Question

1. Consider the following results for independent samples taken from two populations.

Sample 1	Sample 2
n1 = 500	n2= 200
p1= 0.45	p2= 0.34

a. What is the point estimate of the difference between the two population proportions (to 2 decimals)?

b. Develop a 90% confidence interval for the difference between the two population proportions (to 4 decimals). Use z-table.
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c. Develop a 95% confidence interval for the difference between the two population proportions (to 4 decimals). Use z-table.
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2.

Consider the hypothesis test below. H 0: p 1 - p 2  0 H a: p 1 - p 2 > 0 The following results are for independent samples taken from the two populations. Sample 1 Sample 2 n 1 = 100 n 2 = 300 p 1 = 0.23 p 2 = 0.15 Use pooled estimator of p. What is the value of the test statistic (to 2 decimals)? What is the p-value (to 4 decimals)? Use z-table. With  = .05, what is your hypothesis testing conclusion? SelectConclude the difference between the proportions is greater than 0Cannot conclude the difference between the proportions is greater than 0Item 3

Answer 1

Question 1

Part a)

Point Estimate = (p̂1 - p̂2) = 0.11

part b)

(p̂1 - p̂2) ± Z(α/2) * √( ((p̂1 * q̂1)/ n1) + ((p̂2 * q̂2)/ n2) )
Z(α/2) = Z(0.1 /2) = 1.645
Lower Limit = ( 0.45 - 0.34 )- Z(0.1/2) * √(((0.45 * 0.55 )/ 500 ) + ((0.34 * 0.66 )/ 200 ) = 0.0439
upper Limit = ( 0.45 - 0.34 )+ Z(0.1/2) * √(((0.45 * 0.55 )/ 500 ) + ((0.34 * 0.66 )/ 200 )) = 0.1761
90% Confidence interval is ( 0.0439 , 0.1761 )
( 0.0439 < ( P1 - P2 ) < 0.1761 )

part c)

(p̂1 - p̂2) ± Z(α/2) * √( ((p̂1 * q̂1)/ n1) + ((p̂2 * q̂2)/ n2) )
Z(α/2) = Z(0.05 /2) = 1.96
Lower Limit = ( 0.45 - 0.34 )- Z(0.05/2) * √(((0.45 * 0.55 )/ 500 ) + ((0.34 * 0.66 )/ 200 ) = 0.0312
upper Limit = ( 0.45 - 0.34 )+ Z(0.05/2) * √(((0.45 * 0.55 )/ 500 ) + ((0.34 * 0.66 )/ 200 )) = 0.1888
95% Confidence interval is ( 0.0312 , 0.1888 )
( 0.0312 < ( P1 - P2 ) < 0.1888 )

Question 2

Part a)

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 = ( 23 + 45 ) / ( 100 + 300 )
p̂ = 0.17
q̂ = 1 - p̂ = 0.83
Z = ( 0.23 - 0.15) / √( 0.17 * 0.83 * (1/100 + 1/300) )
Z = 1.84

Part b)

P value = P ( Z > 1.8444 ) = 0.0326

Part c)

Reject null hypothesis if P value < α = 0.05
Since P value = 0.0326 < 0.05, hence we reject the null hypothesis
Conclusion :- We Reject H0

We can Conclude the difference between the proportions is greater than 0.