Critical Value Approach to Hypothesis Testing

Usually, we define a test statistic by
\[z = \frac{{\bar x - {\mu_0}}}{{{\sigma_{\bar X}}}} = \frac{{\bar x - {\mu_0}}}{{{\sigma_X}/\sqrt n }}\]
Let's do some practical example. If in the year 2000, the average travel time in town A is 45 minutes, a recent survey in the year 2010 of the same town with n=100 commute times found that the average travel time is 47.2 minutes with the standard deviation $\sigma=15$ minutes. Has the average commute time really increased?

Here we have the statistical hypotheses:
H0: $\mu=45$ minutes
H1: $\mu> 45$ minutes

Thus, based on the above information
\[z = \frac{{47.2 - 45}}{{15/\sqrt {100} }} = \frac{{2.2}}{{1.5}} = 1.47\]

This is an upper tail test: 
\[P_{H_0}(T \le t) = 1- \alpha \]
We want conduct this test at the significant level $\alpha=0.05$. Conceptually, the rejection region is the top 5% of the area under a standard normal curve, as shown by the following MATLAB code

>> cv = norminv(0.95,0,1)

cv =
    1.6449

>> p=normspec([cv,inf],0,1)

p =
    0.0500

The 1.6449 is the critical value for rejecting H0 at $\alpha=0.05$. Since the statistic test result 1.47 is less than the critical value, we do not reject H0. This means the average travel time in 2010 is not really different from 2000.