Question

An automobile manufacturer claims that its cars average more than 410 miles per tankful (mpt).
As evidence, they cite an experiment in which 17 cars were driven for one tankful each and
averaged 420 mpt. Assume σ = 14 is known.

a. Is the claim valid? Test at the 5 percent level of significance.

b. How high could they have claimed the mpt to be? That is, based on this experiment, what is the maximum value for µ which would have been rejected as an hypothesized value?

c. What is the power of the test in part (a) when the true value of µ is 420 mpt? (Hint: Your rejection region for part (a) was stated in terms of comparing Zobs with a cut-off point on the Z distribution. Find the corresponding x̅cut-off and restate your rejection region in
terms of comparing the observed x̅value with the x̅cut-off. Then assume H1 is true (i.e. µ
= 420 mpt) and find the probability that x̅is in the rejection region.)

Answer 1

a)

Ho :   µ =   410
Ha :   µ >   410
      
Level of Significance ,    α =    0.05
population std dev ,    σ =    14
Sample Size ,   n =    17
Sample Mean,    x̅ =   420
      
'   '   '
      
Standard Error , SE =   σ/√n =   3.3955
      
Z-test statistic=   (x̅ - µ )/SE =    2.9451
      
  
p-Value   =   0.0016
Conclusion:     p-value<α, Reject null hypothesis   
so, claim is valid

b)

we reject null hypothesis,

when Z-stat>=Zα

(420-µ)/SE>=1.645

µ≤420-1.645*3.3955=414.4144

so, answer=414.4144

c)

true mean ,    µ =    420
      
hypothesis mean,   µo =    410
significance level,   α =    0.05
sample size,   n =   17
std dev,   σ =    14
      
δ=   µ - µo =    10
      
std error of mean,   σx = σ/√n =    3.3955

Zα =       1.6449   (right tailed test)
We will fail to reject the null (commit a Type II error) if we get a Z statistic <               1.6449
this Z-critical value corresponds to X critical value( X critical), such that              
              
       (x̄ - µo)/σx ≤ Zα      
       x̄ ≤ Zα*σx + µo      
       x̄ ≤    415.585   (acceptance region)
              
now, type II error is ,ß =    P( x̄ ≤    415.585   given that µ =   420
              
   = P ( Z < (x̄-true mean)/σx )          
   = P ( Z <    -1.300   )  
   =   0.097      
              
              
              
power =    1 - ß =   0.9032