estimate the probability

Question

Demand for a product is normally distributed with mean demand equal to 200 units.  There is a 0.95 probability that demand for this produce is between 180-220 units.  Estimate the probability that demand for this product will exceed 195 units?

Answer 1

Okay, so this question is asking you to work backward from a confidence interval to the standard deviation, then to use that to find the z value, which you can then translate to a percentage via the cumulative distribution function.

 

Dang, that's a lot of big words, isn't it?

 

 

Let's take it step by step.  

 

First, 95% of all area under the normal curve are between -1.96 and  +1.96 SDs.  Therefore, we know that 20 units (the difference between 220-200 or 200-180) is 1.96 SDs.  We'll call it 2 SDs, to make the math easy.

 

if 20 units is 2 sds, then 10 units is 1 SD.  (YOU NOW HAVE THE VALUE OF 1 SD.)

 

Now, we want the probability that the demand will exceed 195 units.  That's 5 units less than the mean - when means about 1/5 SD less than the mean.  (YOU NOW HAVE THE Z SCORE)

 

A quick check of the charts tells us that .1915 of the curve is between -.5 SDs and the mean.  Add that to the 0.5 of the curve that's to the right of the mean, and we get .6915 or 69.15% of the time that demand will meet or exceed 195 units. (YOU NOW HAVE THE PROBABILITY.)