Question

5. We want to know whether the difference in the population proportions between group 1 and group 2 is greater than 0. Specifically, we want to test H0 : p1−p2 ≤ 0 versus Ha : p1−p2 > 0. We know that 224 of the 549 observations in group 1 have the characterisitic. And, we know that 185 of the 547 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 = 224/549 = 0.40801

b)

p1cap = X2/N2 = 185/547 = 0.33821

c)

pcap = (X1 + X2)/(N1 + N2) = (224+185)/(549+547) = 0.37318

d)

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.40801 * (1-0.40801)/549 + 0.33821*(1-0.33821)/547)
SE = 0.02914

e)

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.40801-0.33821)/sqrt(0.37318*(1-0.37318)*(1/549 + 1/547))
z = 2.39

f)

P-value Approach
P-value = 0.008

g)

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