Question

Independent random samples of

n₁ = 170

and

n₂ = 170

observations were randomly selected from binomial populations 1 and 2, respectively. Sample 1 had 96 successes, and sample 2 had 103 successes.

You wish to perform a hypothesis test to determine if there is a difference in the sample proportions

p₁

and

p₂.

(a)

State the null and alternative hypotheses.

H₀: (p₁ − p₂) < 0 versus H_a: (p₁ − p₂) > 0

H₀: (p₁ − p₂) = 0 versus H_a: (p₁ − p₂) ≠ 0

    

H₀: (p₁ − p₂) ≠ 0 versus H_a: (p₁ − p₂) = 0

H₀: (p₁ − p₂) = 0 versus H_a: (p₁ − p₂) > 0

H₀: (p₁ − p₂) = 0 versus H_a: (p₁ − p₂) < 0

(b)

Find the test statistic and rejection region, using the α = 0.10 level of significance. (Round your answers to two decimal places. If the test is one-tailed, enter NONE for the unused region.)

test statisticz=rejection regionz>z<

(c)

State your conclusion.

H₀ is rejected. There is insufficient evidence to indicate that p₁ is different from p₂.H₀ is rejected. There is sufficient evidence to indicate that p₁ is different from p₂.    H₀ is not rejected. There is sufficient evidence to indicate that p₁ is different from p₂.H₀ is not rejected. There is insufficient evidence to indicate that p₁ is different from p₂.

Independent random samples of

n₁ = 170

and

n₂ = 170

observations were randomly selected from binomial populations 1 and 2, respectively. Sample 1 had 96 successes, and sample 2 had 103 successes.

You wish to perform a hypothesis test to determine if there is a difference in the sample proportions

p₁

and

p₂.

(a)

State the null and alternative hypotheses.

H₀: (p₁ − p₂) < 0 versus H_a: (p₁ − p₂) > 0

H₀: (p₁ − p₂) = 0 versus H_a: (p₁ − p₂) ≠ 0

    

H₀: (p₁ − p₂) ≠ 0 versus H_a: (p₁ − p₂) = 0

H₀: (p₁ − p₂) = 0 versus H_a: (p₁ − p₂) > 0

H₀: (p₁ − p₂) = 0 versus H_a: (p₁ − p₂) < 0

(b)

Find the test statistic and rejection region, using the α = 0.10 level of significance. (Round your answers to two decimal places. If the test is one-tailed, enter NONE for the unused region.)

test statisticz=rejection regionz>z<

(c)

State your conclusion.

H₀ is rejected. There is insufficient evidence to indicate that p₁ is different from p₂.H₀ is rejected. There is sufficient evidence to indicate that p₁ is different from p₂.    H₀ is not rejected. There is sufficient evidence to indicate that p₁ is different from p₂.H₀ is not rejected. There is insufficient evidence to indicate that p₁ is different from p₂.

Answer 1

a)

H0: (p1 − p2) = 0 versus Ha: (p1 − p2) ≠ 0

b)

p1cap = X1/N1 = 96/170 = 0.5647
p1cap = X2/N2 = 103/170 = 0.6059
pcap = (X1 + X2)/(N1 + N2) = (96+103)/(170+170) = 0.5853

Rejection Region
This is two tailed test, for α = 0.05
Critical value of z are -1.96 and 1.96.
Hence reject H0 if z < -1.96 or z > 1.96

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.5647-0.6059)/sqrt(0.5853*(1-0.5853)*(1/170 + 1/170))
z = -0.77

c)

H0 is not rejected.There is insufficient evidence to indicate that p1 is different from p2.