Question

A bank has recently acquired a new branch and thus has customers in this new territory. They are interested in the default rate in their new territory. They wish to test the hypothesis that the default rate is different from their current customer base. They sample 200 files in area A, their current customers, and find that 20 have defaulted.In area B, the new customers, another sample of 200 files shows 12 have defaulted on their loans. At a 10% level of significance can we accept/reject the null hypothesis H0 : pA = pB against the alternative hypothesis Ha : pA ≠ pB?

Is it A or B?

a. Hull hypothesis CANNOT be rejected at 10% confidence level

b. Hull hypothesis CAN be rejected at 10% confidence level

Answer 1

p1cap = X1/N1 = 20/200 = 0.1
p1cap = X2/N2 = 12/200 = 0.06
pcap = (X1 + X2)/(N1 + N2) = (20+12)/(200+200) = 0.08

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p1 = p2
Alternate Hypothesis, Ha: p1 ≠ p2

Rejection Region
This is two tailed test, for α = 0.1
Critical value of z are -1.64 and 1.64.
Hence reject H0 if z < -1.64 or z > 1.64

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.1-0.06)/sqrt(0.08*(1-0.08)*(1/200 + 1/200))
z = 1.47

P-value Approach
P-value = 0.1416
As P-value >= 0.1, fail to reject null hypothesis.

a. null hypothesis CANNOT be rejected at 10% confidence level