We have a claim about a population mean, the average length of time a fuel injection system lasts before it needs to be replaced, and we are given the results of random sample of 10 fuel injection systems.
Our null hypothesis: Ho: μ = 48 months
Our alternative hypothesis μ ≠ 48 months
Note: We might assume that the consumer agency wants to test the claim that the fuel injection system lasts at least 48 months, in which case the alternative hypothesis would be μ > 48, but we don't want to assume too much, and a two sided test is the more conservative.
Calculate the sample mean and the sample standard deviation. I have used a TI graphing calculator to find:
Sample mean: x-bar = 43.4
Sample standard deviation: s= 10.37
Sample size: n=10
Now we can calculate our test statistic. We do not know the population standard deviation, so this is a t-test, with 9 degrees of freedom. Recall: Degrees of Freedom = n -1
t= (x-bar - µ) / (s/√n-1) = (43.4 - 48) / (10/√10-1) = -1.38
The problem does not specify a significance level, but if we assume α=.05, we will fail to reject the null hypothesis. Notice that even if we used a one tail test and a significance level of 0.01, we would still fail to reject the null hypothesis. The manufacturer's claim is reasonable.
