SOLUTION: 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

Question 1185468: 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 9% five-year-olds, 42% four-year-olds, 20% three-year-olds, and 29% under three years. When the program was assessed in a particular region, it was found that of the 214 randomly selected participants, 21 were 5 years old, 92 were 4 years old, 45 were 3 years old, 56 were under 3 years old. Perform a test to see if there sufficient evidence at α = 0.05 that the region's proportions differ from the national proportions.
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 conduct a Chi-Square Goodness-of-Fit test to compare the regional age distribution to the national distribution:
**1. State the Hypotheses:**
* **Null Hypothesis (H₀):** The age distribution of participants in the region is the same as the national distribution.
* **Alternative Hypothesis (H₁):** The age distribution of participants in the region is different from the national distribution.
**2. Calculate Expected Frequencies:**
Multiply each national proportion by the total number of regional participants (214):
* 5-year-olds: 0.09 * 214 = 19.26
* 4-year-olds: 0.42 * 214 = 89.88
* 3-year-olds: 0.20 * 214 = 42.8
* Under 3: 0.29 * 214 = 62.06
**3. Calculate the Chi-Square Statistic:**
χ² = Σ [(Observed - Expected)² / Expected]
χ² = (21 - 19.26)² / 19.26 + (92 - 89.88)² / 89.88 + (45 - 42.8)² / 42.8 + (56 - 62.06)² / 62.06
χ² ≈ 0.163 + 0.051 + 0.157 + 0.592
χ² ≈ 0.963
**4. Determine Degrees of Freedom:**
df = Number of categories - 1 = 4 - 1 = 3
**5. Find the p-value:**
Consult a Chi-Square distribution table with df = 3. A χ² value of 0.963 is quite low. The p-value will be much greater than 0.05
**6. Make a Decision:**
* **Decision:** Since the p-value is greater than α = 0.05, we fail to reject the null hypothesis.
**7. Conclusion:**
There is not sufficient evidence at the α = 0.05 significance level to conclude that the age distribution of Head Start participants in this region differs from the national distribution.
**Answers:**
* The test value (χ²) is approximately **0.963**.
* The p-value is approximately **> 0.05** (A Chi-Square table will show a p-value > 0.05. Statistical software can give a more precise answer.)

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