Question

A)In a study of distances traveled by buses before the first major engine failure, a sample of 191 buses results in a mean of 96,700 miles and a population standard deviation of 37,500 miles. Calculate the p-value corresponding to the test statistic used to test the claim that the mean distance traveled before a major engine failure is more than 90,000 miles and determine if the null hypothesis can be rejected at α = .01.

b) A major car manufacturer wants to test a new catalytic converter to determine whether it meets new air pollution standards. The mean emission of all converters of this type must be less than 20 parts per million of carbon. Ten (10) converters are manufactured for testing purposes and their emission levels are measured, with a mean of 17.17 and a standard deviation of 2.98. Test the hypotheses at α = .01.

Answer 1

a) Sample size = n = 191

Sample mean = = 96700

Population standard deviation = = 37500

Claim: The mean distance traveled before a major engine failure is more than 90,000 miles.

The null and alternative hypothesis is

Level of significance = 0.01

Here population standard deviation is known so we have to use z-test statistic.

Test statistic is

P-value = P(Z > 2.47) = 0.0068

P-value < 0.01 we reject the null hypothesis.

Conclusion:

The mean distance traveled before a major engine failure is more than 90,000 miles.

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b)  

Sample size = n = 10

Sample mean = = 17.17

Standard deviation = s = 2.98

Claim: The mean emission of all converters of this type must be less than 20 parts per million of carbon

The null and alternative hypothesis is

Level of significance = 0.01

Here population standard deviation is unknown so we have to use t-test statistic.
Test statistic is

Degrees of freedom = n - 1 = 10 - 1 = 9

P-value = P(T < - 3 ) = 0.0074

P-value < 0.01 we reject the null hypothesis.

Conclusion: The mean emission of all converters of this type must be less than 20 parts per million of carbon.