AI Grounds
Open AI Grounds on a desktop
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AI Grounds
These interactive lessons need a larger screen. Please continue on a desktop or laptop computer.
Guided discovery
Change a true effect or study size; compare detection probability.
One toy study plan: independent normal A/B observations, known population SD 4 in each group, 25 observations per group (50 total), and assumed true population gap 1.00. These are planning assumptions, not observed data or collected samples.
H0: true gap = 0. Fixed predeclared two-sided 5% rule: reject if |Z| ≥ 1.9600, including equality.
Assumed population mean B minus A, in score units. A true effect, not a sample gap. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
Independent observation count in each group; both mean variances contribute uncertainty. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
True gap 1.00; size 25 per group. Specified nonzero effect: power ≈ 14.32%. Miss probability ≈ 85.68%.
The curve’s expected Z is 0.8839 and standardized SD stays 1. Orange areas beyond both fixed indigo dashed boundaries are rejection probability. With this nonzero true gap, those rejections detect the specified effect; the unshaded center is misses. There is no observed-statistic marker because this is a study plan.
Expected Z = 0.8839; standardized SD = 1. Fixed boundaries: ±1.9600. Rejection probability ≈ 14.32%; non-rejection ≈ 85.68%.
The finite plot shows −14 to 14 with density scale 0 to 0.45. Numerical probabilities include the full real line. Density height is not point probability. Larger size shrinks raw difference SE; after standardization it shifts expected Z for a nonzero effect, while plotted SD remains 1.
Difference SE = √(16 / 25 + 16 / 25) = 1.1314. Expected Z = true gap / SE = 1.00 / 1.1314 = 0.8839. Standardized effect d = true gap / 4 = 0.2500.
d expresses the signed true gap in population SD units. Expected Z also depends on size: d√(n / 2). Neither is a realized observed statistic or an estimated effect from data.
Correct detection given this true gap and model.
≈ 14.32%
P(reject H0 | assumed true gap)
Signed true gap in population SD units.
0.2500
True gap / SD = 1.00 / 4
Uncertainty of the observed mean difference.
1.1314
√(4² / 25 + 4² / 25)
Power conditions on the assumed true effect and model. It is not posterior truth probability, a p-value, causal proof or practical importance. Percentages are rounded; even power displayed as 100% can have a tiny positive miss probability. Bigger samples do not repair bias.
| Size per group | Difference SE | Expected Z | Power | Non-rejection |
|---|---|---|---|---|
| 25 | 1.1314 | 0.8839 | ≈ 14.32% | ≈ 85.68% |
| 100 | 0.5657 | 1.7678 | ≈ 42.39% | ≈ 57.61% |
| 400 | 0.2828 | 3.5355 | ≈ 94.24% | ≈ 5.76% |
Rejection probability ≈ 0.1431743420; non-rejection probability ≈ 0.8568256580. Expected Z = 0.8838834765. Fixed α ≈ 0.04999986002. Full precision is used before display rounding.
With λ = true gap / SE and c = 1.959964, rejection probability is Q(c − |λ|) + Q(c + |λ|), while non-rejection is Φ(c − |λ|) − Φ(−c − |λ|). Q is the standard-normal upper tail and Φ its CDF. Non-rejection is computed directly, preserving tiny positive values when rejection rounds to 1. A&S 26.2.17 supplies moderate tails; NIST DLMF 7.9.2 supplies a continued fraction above distance 4. Under true gap zero, rejection probability is α, not correct detection of an effect.
The normal known-SD assumptions make the Z law exact for the declared study plan. Unknown spread, paired or unequal groups, non-normal small samples, dependence, biased sampling, effect uncertainty, multiple comparisons and sequential peeking require other analysis. Choose the real test before collecting data. No universal effect magnitude or sample count guarantees practical importance, causal validity or detection. The input effect is a planning assumption; this is not post hoc observed power.
Berkeley SticiGui · Power against a specified alternative