Chi-Square Goodness of Fit Test | Colors Freq | UPSC ISS 2022 Paper-1 | Problem-58 | RitwikMath

This video solves a **Chi-square goodness of fit** problem, a standard application-based question in **UPSC Civil Services (Statistics Optional)** and **ISS examinations**.

A person has balls of five different colors with the following observed frequencies:

| Color | Black | Red | Green | Orange | Blue |
|--------|-------|-----|-------|--------|------|
| Observed (O) | 10 | 8 | 5 | 12 | 15 |

---

### 🔹 Step 1: State the Null Hypothesis

\[
H_0:\ \text{All colors occur with equal frequency}
\]

---

### 🔹 Step 2: Find Expected Frequencies

Total number of balls:
\[
10 + 8 + 5 + 12 + 15 = 50
\]

Number of color categories = 5

So, expected frequency for each color:
\[
E = \frac{50}{5} = 10
\]

---

### 🔹 Step 3: Compute the Chi-Square Statistic

\[
\chi^2 = \sum \frac{(O - E)^2}{E}
\]

| Color | O | E | (O−E)² | (O−E)² / E |
|--------|---|---|--------|-------------|
| Black | 10 | 10 | 0 | 0 |
| Red | 8 | 10 | 4 | 0.4 |
| Green | 5 | 10 | 25 | 2.5 |
| Orange | 12 | 10 | 4 | 0.4 |
| Blue | 15 | 10 | 25 | 2.5 |

\[
\chi^2 = 0 + 0.4 + 2.5 + 0.4 + 2.5 = 5.8
\]

---

### ✅ Final Answer

\[
\boxed{\chi^2 = 5.8}
\]

✅ Therefore, **Option B is the correct answer**.

This problem clearly demonstrates the application of the **Chi-square goodness of fit test**, a must-know topic for **Statistics Optional** and **ISS preparation**.
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