Question

7. We want to know whether the difference in the population proportions between group 1 and group 2 is less than 0. Specifically, we want to test H0 : p1 − p2 ≥ 0 versus Ha : p1 − p2 < 0. We know that 347 of the 628 observations in group 1 have the characterisitic. And, we know that 421 of the 717 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 = 347/628 = 0.55255

b)

p1cap = X2/N2 = 421/717 = 0.58717

c)

pcap = (X1 + X2)/(N1 + N2) = (347+421)/(628+717) = 0.571

d)
Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.55255 * (1-0.55255)/628 + 0.58717*(1-0.58717)/717)
SE = 0.02705

e)

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.55255-0.58717)/sqrt(0.571*(1-0.571)*(1/628 + 1/717))
z = -1.28

f)

P-value Approach
P-value = 0.100

g)

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