In a sample of 200 homes in Lagrange, 40 were found to have unsafe radon levels and of 100 homes in Hyde Park, 30 had unsafe radon levels. Form a 95% confidence interval on the difference in the proportions.
Answer:
Given,
n1 = 200
n2 = 100
X1 = 40
X2 = 30
p1 = X1/n1
p1 = 40/200
p1 = 0.2
p2 = X2/n2
p2 = 30/100
p2 = 0.3
p bar = (n1p1 + n2p2)/(n1 + n2)
p bar = (200*0.2 + 100*0.3)/(200+100)
p bar = 0.2333
q bar = 1 - pbar
= 1 - 0.233
q bar = 0.77
SE(p1 -p2) = sqrt((pbar*q bar (1/n1)+(1/n2)))
substitute values in above expression
SE(p1 - p2) = sqrt(0.2333*0.77*((1/200)+(1/100)))
= sqrt(2.69115*10^-3)
SE(p1 - p2) = 0.052
consider,
1-alpha = 0.95
alpha = 0.05
alpha/2 = 0.025
Zalpha/2 = Z0.025 = 1.96 [since from the standard normal distribution table]
Interval can be given as follows
(p1 - p2) +/- Zalpha/2 SE(p1-p2)
substitute values
= (0.2-0.3) +/- 1.96(0.052)
= (- 0.20192 , 0.00192)
Hence the 95% confidence interval for (p1 - p2)difference are (-0.20192 , 0.00192)
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