9 dollars.


the savings are equal to 988,358,087.3 dollars minus 979,983,234.9 dollars = 8,374,852.358 dollars.


that's 8.3754852358 million dollars, which can also be shown as 8.3754852358 * 10^6 dollars.


i used statistical calculators to confirm the manual calculations.


here's what they showed.


<img src = "http://theo.x10hosting.com/2020/070902.jpg" >


<img src = "http://theo.x10hosting.com/2020/070903.jpg" >


<img src = "http://theo.x10hosting.com/2020/070904.jpg" >


<img src = "http://theo.x10hosting.com/2020/070905.jpg" >


they confirmed the manual calculations were correct.


in both cases, the number of packets weighing less than 800 grams was less than or equal to 1%.


the calculators can be found at:


<a href = "https://www.omnicalculator.com/statistics/normal-distribution" target = "_blank">https://www.omnicalculator.com/statistics/normal-distribution</a>


<a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">http://davidmlane.com/hyperstat/z_table.html</a>