Question

A tire manufacturer produces tires that have a mean life of at least 30000 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 30000 miles.

The testable hypotheses in this situation are H0:μ=30000H0:μ=30000 vs HA:μ<30000HA:μ<30000.

1. Identify the consequences of making a Type I error.
A. The manager stops production when it is not necessary.
B. The manager stops production when it is necessary.
C. The manager does not stop production when it is not necessary.
D. The manager does not stop production when it is necessary.

2. Identify the consequences of making a Type II error.
A. The manager stops production when it is necessary.
B. The manager does not stop production when it is necessary.
C. The manager does not stop production when it is not necessary.
D. The manager stops production when it is not necessary.

To monitor the production process, the operations manager takes a random sample of 30 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 29000, they will stop production and recalibrate the machines. They know based on past experience that the standard deviation of the tire life is 2750 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 28750 miles? That is, what is the probability they will reject H0H0 when the actual average life of the tires is 28750 miles? Round your answer to four decimal places.

Answer 1

1) A. The manager stops production when it is not necessary.

2) B. The manager does not stop production when it is necessary.

3) P(type I error) = P(Xbar <29000|µ = 30000)

Z =   (X - µ )/(σ/√n) = (   29000   -   30000.00   ) / (   2750.000   / √   30   ) =   -1.992  
                                          
P(X < 29000   ) = P(Z ≤   -1.992   ) =   0.0232                       (answer)

4) std error of mean,   σx = σ/√n =    2750.0000   / √    30   =   502.07901

now, type II error is ,ß =    P( x̄ ≥    29000 given that µ =   28750   )
                  
   = P ( Z > (x̄-true mean)/σx )               
=P(Z > (   29000.000   -   28750   ) /   502.0790
                  
   = P ( Z >    0.498   ) =   0.3093   [ Excel function: =1-normsdist(z) ]
                  
                  
power =    1 - ß =   0.6907