Suppose that the demand for a company’s product in weeks 1, 2, and 3 are each normally

Suppose that the demand for a company’s product in weeks 1, 2, and 3 are each normally distributed and the mean demand during each of these three weeks is 50, 45, and 65, respectively. Suppose the standard deviation of the demand during each of these three weeks is known to be 10, 5, and 15, respectively. It turns out that if we can assume that these three demands are probabilistically independent then the total demand for the three week period is also normally distributed. And, the mean demand for the entire three week period is the sum of the individual means. Likewise, the variance of the demand for the entire three week period is the sum of the individual weekly variances. But be careful! The standard deviation of the demand for the entire 3 week period is not the sum of the individual standard deviations. Square roots don’t work that way!

Now, suppose that the company currently has 180 units in stock, and it will not be receiving any further shipments from its supplier for at least 3 weeks. What is the probability that the company will run out of units?

***This is a forum question, so **Please explain how you were able to come to the answer**.  Similarly to the way the answer is explained below.  There is no need for any type of table. Please provide the answer in a word document.***

To determine the probability that a company will run out of units we will let Y equal the demand for three weeks by establishing the mean.

(Y) = 3 week period

(Y) = 50+45+65 = 160 (Mean)

Next we need to find the variance by squaring the standard deviations and adding them together.

Variance (3 week) = 10^2 + 5^2 + 15^2 = 350

Next determine the standard deviation (SD) of the three week interval but utilizing the square root of the variant.

SD of 3 week interval = square root(350) = 18.7 = SD

Information calculated thus far:

mean = 160

Variance = 350

Standard deviation = 18.7  

Z score = ( value-mean ) / Standard Deviation

Using the above formula we come up with the following:

Z score = (180-160) / 18.7 = 1.07

Using the normal distribution table, if we look up 1.07 we get the result of .8577, which translates to 85.77%.  

To find out the probability of the company running out of their stock if they don’t receive a shipment from its supplier for three weeks use the following equation:

100% - 85.77% = 14.23%

Based on the given information, 14.23% of the time the company will run out of stock if it doesn't receive a shipment from its supplier for three weeks.