A manufacturing process for silicon wafers produces wafers with a mean contamina

Question

A manufacturing process for silicon wafers produces wafers with a mean contamination level that is hypothesized to be 0.01 particles/mm2. The contamination level in the wafers follows a Normal distribution with standard deviation 0.2 particle/mm2. To test the quality of the process, a monthly random sample of 100 wafers is tested for contamination levels. If the sample average contamination level is found to be in the range 0.009 lessthanorequalto x lessthanorequalto 0.011, the hypothesis about the mean contamination is concluded to be true. Otherwise, it is believed to be false. (a) Calculate alpha, the probability of error type I in this test. (b) If the mean contamination is actually 0.015, calculate beta, the probability of error type II of the test.

Explanation / Answer

a) here std error of mean =std deviation/(n)1/2 =0.2/(100)1/2 =0.02

hence type I error =1-P(0.009<X<0.011)=1-P((0.009-0.01)/0.02<Z<(0.011-0.01)/0.02)

=1-P(-0.05<z<0.05)=1-(0.5199-0.4801)=0.9601

b) Type II error =P(0.009<X<0.011) when mean =0.015

=P((0.009-0.015)/0.02<Z<(0.011-0.015)/0.02)=P(-0.3<Z<-0.2)=0.4207-0.3821=0.0387

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