Question

A hair dryer manufacturer is testing a new, cheaper motor design, but will not produce it if it is less reliable than the existing motor design. Out of 250 of the new motors, 55 failed. Out of 250 of the current motors, 75 failed. Is it reasonable to assume that the new motors fail less than the current ones? Use an alpha of 0.05

A) No, the p-value is 0.0525 > 0.05

B) Yes, the p-value is 0.0262 < 0.05

C) No, the p-value is 0.9840 > 0.05

D) No, the p-value is 0.0262 < 0.05

E) Impossible without knowing the std deviation

Answer 1

To Test :-

H0 :- P1 = P2
H1 :- P1 < P2

p̂1 = 55 / 250 = 0.22
p̂2 = 75 / 250 = 0.30

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 = ( 55 + 75 ) / ( 250 + 250 )
p̂ = 0.26
q̂ = 1 - p̂ = 0.74
Z = ( 0.22 - 0.3) / √( 0.26 * 0.74 * (1/250 + 1/250) )
Z = -2.0391

P value = P ( Z < -2.0391 ) = 0.0262

Reject null hypothesis if P value < α = 0.05
Since P value = 0.0262 < 0.05, hence we reject the null hypothesis
Conclusion :- We Reject H0

D) No, the p-value is 0.0262 < 0.05