Question

Two different types of injection-moulding machines are used to form plastic parts. A part is considered defective if it has excessive shrinkage or is discolored. Two random samples, each of size 300, are selected, and 10 defective parts are found in the sample from machine 1, and 8 defective parts are found in the sample from machine 2. Construct a 99% lower bound on the difference in the two fractions (p1-p2). Please report your answer to 3 decimals.

Answer 1

Here, x1 = 10 , x2 = 8 , n1 = 300 , n2 = 300
p1cap = 0.0333 , p2cap = 0.0267

pcap = (x1 + x2)/(n1 + n2)
pcap = (10 + 8)/(300 + 300)
pcap = 0.03

Standard Error,
SE = sqrt(pcap * (1-pcap) * (1/n1 + 1/n2))
SE = sqrt(0.03 * (1-0.03) * (1/300 + 1/300))
SE = 0.0139

For 0.99 CI, z-value = 2.58
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.0333 - 0.0267 - 2.58*0.0139, 0.0333 - 0.0267 + 2.58*0.0139)
CI = (-0.0293 , 0.0425)

Hence lower bound = -0.0293