A 2002 article in The American Statistician describes an effort by an engineerin

Question

A 2002 article in The American Statistician describes an effort by an engineering team to reduce the defect rate in the manufacture of printed circuit boards. The team decided to reconfigure the transistor heat sink. A total of 1500 boards were produced the week before the reconfiguration and 345 of these were defective. A total of 1500 boards were produced the week after reconfiguration and 195 of these were defective. Find a 95% confidence interval for the decrease in the defective rate after the reconfiguration. Can you provide a 90% lower bound on the difference in the decrease in the defective rate?

Explanation / Answer

defective rate before reconfiguration=p1=x1/n1=345/1500=0.23

variance of p1=var(p1)=p1(1-p1)/n1=0.23*(1-0.23)/1500=0.000118

defective rate after reconfiguration=p2=x2/n2=195/1500=0.13

variance of p1=var(p2)=p2(1-p2)/n2=0.13*(1-0.13)/1500=0.000075

decrerease in defective rate =p1-p2=0.23-0.13=0.1

variance of (p1-p2)=var(p1)+var(p2)=0.000193

SE(p1-p2)=sqrt(var(p1-p2))=0.0139

(1-alpha)*100% confidence interval for (p1-p2)=(p1-p2)± z(alpha/2)*SE(p1-p2)

95% confidence interval =0.1±z(0.05/2)*0.0139=0.1+1.96*0.0139=0.1±0.0272=(0.0728,0.1272)

90% confidence interval =0.1±z(0.1/2)*0.0139=0.1+1.6449*0.0139=0.1±0.0228=(0.0772,0.1228)

required 90% lower bound =0.0772

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