please explain step by step.. thank you in advance A stamping machine produces c

Question

please explain step by step.. thank you in advance
A stamping machine produces can tops whose diameters arenormally distributed with a standard deviation of 0.01inch. at what"normal"diameter should the machine be set so that no more than 5%of the can tops produced have diameter excedding 3inches? please explain step by step.. thank you in advance
A stamping machine produces can tops whose diameters arenormally distributed with a standard deviation of 0.01inch. at what"normal"diameter should the machine be set so that no more than 5%of the can tops produced have diameter excedding 3inches?

Explanation / Answer

let X be the diameter of the can tops, then X ~ N(u, 0.01) where uis the 'normal' diameter at which the machine needs to be setat. Now u needs to be such that at most 5% have diameter greaterthan 3 inches. Hence we require : P( X > 3 ) = 0.05 or,   P(    (X - u) / 0.01 > (3 - u) / 0.01   ) = 0.05 or, P( Z   > (3 - u) /0.01   ) = 0.05 , where Z = (X -u) / 0.01 ~ N ( 0, 1) using standard normal tables :   P ( Z > 1.645 ) = 0.05   and P( Z > 1.65 ) =0.495    =>   (3 - u) / 0.01 = 1.65 as weneed that at max 5% of the tops should have diameter greater than3 or,   u = 3 - 0.0165 = 2.9835 inches

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