Question

Two random samples of student loans were collected: one from students at for-profit schools and another from students at non-profit schools. The accompanying data show the sample sizes and the number of loans in each sample that defaulted. Complete parts a through c. Click the icon to view the loan data. a. Perform a hypothesis test using a = 0.10 to determine if the proportion of for-profit loans that default is larger than the proportion of loans for nonprofit schools that default. Let P, represent the population proportion of loans for for-profit schools that default and let Pz represent the population proportion of loans for nonprofit schools that default. Identify the null and alternative hypotheses. Select the correct choice below and fill in the answer boxes to complete your choice. (Type integers or decimals. Do not round.) OC. Ho: P1-P22 OB. Ho: P1 - P2 O A. Ho: P1-P27 Hy: P1-P2 H: P1-P2) Hy: P1-P2= OF. Ho: P1-P2 O E. Ho: P1-P2 OD. Ho: P1-P2- Hy: P1 - P22 HP1-P2s Hy: P1-P26 Identify the test statistic, (Round to two decimal places as needed.) Identify the critical value. (Round to two decimal places as needed.) State the conclusions. the null hypothesis. There sufficient evidence to conclude that the proportion of for-profit loans that default larger than the proportion of loans for nonprofit schools that default. b. Determine the p-value and interpret the results. p-value (Round to three decimal places as needed.)

Answer 1

Part a)

H0 :- P1 ≤ P2
H1 :- P1 > P2

p̂1 = 25 / 238 = 0.105
p̂2 = 11 / 212 = 0.0519

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 = ( 25 + 11 ) / ( 238 + 212 )
p̂ = 0.08
q̂ = 1 - p̂ = 0.92
Z = ( 0.105 - 0.0519) / √( 0.08 * 0.92 * (1/238 + 1/212) )
Z = 2.07

Critical value   Z(α) = Z(0.1) = 1.28

Test Criteria :-
Reject null hypothesis if Z > Z(α)
Z(α) = Z(0.1) = 1.28
Z > Z(α) = 2.07 > 1.28, hence we reject the null hypothesis
Conclusion :- We Reject H0

Reject null hypothesis, there is sufficient evidence to support the claim.

P value = P ( Z > 2.0747 ) = 0.019

Reject null hypothesis if P value < α = 0.1
Since P value = 0.019 < 0.1, hence we reject the null hypothesis
Conclusion :- We Reject H0

P value is less than α, therefore, reject null hypothesis, the conclusion is same.

Part c)

(p̂1 - p̂2) ± Z(α/2) * √( ((p̂1 * q̂1)/ n1) + ((p̂2 * q̂2)/ n2) )
Z(α/2) = Z(0.1 /2) = 1.645
Lower Limit = ( 0.105 - 0.0519 )- Z(0.1/2) * √(((0.105 * 0.895 )/ 238 ) + ((0.0519 * 0.9481 )/ 212 ) = 0.012
upper Limit = ( 0.105 - 0.0519 )+ Z(0.1/2) * √(((0.105 * 0.895 )/ 238 ) + ((0.0519 * 0.9481 )/ 212 )) = 0.094
90% Confidence interval is ( 0.012 , 0.094 )
( 0.012 < ( P1 - P2 ) < 0.094 )