Question

Question 6 A major car manufacturer wants to test a new engine to determine if it meets new air pollution standards. The mean emission, u 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 are roughly Normally distributed with o = 3. If the mean emission of all engines is, in fact, u = 22, what is the power of this test? Draw a picture to illustrate the power. Note: Assume 5% significance level.

Answer 1

At 0.05 ssignificance level, the critical value is z_crit = 1.645

z_crit = 1.645

or, ($\bar x_{crit}$ - $\mu$ )/($\sigma/\sqrt n$) = 1.645

or, ($\bar x_{crit}$ - 20)/(3/$\sqrt{10}$) = 1.645

or, $\bar x_{crit}$ = 1.645 * 3/$\sqrt{10}$ + 20

or, $\bar x_{crit}$ = 21.56

$\beta$ = P($\bar x$ < 21.56)

= P(($\bar x$ - $\mu$ )/($\sigma/\sqrt n$) < (21.56 - $\mu$ )/($\sigma/\sqrt n$))

= P(Z < (21.56 - 22)/(3/$\sqrt{10}$))

= P(Z > -0.46)

= 1 - P(Z < -0.46)

= 1 - 0.3228

= 0.6772

Power = 1 - $\beta$

= 1 - 0.6772 = 0.3228