SOLUTION: Random samples of 200 men, all retired were classified according to education and number of children is as shown below
`
No of children
Education
0 to 1
2 to 3
Over 3
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-> SOLUTION: Random samples of 200 men, all retired were classified according to education and number of children is as shown below
`
No of children
Education
0 to 1
2 to 3
Over 3
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Question 1190619: Random samples of 200 men, all retired were classified according to education and number of children is as shown below
`
No of children
Education
0 to 1
2 to 3
Over 3
Elementary
14
37
32
Secondary and above
31
59
27
Test the hypothesis that the size of the family is independent of the level of education attained by fathers. (Use 5% level of significance) Answer by CPhill(1959) (Show Source):
You can put this solution on YOUR website! 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.
