Question 1107404: Just would like to know how I would go about answering this problem. I am really getting some of the concepts down but when it come to a variety of different word problems, i have a hard time setting them up. Just need help setting them up. See questions below
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?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if the mean of the total demand is equal to the sum of the mean of each individual demands, then the mean of the total demand would have to be equal to 50 + 45 + 65 = 160.
the variance is the square of the standard deviation.
therefore, the variance for each of the 3 weeks would be 10^2, 5^2, and 15^2, which would be equal to 100, 25, and 225.
since the problem states that the variance for the 3 weeks is the sum of the variance for each of the 3 weeks, then the sum of the variance has to be 350.
since the standard deviation is equal to the square root of the variance, that makes the standard deviation of the 3 weeks equal to sqrt(350).
you have the mean of the overall demand being equal to 160 and the standard deviation of the overall demand being sqrt(350).
the stock is 180, so you want to know the probability that the overall demand will be greater than 180.
z = (x-m)/s
z is the z-score.
x is the inventory.
m is the mean of the demand.
s is the standard deviation.
you get z = (180 - 160) / sqrt(350).
solve for z to get z = 20/sqrt(350) = 1.069044968.
the area to the right of this z-score is equal to .1425247345.
that's the probability that the demand will be greater than 180 which means that the inventory won't be enough to satisfy the demand.
that's what i think is how you would solve this problem.
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