Question

Whenever you are asked to test a hypothesis, be sure to: (a) state the null and alternative hypotheses; (b) state the relevant sample statistic; (c) give the rejection region; (d) compute the test; (e) give your decision and a conclusion in English.

1. Assume that last year, licensed American drivers drove an average of 10,000 miles, with a standard deviation of 2,000 miles (these are population figures).  This year, the government campaigned to get people to save gas by driving less.  To test the effectiveness of the campaign, a study is conducted.  A sample of 100 drivers is drawn at random from the general population and the number of miles driven by each person is recorded.  On the average, these 100 drivers drove 11,000 miles.  Was the campaign effective?  Use alpha = .01.

Answer 1

The provided sample mean is = 11000 and the known population standard deviation is = 2000, and the sample size is n = 100.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: ≤ 10000

Reduction in driving miles than before

Ha: >10000

No reduction in driving miles than before

This corresponds to a right-tailed test, for which a z-test for one mean, with known population standard deviation, will be used.

(2) Rejection Region

Based on the information provided, the significance level is = 0.01, and the critical value for a right-tailed test is = 2.33.

(3) Test Statistics

The z-statistic is computed as follows:

z = 5

(4) The decision about the null hypothesis

Since it is observed that z = 5 > = 2.33, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0, and since p = 0 < 0.01p=0<0.01, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is greater than 10000, at the 0.01 significance level.

Hence, there is no reduction in the driving miles

No effect of the campaign.