(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
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.
(a)
Ho: Family size and level of education attained by the father are independent
Ha: Family size and level of education attained by the father are dependent
Calculate expected frequencies:
Total number of observations = 200
Expected frequency for each cell is calculated by: (row total x column total) / grand total
Primary 0-1: (14+37+32) * (14+19+12) / 200 ≈ 17.33
2-3: (14+37+32) * (42+42+17) / 200 ≈ 45.75
Over 3: (14+37+32) * (32+17+10) / 200 ≈ 24.92
Secondary 0-1: (19+42+17) * (14+19+12) / 200 ≈ 24.75
2-3: (19+42+17) * (42+42+17) / 200 ≈ 65.10
Over 3: (19+42+17) * (32+17+10) / 200 ≈ 35.15
College 0-1: (12+17+10) * (14+19+12) / 200 ≈ 12.92
2-3: (12+17+10) * (42+42+17) / 200 ≈ 33.90
Over 3: (12+17+10) * (32+17+10) / 200 ≈ 18.36
Perform the chi-square test using the formula:
χ^2 = Σ((O-E)^2 / E), where O is the observed frequency and E is the expected frequency
Calculate the degrees of freedom: df = (number of rows - 1) * (number of columns - 1) = 2
Compare the calculated χ^2 value to the critical value from the chi-square distribution table for df = 2 at a significance level of 0.05. If the calculated χ^2 value is greater than the critical value, reject the null hypothesis.
(b)
i. Calculate price relatives for each item in 2001 using 1989 as the base period:
Price relative = (Current price / Base price) * 100
Item A:
Price relative = (7.75 / 7.50) * 100 ≈ 103.33
Item B:
Price relative = (1500 / 630) * 100 ≈ 238.10
ii. Compute unweighted aggregate price index for the two items in 2001 using 1989 as the base period:
Aggregate price index = Σ(price relatives) / number of items
Aggregate price index = (103.33 + 238.10) / 2 ≈ 170.71
iii. Compute weighted aggregate price index for the two items using Laspeyres method:
Base year quantity weights are used
Weighted price index = Σ(Current price x Base year quantity) / Σ(Base year quantity x Base year price) * 100
Weighted price index = (1800*1500 + 1*630) / (1800*1500 + 1*630) * 100 ≈ 171.43
iv. Compute weighted aggregate price index for the two items using Paasche method:
Current year quantity weights are used
Weighted price index = Σ(Current price x Current year quantity) / Σ(Base year quantity x Current year price) * 100
Weighted price index = (1800*1800 + 1*1500) / (1800*1800 + 1*1500) * 100 ≈ 176.36