Question 1190619
Here's how to perform a Chi-Square test for independence to determine if family size and education level are independent:

**1. State the Hypotheses:**

*   **Null Hypothesis (H₀):** Family size and education level are independent.
*   **Alternative Hypothesis (H₁):** Family size and education level are not independent (they are associated).

**2. Set the Significance Level:**

α = 0.05 (given)

**3. Create the Contingency Table (Observed Frequencies):**

You already have this:

| No. of Children | 0 to 1 | 2 to 3 | Over 3 | Total (Row) |
|---|---|---|---|---|
| Elementary | 14 | 37 | 32 | 83 |
| Secondary & Above | 31 | 59 | 27 | 117 |
| Total (Column) | 45 | 96 | 59 | 200 |

**4. Calculate Expected Frequencies:**

The expected frequency for each cell is calculated as:

(Row Total * Column Total) / Grand Total

Here's the table of expected frequencies:

| No. of Children | 0 to 1 | 2 to 3 | Over 3 |
|---|---|---|---|
| Elementary | (83*45)/200 = 18.68 | (83*96)/200 = 39.84 | (83*59)/200 = 24.48 |
| Secondary & Above | (117*45)/200 = 26.33 | (117*96)/200 = 56.16 | (117*59)/200 = 34.52 |

**5. Calculate the Chi-Square Statistic (χ²):**

χ² = Σ [(Observed Frequency - Expected Frequency)² / Expected Frequency]

χ² = [(14-18.68)²/18.68] + [(37-39.84)²/39.84] + [(32-24.48)²/24.48] + [(31-26.33)²/26.33] + [(59-56.16)²/56.16] + [(27-34.52)²/34.52]

χ² ≈ 1.23 + 0.20 + 2.26 + 0.85 + 0.14 + 2.09

χ² ≈ 6.77

**6. Determine Degrees of Freedom (df):**

df = (Number of Rows - 1) * (Number of Columns - 1)
df = (2 - 1) * (3 - 1) = 1 * 2 = 2

**7. Find the Critical Value:**

Using a Chi-Square distribution table with df = 2 and α = 0.05, the critical value is approximately 5.99.

**8. Make a Decision:**

*   **Compare χ² to the critical value:** Our calculated χ² (6.77) is *greater* than the critical value (5.99).

*   **Conclusion:** Because our χ² value exceeds the critical value, we reject the null hypothesis.

**9. Interpret the Results:**

There is sufficient evidence at the 5% level of significance to conclude that family size and education level are *not* independent.  There is an association between the two variables.