In: Statistics and Probability
CHAPTER 10 LECTURE NOTES EXAMPLE #3
A car manufacturer wants to test a new engine to see whether it
meets new air pollution standards. The mean emission, μ, of all
engines of this type must be less than 20 parts per million of
carbon. Ten engines are manufactured for testing purposes, and the
mean and standard deviation of the emissions for this sample of
engines are determined to be:
X¯¯¯=17.1 parts per million s=3.0 parts per million n = 10X¯=17.1 parts per million s=3.0 parts per million n = 10
Do the data supply sufficient evidence to allow the manufacturer to
conclude that this type of engine meets the pollution standard?
Assume that the manufacturer is willing to risk a Type I error with
probability equal to α = .01. Do a complete and appropriate
hypothesis test.
Step 1 (Hypotheses)
H0: μ Correct ≥ Correct 20 20 Correct
HA: μ Correct < Correct 20 20 Correct
Step 2 (Decision rule)
Using only the appropriate statistical table in your textbook, the critical value for rejecting H0 is - Correct3.056 3.056 Incorrect . (report your answer to 3 decimal places, using conventional rounding rules)
Step 3 (Test statistic)
Using the sample data, the calculated value of the test statistic is - Correct3.25 3.25 Incorrect . (report your answer to 4 decimal places, using conventional rounding rules)
Step 4 (Evaluate the null hypothesis)
Should the null hypothesis be rejected? yes Correct
Step 5 (Practical conclusion)
Can the manufacturer conclude that this type of engine meets the pollution standard? yes Correct
Using only the appropriate statistical table in your textbook, what is the most accurate statement you can make about the numerical value of the p-value of this hypothesis test?
Answer: The p-value is less than 0.03 so the results are significant. The p-value is less than 0.03 so the results are significant. Incorrect (provide a one-sentence statement about the p-value)
Given ,
Sample size = n = 10
sample mean = = 17.1 parts per million
Standard deviation = s = 3 parts per million
Population standard deviation is not known , so we have to perform one sample t test.
1) Hypothesis :
( Claim )
Left tailed test .
2) Decision rule :
Significance level = = 0.01
df = n - 1 = 10 - 1 = 9
Critical value for this left tailed test is ,
{ Using Excel function , =T.INV( 0.01 , 9 ) = -2.831 }
Rejection region = { t : t < -2.821 }
3) Test statistic :
4) Decision about null hypothesis :
It is observed that test statistic ( -3.0569 ) is less than critical value (-2.821 )
So reject null hypothesis .
5) Conclusion :
There is sufficient evidence to conclude that this type of engine meets the pollution standard.
P-value :
P-value for this left tailed test is given by,
P-value = P( t < test statistic ) = P( t < -3.0569 )
Using Excel function , = T.DIST( t , df , 1 )
P( t < -3.0569 )= T.DIST( -3.0569 , 9 , 1 ) = 0.0068
P-value = 0.0068
P-value ( 0.0068 ) is less than significance level = 0.01 , so the results are significant.