P-value Approach to Hypothesis Testing

Recall that in the critical value approach to hypothesis testing, one must set the alpha in order to get the critical value. Then, if a test statistic is higher than this critical value, the null hypothesis will be rejected. Alternatively, the p-value is the probability of observing a value of the test statistics (ie. the mean) as extreme as that is observed. For example, the p-value for lower-tail test is

\[p-value = {P_{{H_0}}}(T \le {t_0})\]
And for upper tail test is
\[p-value = {P_{{H_0}}}(T \ge {t_0})\]
Where $t_0$ is the observed value of the test statistic $T$, and $P_{H_0}$ denotes the probability under the null hypothesis. The p-value is sometime referred to as the observed significance level.

For example, The p-value of average traval time 47.2 from the previous average 45 with standard deviation 1.5 is

>> mu=45;

>> sig=1.5;

>> xbar=47.2;

>> zobs = (xbar-mu)/sig;

>> pval=1-normcdf(zobs,0,1)

pval =

0.0712

The p-value is 0.0712. If we do the hypothesis test at 0.05 significant level, we would not reject the null hypothesis.