The Language of Hypothesis Testing
Master the foundational vocabulary and concepts that drive statistical inference and decision-making.
Null & Alternative Hypotheses
Hypothesis: A statement regarding a characteristic of one or more populations.
Hypothesis Testing: A process, based on sample evidence and probability, used to test statements regarding a characteristic of a population.
Null Hypothesis
A statement to be tested that is assumed to be true until evidence indicates otherwise. It typically represents "no effect" or "no difference."
Alternative Hypothesis
A statement that we are trying to find evidence to support. It represents what we believe might be true.
Think of it like a trial: The null hypothesis is "innocent until proven guilty." We assume it's true unless we have strong enough evidence (data) to reject it.
Types of Tests
The form of the alternative hypothesis determines the type of test:
Two-Tailed Test
Tests if the parameter is different from the hypothesized value (either direction).
Left-Tailed Test
Tests if the parameter is less than the hypothesized value.
Right-Tailed Test
Tests if the parameter is greater than the hypothesized value.
Type I & Type II Errors
When making a decision based on sample data, there are four possible outcomes:
| Reject | Type I Error (False Positive) | Correct Decision |
| Fail to Reject | Correct Decision | Type II Error (False Negative) |
Type I Error ()
Rejecting when it is actually true. The probability of this error is the Level of Significance ().
Type II Error ()
Failing to reject when is actually true. The probability of this error is denoted by .
Stating Conclusions
We NEVER "accept" the null hypothesis!
If We Reject
"There is sufficient evidence at the α level of significance to support the claim that [alternative hypothesis in words]."
If We Fail to Reject
"There is not sufficient evidence at the α level of significance to support the claim that [alternative hypothesis in words]."
Common Pitfalls
Saying "Accept H₀"
We never prove the null hypothesis true. We can only fail to find evidence against it.
Confusing α and β
α (Type I) is rejecting a true null; β (Type II) is failing to reject a false null.
Putting the Wrong Symbol in H₀
The null hypothesis always contains the equality (=, ≤, or ≥). The claim goes in the alternative if it involves <, >, or ≠.
Real-World Applications
💊 Clinical Trials
H₀: A new drug is no more effective than a placebo.
A Type I error would mean approving an ineffective drug. A Type II error would mean rejecting an effective one.
⚖️ Legal System
H₀: The defendant is innocent.
A Type I error is convicting an innocent person. A Type II error is letting a guilty person go free.
🏭 Quality Control
H₀: A manufacturing batch meets quality standards.
A Type I error is discarding a good batch. A Type II error is shipping a defective batch.
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