Answer to Question #208224 in Statistics and Probability for Divya Kotla

Question

Find the following data calculate fisher's ideal inderx and proves that it satisfies TRT and FRT

baseyear current year

Commodity price Qty price Qty

A 4 60 6 60

B 10 30 12 24

C 6 50 10 56

D 2 100 2 120

Answer 1

Fisher's ideal Index is

$=\sqrt{\dfrac{\sum P_1Q_o}{\sum P_oQ_o}\times \dfrac{\sum P_1Q_1}{\sum P_oQ_1}}\times 100\\[9pt]=\sqrt{\dfrac{1420}{1040}\times \dfrac{1148}{1056}}\times 100=1.3679\times 100=136.79$

$TRT\implies P_{o1}\times P_{1o}=1$

LHS-

$=\sqrt{\dfrac{\sum P_1Q_o}{\sum P_oQ_o}\times \dfrac{\sum P_1Q_1}{\sum P_oQ_1}}\times \sqrt{\dfrac{\sum P_oQ_1}{\sum P_1Q_1}\times \dfrac{\sum P_o Q_o}{P_1 Q_o}}\\[9pt]=\sqrt{1}=1=RHS$

TRT satisfied.

$FRT\implies P_{o1}\times Q_{o1}=\dfrac{\sum P_1Q_1}{\sum P_o Q_o}$

So LHS $=\sqrt{\dfrac{\sum P_1Q_o}{\sum P_oQ_o}\times \dfrac{\sum P_1Q_1}{\sum P_oQ_1}}\times \sqrt{\dfrac{\sum Q_1P_o}{\sum P_oQ_o}\times \dfrac{\sum Q_1P_1}{\sum Q_oP_1}}$

$=\sqrt{\dfrac{( 1448)^2}{(1040)^2}}=\dfrac{1148}{1040}=\dfrac{\sum P_1Q_1}{\sum P_oQ_o}$ =RHs

FRT Satisfied.