Question

A major car manufacturer wants to test a new engine to determine if it meets new air pollution standards. The mean emission, mu of all engines of this type must be approximately 20 parts per million of carbon. If it is higher than that, they will have to redesign parts of the engine. Ten engines are manufactured for testing purposes and the emission level of each is determined. Based on data collected over the years from a variety of engines, it seems reasonable to assume that emission levels of all new engines are roughly Normally distributed with sigma = 3. If the mean emission of all engines is, in fact, mu=22, what is the power in general and what is power of this test? (Assume 5% significance level)

Answer 1

Power is the probability of rejecting the null hypothesis, when in fact the null hypothesis is false

First, we need to find the sample mean corresponding mu = 20 with z scor 1.645 (for 0.05 alpha level using z distribution table)

sample mean x(bar) =

where mu = 20, z = 1.645 (right tailed), sigma= 3 and n = 10

= 20 + 1.56

= 21.56

Now, Power calculation

using x(bar) = 21.56, mu = 22, sigma = 3 and n = 10

this implies

= P(z>-0.464)

= P(z<0.464)

= 0.6772