You are sampling a dimension of a part on a production line with X_i ~ N(100, 5)

Question

You are sampling a dimension of a part on a production line with X_i ~ N(100, 5) lbs. A sample (n = 30) is drawn with X = 96.8. Do you believe the items in this sample come from the distribution above? Justify your answer. Samples of size n = 30 is pulled from a production line once per day. You want to construct a X control chart with limits +/-3 sigma from the mean: if a X falls outside these limits it is suspect and requires investigation. Calculate the control limits for X. Calculate the standard error for X for n = 30. What would n have to be such that 3 sigma control limits for Q#2 above were no more than +/-0.5 lbs?

Explanation / Answer

1) here std error of mean =std deviation/(n)1/2 =5/(30)1/2 =0.9129

therefore probabilty of having mean 96.8 or less then that =P(X<96.8)=P(Z<(96.8-100)/0.9129)=P(Z<-3.5054)=0.0002

as above probabilty is very less ; we should be suspicious that this sample comes from distribution above.

2) control limits =population mean -/+ 3*std error =100 -/+ 3*0.9129 =97.26 ; 102.74

3) std error of mean =std deviation/(n)1/2 =5/(30)1/2 =0.9129

4) here margin of error E=0.5

therefore sample size n=(3*std deviation/E)2 =(3*5/0.5)2 =~900

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