Final answer:
When the area corresponding to the critical value is in the lower tail of the sampling distribution, 1) The p-value is the area under the curve.
Step-by-step explanation:
When the area corresponding to the critical value is in the lower tail of the sampling distribution, the 1)p-value is the area under the curve. The p-value represents the probability of observing a test statistic as extreme or more extreme than the one obtained from the sample, assuming the null hypothesis is true.
In the context of a hypothesis test, the critical value is a threshold beyond which we reject the null hypothesis. If the critical value is in the lower tail, the p-value is the cumulative probability of obtaining a test statistic as extreme or more extreme than the observed value, considering the entire area under the curve to the left of the critical value.
Mathematically, the p-value is calculated by finding the probability of the observed test statistic or a more extreme value occurring under the null hypothesis. This involves integrating the probability density function (PDF) of the sampling distribution.
The lower the p-value, the stronger the evidence against the null hypothesis. In the case where the critical value is in the lower tail, the p-value is the cumulative probability in that tail, indicating the likelihood of observing a value as extreme as, or more extreme than, the observed test statistic.
In summary, understanding the placement of the critical value in the sampling distribution helps determine where to look for the p-value. When it is in the lower tail, the p-value is the area under the curve to the left of the critical value, indicating the probability of obtaining a test statistic as extreme or more extreme under the null hypothesis.