Question

A consumer research group is interested in testing an automobile manufacturer's claim that a new economy model will travel at least 27 miles per gallon of gasoline (H ₀:   27).

1. With a .02 level of significance and a sample of 40 cars, what is the rejection rule based on the value of x for the test to determine whether the manufacturer's claim should be rejected (to 2 decimals)? Assume that x is 5 miles per gallon.
   
   Reject H ₀ if x is less than or equal to ______  
   
2. What is the probability of committing a Type II error if the actual mileage is 25 miles per gallon (to 4 decimals)?
   
3. What is the probability of committing a Type II error if the actual mileage is 26 miles per gallon (to 4 decimals)?

Answer 1

a)

assuming that x is sigma = 5

Ho : mu >= 27

Ha : mu < 27

at alpha = 0.02

We reject if P(Z < z) = 0.02

z = -2.054

(Xbar - mu)/(s/sqrt(n)) = -2.054

Xbar < 27 - 2.054 * 5/sqrt(40) = 25.3762

b)

type ii error = fail to reject the null when null is false

P(Xbar > 25.3762| mu = 25)

= P(Z > (25.3762 - 25)/(5/sqrt(40) )

= P(Z > 0.4758595 )

= 0.3171

c)

P(Xbar > 25.3762| mu = 26)

= P(Z > (25.3762 - 26)/(5/sqrt(40) )

= P(Z >   -0.78905 )

= 0.7849