Question

A quality engineer samples 100 steel rods made on mill A and 150 rods made on mill B. Of the rods from mill A, 88 meet specifications, and the rods from mill B, 135 meet specifications.

a) Estimate the proportion of rods from mill A that meet specifications, and find the uncertainty in the estimate.

b) Estimate the proportion of rods from mill B that meets specifications, and find the uncertainty in the estimate.

c) Estimate the difference between the proportions, and find the uncertainty in the estimate.

Answer 1

a)
sample proportion, pcap = 88/100 = 0.88

sample size, n = 100

Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.88 * (1 - 0.88)/100) = 0.0325

b)
sample proportion, pcap = 135/150 = 0.9

sample size, n = 150

Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.9 * (1 - 0.9)/150) = 0.0245

c)
Here, , n1 = 100 , n2 = 150
p1cap = 0.88 , p2cap = 0.9

p1cap - p2cap = 0/88 - 0.9 = -0.02

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.88 * (1-0.88)/100 + 0.9*(1-0.9)/150)
SE = 0.0407