Question 1185389
Here's how to perform a Chi-Square Goodness of Fit test to determine if the region's proportions differ from the national proportions:

**1. State the Hypotheses:**

*   **Null Hypothesis (H₀):** The distribution of ages in the region is the same as the national distribution.
*   **Alternative Hypothesis (H₁):** The distribution of ages in the region is different from the national distribution.

**2. Set the Significance Level:**

α = 0.10

**3. Calculate Expected Frequencies:**

Multiply the national proportions by the total number of participants in the region (202):

*   5-year-olds: 0.12 * 202 = 24.24
*   4-year-olds: 0.43 * 202 = 86.86
*   3-year-olds: 0.20 * 202 = 40.4
*   Under 3: 0.25 * 202 = 50.5

**4. Calculate the Chi-Square Statistic:**

Use the formula:  χ² = Σ [(Observed - Expected)² / Expected]

| Age Group | Observed (O) | Expected (E) | (O - E)² | (O - E)² / E |
|---|---|---|---|---|
| 5-year-olds | 30 | 24.24 | 33.86 | 1.396 |
| 4-year-olds | 83 | 86.86 | 14.89 | 0.171 |
| 3-year-olds | 51 | 40.4 | 112.36 | 2.781 |
| Under 3 | 38 | 50.5 | 156.25 | 3.094 |
| **Total** | 202 | 202 |  | **7.442** |

χ² = 7.442

**5. Determine the Degrees of Freedom:**

Degrees of freedom (df) = Number of categories - 1 = 4 - 1 = 3

**6. Find the Critical Value (or P-value):**

*   **Using a Chi-Square Table:** Look up the critical value for df = 3 and α = 0.10. The critical value is approximately 6.251.
*   **Using a Calculator or Software:** A calculator or statistical software can provide a more precise p-value.

**7. Calculate the P-value:**

Using statistical software or a calculator, with χ² = 7.442 and df = 3, the p-value is approximately 0.059.

**8. Make a Decision:**

*   **Using the Critical Value:** Since our calculated χ² (7.442) is greater than the critical value (6.251), we reject the null hypothesis.
*   **Using the P-value:** Since the p-value (0.059) is less than the significance level (0.10), we reject the null hypothesis.

**9. Conclusion:**

There is sufficient evidence at the α = 0.10 level to conclude that the region's proportions of children in the Head Start program differ from the national proportions.

**Answers:**

*   **The test value is 7.442.**
*   **The p-value is 0.059.**