Question

a) An appliance company advertises that its brand of washing machine will last a lifetime. Well, not really a lifetime, but longer than 10 years. A random sample of 50 machines manufactured in the past 20 years is selected to test this claim at a significance level of .1. If the sample mean is 11 years and the sample standard deviation is 4. Conduct the test of hypothesis.

b) A school states that over 85% of its graduates find employment within the first year after graduating. A recent sample of 400 graduates was selected at random and asked the question, “Did you find employment within the first year after graduating?” and 348 of those responded “Yes”. Test the schools claim at α = .05.

Answer 1

Part a)

H0 :- µ = 10
H1 :- µ > 10

Test Statistic :-
t = ( X̅ - µ ) / (S / √(n) )
t = ( 11 - 10 ) / ( 4 / √(50) )
t = 1.7678

Test Criteria :-
Reject null hypothesis if t > t(α, n-1)
Critical value t(α, n-1) = t(0.1 , 50-1) = 1.299
t > t(α, n-1) = 1.7678 > 1.299
Result :- Reject null hypothesis

Decision based on P value
P - value = P ( t > 1.7678 ) = 0.0417
Reject null hypothesis if P value < α = 0.1 level of significance
P - value = 0.0417 < 0.1 ,hence we reject null hypothesis
Conclusion :- Reject null hypothesis

There is sufficient evidence to support the claim that washing machine will last a lifetime i.e more than 10 years.

Part b)

To Test :-

H0 :- P = 0.85
H1 :- P > 0.85

P = X / n = 348/400 = 0.87

Test Statistic :-
Z = ( P - P0 ) / √ ((P0 * q0)/n))
Z = ( 0.87 - 0.85 ) / √(( 0.85 * 0.15) /400))
Z = 1.1202

Test Criteria :-
Reject null hypothesis if Z > Z(α)
Z(α) = Z(0.05) = 1.645
Z < Z(α) = 1.1202 < 1.645, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

Decision based on P value
P value = P ( Z > 1.1202 )
P value = 0.1313
Reject null hypothesis if P value < α = 0.05
Since P value = 0.1313 > 0.05, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

There is insufficient evidence to support the claim that over 85% of its graduates find employment within the first year after graduating.