Question

A tire manufacturer produces tires that have a mean life of at least 25000 miles when the production process is working properly. The operations manager stops the production process if there is evidence that the mean tire life is below 25000 miles.

The testable hypotheses in this situation are ?0:?=25000H0:μ=25000 vs ??:?<25000HA:μ<25000.

To monitor the production process, the operations manager takes a random sample of 20 tires each week and subjects them to destructive testing. They calculate the mean life of the tires in the sample, and if it is less than 24500, they will stop production and recalibrate the machines. They know based on past experience that the standard deviation of the tire life is 3000 miles.

3. What is the probability that the manager will make a Type I error using this decision rule? Round your answer to four decimal places.

4. Using this decision rule, what is the power of the test if the actual mean life of the tires is 24350 miles? That is, what is the probability they will reject ?0H0 when the actual average life of the tires is 24350 miles? Round your answer to four decimal places.

Answer 1

3)

for normal distribution z score =(X-μ)/σ
here mean=       μ=	25000
std deviation   =σ=	3000.000
sample size       =n=	20
std error=σx̅=σ/√n=	670.82039

probability =P(X<24500)=(Z<(24500-25000)/670.82)=P(Z<-0.75)=0.2266

(please try 0.2280 if this comes wrong)

4)

probability =P(X<24500)=(Z<(24500-24350)/670.82)=P(Z<0.22)=0.5871

(please try 0.5885 if this comes wrong)