A consumer research group is interested in testing an automobile manufacturer\'s

Question

A consumer research group is interested in testing an automobile manufacturer's claim that a new economy model will travel at least 28 miles per gallon of gasoline (H0: 28).

With a .02 level of significance and a sample of 40 cars, what is the rejection rule based on the value of for the test to determine whether the manufacturer's claim should be rejected (to 2 decimals)? Assume that is 4 miles per gallon.

Reject H0 if is Selectless than or equal togreater than or equal toequal tonot equal toItem 1

What is the probability of committing a Type II error if the actual mileage is 26 miles per gallon (to 4 decimals)?

What is the probability of committing a Type II error if the actual mileage is 27 miles per gallon (to 4 decimals)?

What is the probability of committing a Type II error if the actual mileage is 28.5 miles per gallon?
SelectThe probability is greater than .5The probability is between .1 and .5The probability is .02A Type II error cannot be made because the null hypothesis is true.Item 5

Explanation / Answer

Solution:

Null hypothesis:
H0: 28
Ha: < 28

At = 0.02
From the Z- table Z = 2.054
Our test is left-tailed, so
Reject H0 if Z0 -2.054
Reject H0 if (x - 28)/(4/sqrt(40)) -2.054
Reject H0 if x 28 - 2.054 * 4/sqrt(40)
Reject H0 if x 26.70
------------------------------------------------
Need to find probability of Type II error if the true population mean is 26
Reject H0 if x 26.70
imples, fail to reject H0 if x 26.70
P(Type II error) = P(Fail to reject H0|HA is true )
= P(x > 26.70 | a = 26)
= P(x - /(/n) > 26.70 - 26/(4/sqrt(40))
= P( Z > 1.10)
= 0.8643
Therfore, the probability of commiting a Type II error if the actual mileage is 26 miles per gallon is 0.8643.
-------------------------------------------   
Need to find probability of Type II error if the true population mean is 27.
Fail to reject H0 if x > 26.70
P(Type II error) = P( Fail to reject H0|Ha is true)
= P(x > 26.70 | a = 27)
= P(x - /(/n) > 26.70 - 27/(4/sqrt(40))
= P( Z > -0.47)
= P ( Z < 0.47 ) = 0.6808
Therfore, the probability of commiting a Type II error if the actual mileage is 27 miles per gallon is 0.6808.
-----------------------------------------------
Need to find probability of Type II error if the true population mean is 28.5.
Fail to reject H0 if x > 26.70
P(Type II error) = P( Fail to reject H0|Ha is true)
= P(x > 26.70 | a = 28.5)
= P(x - /(/n) > 26.70 - 28.5/(4/sqrt(40))
= P( Z > -2.85)
= P( Z < 2.85 ) = 0.9978
Therfore, the probability of commiting a Type II error if the actual mileage is 28.5 miles per gallon is 0.9978.

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