Question

At a manufacturing plant that produces 12-volt batteries, a quality assurance technician regularly performs a standard test that records how long each battery will produce 400 amperes. The test results for the batteries are known to be normally distributed with a population mean of 11.05 minutes and a population standard deviation of 1.87 minutes. Every half-hour during a production run, a random sample of 4 batteries is selected and tested. If the sample mean for the three batteries is less than 9.25 minutes, the production of the batteries will be stopped and the machinery will be inspected for problems. The hypotheses for this testing situation are:

H0:μ=11.05 versus HA:μ<11.05H0:μ=11.05 versus HA:μ<11.05

Using this decision rule, what is the power of the hypothesis test if the true mean for the batteries is 9.55 minutes?Give your answer to 4 decimal places.

Answer 1

true mean ,    µ =    9.55
      
hypothesis mean,   µo =    11.05
significance level,   α =    0.05
sample size,   n =   4
std dev,   σ =    1.8700
      
δ=   µ - µo =    -1.5
      
std error of mean=σx = σ/√n =    1.87/√4=   0.9350

Zα =   -1.6449   (left tail test)
P(type II error) , ß =   P(Z > Zα - δ/σx)  
= P(Z >    -1.6449-(-1.5)/0.935)  
=P(Z>    -0.041   ) =
type II error, ß =   0.5162  

power =    1 - ß =   0.4838