### Part (a): Chi-Square Test for Independence
To test the hypothesis that family size and level of education attained by the father are independent, we will use the Chi-Square test for independence.
#### Step-by-Step Process:
1. **State the Hypotheses:**
- Null Hypothesis (\(H_0\)): Family size and level of education are independent.
- Alternative Hypothesis (\(H_A\)): Family size and level of education are not independent.
2. **Create the Contingency Table:**
```
| Primary | Secondary | College | Total
------------------------------------------------------
0-1| 14 | 19 | 12 | 45
2-3 | 37 | 42 | 17 | 96
Over 3 | 32 | 17 | 10 | 59
------------------------------------------------------
Total | 83 | 78 | 39 | 200
```
3. **Calculate the Expected Frequencies:**
Expected frequency for each cell can be calculated using the formula:
\[
E_{ij} = \frac{(Row\ Total_i \times Column\ Total_j)}{Grand\ Total}
\]
For example, the expected frequency for the cell (Primary, 0-1) is:
\[
E_{11} = \frac{(45 \times 83)}{200} = 18.675
\]
Similarly, compute the expected frequencies for all cells.
4. **Calculate the Chi-Square Statistic:**
\[
\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}
\]
where \( O_{ij} \) is the observed frequency, and \( E_{ij} \) is the expected frequency.
For brevity, let's calculate one cell (Primary, 0-1):
\[
\chi^2 = \frac{(14 - 18.675)^2}{18.675} = 1.169
\]
Repeat the calculation for all cells and sum them to get the chi-square statistic.
5. **Determine the Degrees of Freedom:**
\[
df = (r - 1) \times (c - 1)
\]
where \( r \) is the number of rows, and \( c \) is the number of columns.
Here, \( df = (3-1) \times (3-1) = 4 \).
6. **Compare the Chi-Square Statistic to the Critical Value:**
Using a chi-square distribution table or calculator with \( df = 4 \) and \( \alpha = 0.05 \), the critical value is \( 9.488 \).
If the calculated chi-square statistic is greater than 9.488, we reject the null hypothesis; otherwise, we do not reject it.
### Part (b): Computing Price Indices
#### Step-by-Step Process:
1. **Price Relatives for each item in 2001 using 1989 as the base period:**
\[
\text{Price Relative for Item A} = \left( \frac{7.75}{7.50} \right) \times 100 = 103.33
\]
\[
\text{Price Relative for Item B} = \left( \frac{1500}{630} \right) \times 100 = 238.10
\]
2. **Unweighted Aggregate Price Index for 2001 using 1989 as base:**
\[
\text{Unweighted Aggregate Price Index} = \frac{(103.33 + 238.10)}{2} = 170.72
\]
3. **Weighted Aggregate Price Index using Laspeyre’s Method:**
\[
\text{Laspeyre’s Index} = \left( \frac{\sum (P_{1} \times Q_{0})}{\sum (P_{0} \times Q_{0})} \right) \times 100
\]
Calculation for Laspeyre’s Index:
\[
\text{Laspeyre’s Index} = \left( \frac{(7.75 \times 1500) + (1500 \times 2)}{(7.50 \times 1500) + (630 \times 2)} \right) \times 100
\]
\[
= \left( \frac{11625 + 3000}{11250 + 1260} \right) \times 100
= \left( \frac{14625}{12510} \right) \times 100
= 116.94
\]
4. **Weighted Aggregate Price Index using Paasche’s Method:**
\[
\text{Paasche’s Index} = \left( \frac{\sum (P_{1} \times Q_{1})}{\sum (P_{0} \times Q_{1})} \right) \times 100
\]
Calculation for Paasche’s Index:
\[
\text{Paasche’s Index} = \left( \frac{(7.75 \times 1800) + (1500 \times 1)}{(7.50 \times 1800) + (630 \times 1)} \right) \times 100
\]
\[
= \left( \frac{13950 + 1500}{13500 + 630} \right) \times 100
= \left( \frac{15450}{14130} \right) \times 100
= 109.35
\]
These steps provide the required chi-square test and price indices calculations for both problems.
a) A random sample of 200 married men, all retired, were classified according to education
and number of children:
Education Number of Children
0 – 1 2 – 3 Over 3
Primary 14 37 32
Secondary 19 42 17
College 12 17 10
Use the Chi –square to test the hypothesis that family size and level of education attained
by the father are independent. Use 0.05 level of significance.
(b) The following table reports prices and usage quantities for two items in 1989 and 2001
Quantity Unit Price
Item 1989 2001 1989 2001
A 1500 1800 7.50 7.75
B 2 1 630 1500
Please make sure you clearly answer these questions below step by step
i. Compute price relatives for each item in 2001 using 1989 as the base period
ii. Compute an unweighted aggregate price index for the two items in 2001 using 1989
as the base period
iii. Compute a weighted aggregate price index for the two items using the Laspeyre’s
method.
iv. Compute a weighted aggregate price index for the two items using the Paasche
method.
1 answer