Sweetness of Magic Potion (Two-Side Hypothesis Testing)

Let’s say you work as a wizard’s apprentice and you are responsible for brewing a magic potion. The master checked and claimed that the sweetness of the potion is just right with a mean sweetness level of 8 on a scale from 1 to 10. However, you’ve recently made a change to one element of the potion recipe and you suspect it might have affected the sweetness.

Here’s an example of hypothesis testing to determine if the sweetness of the magic potion has changed after the recipe modification. First, we set up the hypotheses:
– Null Hypothesis (): The mean sweetness of the potion is still 8 after the recipe modification.
– Alternative Hypothesis (): The mean sweetness of the potion has changed (it’s not equal to 8).

In mathematical terms, if we denote $\mu$ as the population mean of sweetness level, then the hypotheses can be written as:

and

This is a hypothesis for the mean. So if we can collect at least 30 observations, we can use the z-test. Suppose that you collect a random sample of 30 potions, measure their sweetness levels, and find that the mean sweetness level is and the standard deviation is .

We set the significance level .

The test statistic is

Note that unlike in previous examples, our alternative hypothesis in this case is . Therefore, the abnormal events can fall on either the left or the right tail of the distribution. So we can visualize the critical regions in relation to critical values like this:

In the figure, note that since and the abnormal events can fall on either the left or the right tail of the distribution, the critical region for each side (left/right) is based on to ensure that the sum of the dotted areas is equal to . Also, the standard normal distribution has a bell shape that is symmetric around the mean 0. Therefore, as the critical value for the right tail is , the critical value for the left tail is .

Now since , we see that does not fall into any of the rejection regions. So from this experiment, you cannot conclude that the sweetness level has changed yet.