Welcome back to Professor Baker's Math Class! In this session, we are tackling one of the most important topics in inferential statistics: Hypothesis Testing (Chapters 9.1 to 9.3). This is where we transition from simply describing data to making decisions about populations based on sample evidence.

1. Setting the Stage: Null vs. Alternative Hypotheses

Every hypothesis test begins with two opposing claims about a population parameter (like the mean, $\mu$).

  • The Null Hypothesis ($H_0$): This is the status quo hypothesis to be tested. It always specifies a single value using an equal sign. Mathematically, we express it as: $$H_0: \mu = \mu_0$$
  • The Alternative Hypothesis ($H_a$): This is the claim we are testing for—it acts as the alternative to the null. The direction of the inequality determines the "tail" of the test:

We encountered three specific scenarios in our class notes:

  1. Two-Tailed Test ($H_a: \mu \neq \mu_0$): Used when checking for any difference (e.g., the snack food company checking if bags are not 454g).
  2. Left-Tailed Test ($H_a: \mu < \mu_0$): Used when checking if the mean is less than a value (e.g., is calcium intake below 1000mg?).
  3. Right-Tailed Test ($H_a: \mu > \mu_0$): Used when checking if the mean is greater than a value (e.g., are women today taller than 62.6 inches?).

2. The Test Statistic

To make a decision, we compare our sample data to the null hypothesis. We do this by calculating a Z-score, which serves as our test statistic. The formula is:

$$z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

Where $\bar{x}$ is our sample mean, $\mu_0$ is the hypothesized population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size.

3. Making a Decision: Two Approaches

Once we have our test statistic, how do we know if it is statistically significant? We discussed two methods.

The Critical Value Approach

This method involves defining a Rejection Region based on your significance level ($\alpha$). The boundary of this region is called the Critical Value.

  • If your test statistic falls inside the rejection region, you Reject $H_0$.
  • If it falls in the non-rejection region, you Do Not Reject $H_0$.

Example from class: For a standard two-tailed test with $\alpha = 0.05$, the critical values are $z = \pm 1.96$.

The P-Value Approach

The P-value represents the probability of obtaining sample data as extreme as ours, assuming the null hypothesis is true. This leads to a simple rule of thumb:

If $P \le \alpha$, reject $H_0$.
If $P > \alpha$, do not reject $H_0$.

In our Golf Driving Distance example (Slide 13), we calculated a Z-score of $-2.65$. This resulted in a P-value of roughly $0.4\%$. Since $0.4\% < 5\%$ (our significance level), we rejected the null hypothesis.

Summary

Whether you look at the critical region or calculate the exact probability (P-value), the logic remains the same: if the sample data is incredibly unlikely to happen by chance under the null hypothesis, we conclude that the null hypothesis is likely false. Keep practicing identifying the correct tails for your tests—it is the first step to getting the right answer!