find the z statistic for the hypothesis test

Question

Assume that you plan to use a significance level of alpha=0.05 to test the claim that p1=p2. Use the given sample sizes and number of successes to find the z statistic for the hypothesis test.

A report on the nightly news broadcast stated that 13 out of 103 households with pet dogs were burglarized and 23 out of 177 without pet dogs were burglarized.

Answer 1

P1 = 13/103 = 0.1262 = proportion of households with a dog that were burglarized.

P2 = 23/177 = 0.1299 = proportion of households without a dog that were burglarized.

Test H₀: P1 = P2 vs H₁+ P1 ^= P2 at alpha=0.05

P = proportion of households burglarized = (13+23) / (103+177) = 26/280 = 0.1286

Q = 1-P = 1-0.1286 = 0.8714.

n1=103, n2=177

z = (P1-P2)/[sqrt(PQ) * sqrt(1/n1 + 1/n2)]

z = -0.0037/[sqrt(0.1121)sqrt(1/103 + 1/177)]

z = -0.0037/[(0.3348)*(0.1239)]

z = -0.0892

Pr(z=-0.0892) ~ 0.4641

Answer 2

First you need to find the sample proportions p1^ and p2^.

p1^ = 13/103 = .126

p2^ = 23/177 = .130

Test statistic z = (p1^ - p2^)/(square root [(p-bar* (1-p-bar)/n1 + (p-bar*(1-p-bar)/n2]

p-bar = (13 + 23)/(103 + 177) = 36/280 = .12857

We know n1 = 103 and n2 = 177

Therefore, z = (.126 - .130)/(sqrt(.12857*.87143/103 +.12857*.87143/177))

z = -.1

Since the alternate hypothesis is that p1 does not equal p2, this is a two-tailed test, so at alpha = .05, there is .025 in each tail.

z critical = -1.96 and 1.96

Since the test statistic z = -.1 falls in between the critical values, do not reject Ho. There is not enough evidence to conclude that p1 does not equal p2.