Question 1185389: The head start program provides a wide range of services to low-income children up to the age of 5 years and their families. Its goals are to provide services to improve social and learning skills and to improve health and nutrition status so that the participants can begin school on an equal footing with their more advantaged peers. The distribution of ages for participating children is as follows 12% five-year-olds, 43% four-year-olds, 20% three-year-olds, and 25% under three years. When the program was assessed in a particular region, it was found that of the 202 randomly selected participants, 30 were 5 years old, 83 were 4 years old, 51 were 3 years old, 38 were under 3 years old. Perform a test to see if there sufficient evidence at α = 0.10 that the region's proportions differ from the national proportion.
The test value is_____
The p-value is____
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.**
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