4) A salt solution has mean salt content, by weight, of 18% with standard deviat

Question

4) A salt solution has mean salt content, by weight, of 18% with standard deviation 2%. Assume that the salt concentration is (at least approximately) normally distributed. The solution is used in an etching process which is optimal for salt concentrations around 18%, but which has undesirable side effects when the salt concentration exceeds 21%. In an attempt to avoid the side effects, the salt solution is dumped if the average salt concentration of 4 samples of the solution exceeds 20%. a) Suppose that the mean salt concentration is actually 18% (and so the process is optimal on average). What is the probability that the salt solution will end up being dumped? b) Assuming that the mean salt concentration is held at 18%, what standard deviation would be required to ensure that the probability that the salt solution is dumped is held to less than 0.001? (Note: manufacturers of such a salt solution would be interested in determining this required quality control level.)

Explanation / Answer

here for 0.001 probability at (1-0.001=0.999)~ 99.9 percentile ; critical value of z =3.0902

for we know that z score=(X-mean)/std error

hence std error =(X-mean)/z =(20-18)/3.0902=0.6472

therefore std deviation =std error*(n)1/2 =0.6472*(4)1/2 =0.6472*2=1.2944

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