Question

A researcher investigated determinants that account for individuals' making a transition from having a home (domiciled) but using meal programs to becoming homeless. The following table contains the data obtained in the study.

	Homeless Men	Domiciled Men
Sample size	117	267
Number currently working	33	96

Is there sufficient evidence to indicate that the proportion of those currently working is larger for domiciled men than for homeless men? (Use

α = 0.01.

Use

p₁

for the population proportion of homeless men who are currently working and

p₂

for the population proportion of domiciled men who are currently working.)

State the null and alternative hypotheses.

H₀: p₁ ≠ p₂
H_a: p₁ = p₂

H₀: p₁ > p₂
H_a: p₁ = p₂    

H₀: p₁ = p₂
H_a: p₁ ≠ p₂

H₀: p₁ = p₂
H_a: p₁ > p₂

H₀: p₁ = p₂
H_a: p₁ < p₂

State the rejection region. (Assume the test statistic will be calculated using homeless men − domiciled men. Round your answers to two decimal places. If the test is one-tailed, enter NONE for the unused region.)

z > ___

z < ___

Calculate the appropriate test statistic. (Calculate the test statistic using homeless men − domiciled men. Round your answer to two decimal places.)

z =

What is the conclusion of your test?

Fail to reject H₀. There is not sufficient evidence to indicate that the proportion of those currently working is larger for domiciled men than for homeless men.

Reject H₀. There is not sufficient evidence to indicate that the proportion of those currently working is larger for domiciled men than for homeless men.    

Fail to reject H₀. There is sufficient evidence to indicate that the proportion of those currently working is larger for domiciled men than for homeless men.

Reject H₀. There is sufficient evidence to indicate that the proportion of those currently working is larger for domiciled men than for homeless men.

Answer 1

H0: p1 = p2
Ha: p1 < p2

This is left tailed test, for α = 0.01
Critical value of z is -2.33.
Hence reject H0 if z < -2.33

z > NONE

p1cap = X1/N1 = 33/117 = 0.2821
p1cap = X2/N2 = 96/267 = 0.3596
pcap = (X1 + X2)/(N1 + N2) = (33+96)/(117+267) = 0.3359
Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.2821-0.3596)/sqrt(0.3359*(1-0.3359)*(1/117 + 1/267))
z = -1.48

Fail to reject H0. There is not sufficient evidence to indicate that the proportion of those currently working is larger for domiciled men than for homeless men.