Question

6. We want to know whether the difference in the population proportions between group 1 and group 2 differs from 0. Specifically, we want to test H0 : p1 − p2 = 0 versus Ha : p1 − p2 6= 0. We know that 267 of the 502 observations in group 1 have the characterisitic. And, we know that 266 of the 439 observations in group 2 have the 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 pooled estimator for p? (round to 5 digits after the decimal place)

(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 value of the test statistic? (Round to 2 digits after the decimal place.)

(f) What is the p-value of the test? (Round to 3 digits after the decimal place.)

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

Answer 1

a)

p1cap = X1/N1 = 267/502 = 0.53187

b)

p1cap = X2/N2 = 266/439 = 0.60592

c)

pcap = (X1 + X2)/(N1 + N2) = (267+266)/(502+439) = 0.56642

d)
Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.53187 * (1-0.53187)/502 + 0.60592*(1-0.60592)/439)
SE = 0.03225

e)

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.53187-0.60592)/sqrt(0.56642*(1-0.56642)*(1/502 + 1/439))
z = -2.29

g)

P-value Approach
P-value = 0.022

g)

As P-value < 0.05, reject the null hypothesis.