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 264 of the 514 observations in group 1 have the characterisitic. And, we know that 245 of the 518 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 = 264/514 = 0.51362

b)

p1cap = X2/N2 = 245/518 = 0.47297

c)

pcap = (X1 + X2)/(N1 + N2) = (264+245)/(514+518) = 0.49322

d)

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.51362 * (1-0.51362)/514 + 0.47297*(1-0.47297)/518)
SE = 0.03110

e)

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.51362-0.47297)/sqrt(0.49322*(1-0.49322)*(1/514 + 1/518))
z = 1.31

f)

P-value Approach
P-value = 0.095

g)

As P-value >= 0.05, fail to reject null hypothesis.