Question

A major car manufacturer wants to test a new engine to determine whether it meets new air pollution standards. The mean emission of all engines of this type must be lower than 20 parts per million of carbon. A number of engines are manufactured for testing purposes, and the emission level of each is determined. The data (in parts per million) are listed below:

15.6, 16.2, 22.5, 20.5, 16.4, 19.4, 16.6, 17.9, 12.7, 13.9

a). At 5% level of significance, is there sufficient evidence to allow the manufacturer to conclude that this type of engine meets the pollution standard? Your conclusion must be in terms of the P-Value as well as setting up a Rejection Region. Please show work.

b). Which statistical distribution should be applied in this situation and why? Explain carefully.

c). Knowing that a significant amount of capital investment were required to manufacture the engine, what, if anything, does the manufacturer have to be concerned about with respect to the conclusion of part (a)? Explain

d). Based on a 95% confidence level, estimate the mean emission of all engines.

e). Carefully interpret this interval estimation.

f). Explain carefully whether or not there is sufficient evidence to allow the manufacturer to conclude that this type of engine meets the pollution standard using the estimation in

Answer 1

a).

The Null and Alternative Hypotheses are defined as,

This is a left tailed test.

The significance level = 0.05

Rejection Region: P-value<0.05

The t statistic is obtained using the formula,

For calculation purpose, the mean and standard deviation are obtained in excel using the function =Average() and =STDEV(). The screenshot is shown below,

Now,

P-value

The p-value is obtained from t distribution table for t = -3.002 and degree of freedom = n - 1 = 10 - 1 = 9. (In excel use function =T.DIST.RT(3.002,9))

Conclusion

Since the P-value is less than the significance level = 0.05 at 5% significance level, the null hypothesis is rejected. Hence we can conclude that the mean emission level significantly lower than 2.0

b)

Since we are comparing one sample mean with the population mean and the population standard deviation is not known, the t-distribution is be used to test the hypothesis that the mean emission level is less than 20.

c)

Since the result is significant in part a, the manufacturer can invest in the new car engine.

d)

The confidence interval for the mean is obtained using the formula,

The t critical value is obtained from t distribution table for significance level = 0.05 and degree of freedom = n -1 = 10 - 1 = 9

e)

There is a 95% chance that the population mean will lie in the interval (15.037,19.303)

f)

Since the hypothesized population mean doesn't lie in the 95% confidence interval we can say that there is significantly less emission from a new car engine.