Determining probability using mean and standard deviation.

A food processing company packages a product that is periodically inspected by the Food and Drug Administration. The FDA has ruled that the company's product may have no more than 2.00 grams of a certain toxic substance in it. Past records of the company show that packages of this product have a mean weight of toxic substance equal to 1.25 grams per package and that the weights are normally distributed around 1.25 grams with a standard deviation of 0.25 gram.

What proportion of the individual packages exceed the FDA limit?

A team of FDA inspectors is on its way to inspect a random sample of the company's output. The inspectors plan to take preliminary sample of 25 packages. If they find that the mean weight of toxic substance in the sample exceeds 1.75 grams, they will close down the plant and have an extensive inspection of the company's entire inventory. What is the probability that they will close the plant?

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...eneral Z=(X-mu)/sigma, so in our case we have Z=(X-1.25)/0.25

Therefore P(X>2)=P(Z>(2-1.25)/0.25)=P(Z>3)=1-Phi(3)

We get Phi(3) (the cdf of the standard normal computed in 3) from a table so Phi(3)=0.9987

Then P(X>2) = 1-0.9987 = 0.0013 which is our answer. (Very small probability)

Part 2) Now we want to compute ...