Question

1. Consider this hypothesis test:

H0: p1 - p2 = < 0

Ha: p1 - p2 > 0

Here p1 is the population proportion of “happy” of Population 1 and p2 is the population proportion of “happy” of Population 2. Use the statistics data from a simple random sample of each of the two populations to complete the following:​​​​​​

Population 1

Population 2

Sample Size (n)

1000

1000

Number of “yes”

600

280

a. Compute the test statistic z.

b. What is the p-value?

c. Should H0 be rejected? Use the critical value approach and a level of significance of 0.05 to justify your answer.

d. Use the above data to construct a 95% confidence interval for p1 - p2

Answer 1

Part a)

p̂1 = 600 / 1000 = 0.6
p̂2 = 280 / 1000 = 0.28
Test Statistic :-
Z = ( p̂1 - p̂2 ) / √( p̂ * q̂ * (1/n1 + 1/n2) ))
p̂ is the pooled estimate of the proportion P
p̂ = ( x1 + x2) / ( n1 + n2)
p̂ = ( 600 + 280 ) / ( 1000 + 1000 )
p̂ = 0.44
q̂ = 1 - p̂ = 0.56
Z = ( 0.6 - 0.28) / √( 0.44 * 0.56 * (1/1000 + 1/1000) )
Z = 14.415

Part b)

P value = P ( Z > 14.415 ) = 0
Looking for Z = 14.415 in standard normal table to find the P value.

Part c)

Test Criteria :-
Reject null hypothesis if Z > Z(α)
Z(α) = Z(0.05) = 1.645
Z > Z(α) = 14.415 > 1.645, hence we reject the null hypothesis
Conclusion :- We Reject H0

Yes, we reject null hypothesis

Part d)

p̂1 = 0.6
p̂2 = 0.28
q̂1 = 1 - p̂1 = 0.4
q̂2 = 1 - p̂2 = 0.72
n1 = 1000
n2 = 1000
(p̂1 - p̂2) ± Z(α/2) * √( ((p̂1 * q̂1)/ n1) + ((p̂2 * q̂2)/ n2) )
Z(α/2) = Z(0.05 /2) = 1.960
Lower Limit = ( 0.6 - 0.28 )- Z(0.05/2) * √(((0.6 * 0.4 )/ 1000 ) + ((0.28 * 0.72 )/ 1000 ) = 0.2788
upper Limit = ( 0.6 - 0.28 )+ Z(0.05/2) * √(((0.6 * 0.4 )/ 1000 ) + ((0.28 * 0.72 )/ 1000 )) = 0.3612
95% Confidence interval is ( 0.2788 , 0.3612 )
( 0.2788 < ( P1 - P2 ) < 0.3612 )