Decision Making and P-Values for a Left-Tailed Hypothesis

Below is an interactive exercise that you can use to calculate Type I and II errors, and p-values. Just to jog your memory, here are some definitions:

• P-values: The chance of observing the sample result or something more extreme under the null hypothesis
• Type I error (a): The chance of rejecting the null, when in fact it is true.
• Type II error (b): The chance of concluding the null hypothesis when the alternative is true.

The problem is based on the following hypothesis:

H₀: Bag C
H₁: Bag D

Remember that smaller values support Bag D, the alternative hypothesis. Therefore, the direction of extreme is to the left. This is an example of a left tailed hypothesis.

• So, when calculating the Type I error, we will be calculating the area to the left.
• As the p-value calculations are also based on the null hypothesis and direction of extreme, that area is also calculated to the left.
• As the p-value represents the chance of observing the sample result under the null, the smaller the p-value, the more it supports the alternative hypothesis.
• If the p-value <= a, we reject the null hypothesis in favour of the alternative.

How to use the interactive exercise:

• The red line represents your decision rule. All values to the left of the red line represent the areas where you would reject the null hypothesis in favour of the alternative hypotheis. That is, you would choose Bag D instead of Bag C. Therefore, when calculating the Type I error, we are calculating this area under the null hypothesis (the number of values under Bag C that is to the left of this line). As you move the line, you will see the change in the Type I error.
• Once a decision rule is made (i.e., you have moved your red line), the area to the right represents the region where you would conclude Bag C. As the Type II error is when you conclude Bag C under the alternative, you can calculate this from the graph. Just count the number of values under Bag D to the right of this line.
• Finally, p-value is the area to the left of the selected sample point under the null. Move your cursor so that it points to a value under the null hypothesis, and you will see the p-value. Note that whenever p-value > a, your conclusion is not to reject the null hypothesis. Try it.