Please show work. Thanks A cement manufacturing factory is trying to control the

Question

Please show work. Thanks

A cement manufacturing factory is trying to control the gypsum content in the produced cement in the range of 4.5 to 5.5% (these two numbers are the lower and upper specification limits). Note that the presence of gypsum in cement is necessary for controlling the setting time of cement. A long-term investigation on the gypsum content of the produced cement samples suggests that the gypsum content (G) in the final product is a random variable having a normal distribution with an average and standard deviation of 5.2% and 0.3%. Find out what proportion (percentage) of the produced cement will have gypsum content outside the specification limits. In the previous question, if the factory decides to limit the percentage of cement with excessively high or low gypsum content to 3%, assuming the average stays constant (5.3%), what is the maximum allowable standard deviation of G? In question three, if the factory decides to only limit the percentage of cement with excessive gypsum content to 2%, what will be the percentage of produced cement having gypsum content less than the lower specification limit?

Explanation / Answer

(2)

= 5.2, = 0.3, x1 = 4.5, x2 = 5.5

z1 = (x1 - )/ = (4.5 - 5.2)/0.3 = -2.33 and z2 = (x2 - )/ = (5.5 - 5.2)/0.3 = 1

P(4.5 < x < 5.5) = P(-2.33 < z < 1) = 0.8315

83.15% of the production lies within the specification limits. So 100 - 83.15 = 16.85% of the production lies outside the specification limits.

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