Question

A statistics professor has stated that 90% of his students pass the class. To check this claim, a random sample of 150 students indicated that 129 passed the class. If the professor's claim is correct, what is the probability that 129 or fewer will pass the class this semester?

A) 0.0516

B) 0.9484

C) 0.5516

D) 0.4484

please show work

Answer 1

To Test :-

H0 :- P = 0.90
H1 :- P ≠ 0.90

P = X / n = 129/150 = 0.86

Test Statistic :-
Z = ( P - P0 ) / ( √((P0 * q0)/n)
Z = ( 0.86 - 0.9 ) / ( √(( 0.9 * 0.1) /150))
Z = -1.633

Test Criteria :-
Reject null hypothesis if Z < -Z(α/2)
Z(α/2) = Z(0.05/2) = 1.96
Z > -Z(α/2) = -1.633 > -1.96, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

There is sufficient evience to support the claim.

Sampling distribution of p̂ is approximately normal if np >=10 and n (1-p) >= 10
n * p = 150 * 0.9 = 135
n * (1 - p ) = 150 * (1 - 0.9) = 15
Mean =   = p = 0.9
Standard deviation = = 0.024495

X ~ N ( µ = 0.9 , σ = 0.024495 )
P ( X < ( 129 / 150 ) = 0.86 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 0.86 - 0.9 ) / 0.024495
Z = -1.63
P ( ( X - µ ) / σ ) < ( 0.86 - 0.9 ) / 0.024495 )
P ( X < 0.86 ) = P ( Z < -1.63 )
P ( X < 0.86 ) = 0.0516