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10 changes: 9 additions & 1 deletion docs/stats/hypothesis.md
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Expand Up @@ -380,11 +380,19 @@ $$ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} $$
If $H_0$ is true, both groups should convert at the Global Rate (15%).
* Blue Expected Buys: $100 \times 0.15 = 15$ ($E_1$).
* Red Expected Buys: $100 \times 0.15 = 15$ ($E_2$).

| Group | Bought (E) | Didn't Buy (E) | Total |
| :--- | :--- | :--- | :--- |
| **Blue** | 15 | 85 | 100 |
| **Red** | 15 | 85 | 100 |

4. **Calculate $\chi^2$:**
* Blue Buys: $(10 - 15)^2 / 15 = 25/15 = 1.66$
* Red Buys: $(20 - 15)^2 / 15 = 25/15 = 1.66$
* *Do the same for "Didn't Buy" cells...*
* Blue didn't buy: $(90−85)^2/85 = 0.29$
* Red didn't buy: $(80−85)^2/85 = 0.29$
* Sum it up
* $\chi^2 \approx 3.92$.

5. **Compute P-Value:**
Expand Down Expand Up @@ -486,4 +494,4 @@ We started with a promise: to learn the universal algorithm of inference.
$$ \text{Score} = \frac{\text{Observed Signal}}{\text{Expected Noise}} $$
4. **The Decision:** If the Signal is much stronger than the Noise (High Score, Low P-value), we accept the new reality.

**You have now graduated from "guessing" to "inference."**
**You have now graduated from "guessing" to "inference."**