Lesson 1: The Logic of Hypothesis Testing

The Logic of Hypothesis Testing

Hypothesis testing is like a criminal trial. We start with an assumption of innocence (Null Hypothesis) and only reject it if we have overwhelming evidence to the contrary.

1. Setting the Stage

Null Hypothesis ( $H_0$ )

The status quo. “Nothing is happening.” “There is no difference.”

Example

The new drug has no effect on recovery time.

$\mu_{new} = \mu_{old}$

Alternative Hypothesis ( $H_1$ )

The claim we want to test. “Something is happening.”

Example

The new drug reduces recovery time.

$\mu_{new} < \mu_{old}$

2. The P-Value

The p-value is the probability of observing data at least as extreme as what we saw, assuming the Null Hypothesis is true.

Rule of Thumb:

Low p-value (< 0.05): The data is unlikely under $H_0$ . We Reject $H_0$ . (Statistically Significant)
High p-value (> 0.05): The data is consistent with $H_0$ . We Fail to Reject $H_0$ .

3. Errors

We can make mistakes:

Type I Error ( $\alpha$ ): Rejecting a true Null Hypothesis (False Positive).
Type II Error ( $\beta$ ): Failing to reject a false Null Hypothesis (False Negative).

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Lesson 1