Question

# In a sample of 200 homes in Lagrange, 40 were found to have unsafe radon levels...

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.

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)