Question

4. Suppose that 326 of the 666 observations in group 1 have a characterisitic of interest. And, suppose that 255 of the 570 observations in group 2 have this characterisitic.

(a) What is the sample proportion for group 1? (round to 5 digits after the decimal place)

(b) What is the sample proportion for group 2? (round to 5 digits after the decimal place)

(c) What is the difference in the sample proportions between groups 1 and 2? (round to 5 digits after the decimal place) 1

(d) What is the standard error for the difference in the sample proportions? (Use σp1−p2 and round to 5 digits after the decimal place.)

(e) What is the estimate of the left end of a 95% confidence interval? (Round to 5 digits after the decimal place.)

(f) What is the estimate of the right end of a 95% confidence interval? (Round to 5 digits after the decimal place.)

(g) Do we reject or not reject the null hypothesis that the population proportions are the same at the .05 level of significance? Reject Not reject

Answer 1

a)

p1cap = X1/N1 = 326/666 = 0.48949

b)

p1cap = X2/N2 = 255/570 = 0.44737

c)

difference = 0.48949 - 0.44737 = 0.04212

d)

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.48949 * (1-0.48949)/666 + 0.44737*(1-0.44737)/570)
SE = 0.02844

e)

For 0.95 CI, z-value = 1.96
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.48949 - 0.44737 - 1.96*0.02844, 0.48949 - 0.44737 + 1.96*0.02844)
CI = (-0.01362 , 0.09786)

left end = -0.01362

f)

right end = 0.09786

g)

Not reject