With individual lines at its various windows, a post office finds that the standard deviation for waiting times for customers on Friday afternoon is 7.2 minutes. The post office experiments with a single, main waiting line and finds that for a random sample of 25 customers, the waiting times for customers have a standard deviation of 3.5 minutes on a Friday afternoon.
With a significance level of 5%, test the claim that a single line causes lower variation among waiting times for customers.
Id like to test the hypothesis that the waiting times for a single line are longer.
For this problem, I get Chi critical value of 36.41502850. Which is much larger than noted in the solution. If Chi squared test statistic value is larger than the critical value, then the null hypothesis ( that the waiting times for singe line has a more variation could be rejected? ) What if one would like to determine if one has to wait longer than 7.2 minutes? Then I guess the appropriate F score would be 1-$/alpha$ quantile?
Best Answer
First you have to assume Normality...
Then your Hypothesis' System to be verified is the following
$$\begin{cases} \mathcal{H}_0: & \sigma^2=51.84 \\ \mathcal{H}_1: & \sigma^2<51.84 \end{cases}$$
thus your critical chi value at 5% is $13.8484$
As your Observed Statistic is $\frac{24\times3.5^2}{7.2^2}\approx5.67$ this means that you are at the left of your critical value: you reject $\mathcal{H}_0$ that is: there are enough evidences against the null hypothesis to agree with the claim that "a single line causes lower variation among waiting times for customers".