Hypothesis Testing
Null vs Alternative
Two competing explanations for the data
The distributions are separating. H₀ says the data comes from μ=0; H₁ says μ=1.5. The shaded overlap shows where we can't easily tell which hypothesis generated the data.
Step 1 of 6: Null vs Alternative
Two competing explanations for the data
The distributions are separating. H₀ says the data comes from μ=0; H₁ says μ=1.5. The shaded overlap shows where we can't easily tell which hypothesis generated the data.
Step 2 of 6: Test Statistics
Measuring how extreme the observed data is under H₀
z = 1.20 — the shaded tails (pink) show the probability of seeing a value this extreme or more under H₀. The p-value is 0.230 — the two-tailed probability.
Step 3 of 6: P-Values
The probability of seeing data this extreme if H₀ is true
p = 0.072 — getting smaller. The tails are narrowing, but p > 0.05 means we'd see data this extreme 7.2% of the time under H₀. Suggestive but not conclusive.
Step 4 of 6: Significance Level α
The threshold for rejecting H₀ — you choose it before looking at data
At α = 0.050, the critical value is ±1.96. Our z = 2.1 falls in the pink rejection region (p = 0.0357 < α). We reject H₀.
Step 5 of 6: Type I & Type II Errors
False positives vs false negatives — you can't minimize both
Low power (32%). The alternative (purple) overlaps heavily with the acceptance region. β = 0.68 — we'd miss the effect most of the time. Increase the effect size or lower α less to improve power.
Step 6 of 6: Power & Sample Size
How many samples you need to reliably detect an effect
Power = 61% — getting better but below the conventional 80% threshold. The standard error shrinks as √n, so each additional sample helps. Need n ≈ 32 for 80%.