Question

The Boston public school district has had difficulty maintaining on-time bus service for its students. Suppose the district develops a new bus schedule to help combat chronic lateness on a particularly woeful route. After the schedule adjustment, the first 36 runs were an average of 8 minutes late. As a result, the Boston public school district claimed that the schedule adjustment was an improvement—students were late less than 15 minutes. Assume a population standard deviation for bus arrival time of 12 minutes. At a 1% significance level, the decision is to ________.

		reject H0 we can conclude that the average lateness of Boston public school is less than 15 minutes.
		reject H0 we cannot conclude that the average lateness of Boston public school is less than 15 minutes.
		Not reject H0 we cannot conclude that the average lateness of Boston public school is less than 15 minutes.
		Not reject H0 we cannot conclude that the average lateness of Boston public school is less than 15 minutes.

Answer 1

The answer is:

reject H0 we can conclude that the average lateness of Boston public school is less than 15 minutes.

The test:

One-Sample Z test

The sample mean is Xˉ=8, the population standard deviation is σ=12, and the sample size is n=36. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: μ =15 Ha: μ <15 This corresponds to a Left-tailed test, for which a z-test for one mean, with known population standard deviation will be used. (2a) Critical Value Based on the information provided, the significance level is α=0.01, therefore the critical value for this Left-tailed test is Zc​=-2.3263. This can be found by either using excel or the Z distribution table. (2b) Rejection Region The rejection region for this Left-tailed test is Z<-2.3263 (3) Test Statistics The z-statistic is computed as follows: (4) The p-value The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true. In this case, the p-value is p =P(Z<-3.5)=0.0002 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that Z=-3.5 < Zc​=-2.3263, it is then concluded that the null hypothesis is rejected. (b) Using p-value method Using the P-value approach: The p-value is p=0.0002, and since p=0.0002≤0.01, it is concluded that the null hypothesis is rejected. (6) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ  is less than 15, at the 0.01 significance level.