Question

Problem 5. Of a random sample of 381 high-quality investment equity options, 191 had less than 30% debt. Ofan independent random sample of 166 high-risk investment equity options, 145 had less than 30% debt. (a) Test, against a two-sided alternative, the null hypothesis that the two population proportions are equal. Use 5% significance level. (b) Find 95% two-sided confidence interval for the difference of the two population proportions.

Answer 1

(a)

H0: Null Hypothesis: P1 = P2

HA: Alternative Hypothesis: P1 P2

n1 = 381

p1 = 191/381 = 0.5013

n2 =166

p2 = 145/166 =0.8735

131 160-0.6143 381 166

Q = 1 - P = 0.3857

Test statistic is:

Z = (0.5013 - 0.8735)/0.0452 = - 8.2345

= 0.05

From Table, critical values of Z = 1.96

Since calculated value of Z = - 8.2345 is less than critical value of Z = - 1.96, the difference is significant. Reject null hypothesis.

Conclusion:

The data do not support the claim that the two population proportions are equal.

(b)

Confidence interval:

(0.5013 - 0.8735) (1.96 X 0.0452)

= - 0.3722 0.0886

= (- 0.4608 , - 0.2836)

So,

Confidence Interval:

- 0.4608 < P1 - P2< - 0.2836