The Los Angeles County Metropolitan Transportation Authority has set a bus mechanical reliability goal of 3,900 bus miles. Bus mechanical reliability is measured specifically as the number of bus miles between mechanical road calls. Suppose a sample of 100 buses resulted in a sample mean of 3,975 bus miles and a sample standard deviation of 275 bus miles.
a. Is there evidence that the population mean bus miles is more than 3,900 bus miles? (Use a 0.05 level of significance.)
b. Determine the p-value and interpret its meaning.
a.
State the null and alternative hypothesis.
Null hypothesis:
That is, the population mean bus miles is 3,900 bus miles.
Alternative hypothesis:
That is, the population mean bus miles is more than 3,900 bus miles.
Decision rule:
If , then reject the null hypothesis .
Use MINITAB to compute .
The formula for is given below:
Where, is the sample mean, is the population mean, S is the sample standard deviation, and n is the sample size.
MINITAB procedure:
Step 1: Choose Stat > Basic Statistics > 1-Sample t.
Step 2: In Summarized data, enter the sample size 100 and mean 3,975.
Step 3: In Standard deviation, enter a value 275.
Step 4: In Perform hypothesis test, enter the test mean 3,900.
Step 5: Check Options, and enter Confidence level as 95.0.
Step 6: Choose greater than in Alternative.
Step 7: Click OK in all dialog boxes.
MINITAB output:
From the above MINITAB output, the value of is 2.73 and the p-value is 0.004.
Interpretation:
Use the significance level,
Here, p-value is lesser than the level of significance.
That is, .
Therefore, by the rejection rule, it can be concluded that there is an evidence to reject the null hypothesis at . That is, the population mean bus miles is more than 3,900 bus miles.
b.
Find the p -value and interpret:
From the above MINITAB output, the p-value is 0.004. When the population mean bus miles is 3,900 bus miles at 5% significance level then the probability of the test statistic which is greater than 2.73 standard error units away from zero is 0.004.