GSEB Class 12 Statistics Solutions Chapter 1 Index Number Exercise 1.3

Question 1. The information about six different items used in the production of an electronics item is as follows. Find the index number and interpret it.

Index number I = \( \frac{\Sigma \mathrm{IW}}{\Sigma \mathrm{W}} \)
\( = \frac{30700}{100} \)
\( = 307 \)
Hence, the index number is 307.

Interpretation: When compared to the base year, there is a \( (307 - 100 =) 207\% \) increase in the current year.

🎯 Exam Tip: Remember to clearly show the calculation steps for \(\Sigma IW\) and \(\Sigma W\). Also, always provide a clear interpretation of the final index number, explaining what the percentage change means.

Question 2. The information about six different items used in the furniture items is as follows. Find the index number for the year 2015 with the base year 2014 and interpret it.

Base Year: 2014

Index number I = \( \frac{\Sigma \mathrm{IW}}{\Sigma \mathrm{W}} \)
\( = \frac{12380}{100} \)
\( = 123.80 \)

Interpretation: Compared to the base year 2014, the index number for 2015 is 123.80. This means there is a \( (123.80 - 100 =) 23.80\% \) increase in the price of furniture.

🎯 Exam Tip: When given prices for different years, correctly identify the base year price (\(P_0\)) and the current year price (\(P_1\)) to calculate price relatives accurately. Ensure your interpretation clearly states the percentage increase or decrease relative to the base year.

Question 3. Find the Laspeyre's, Paasche's and Fisher's index numbers for the year 2015 with the base year 2014 using the following information:

Here, \(P_0\) = Price in 2014, \(q_0\) = Quantity in 2014,
\(P_1\) = Price in 2015, \(q_1\) = Quantity in 2015.

To calculate Laspeyre's, Paasche's, and Fisher's index numbers for 2015, based on 2014, we prepare the following table:

Laspeyre's index number:

\(I_L = \frac{\Sigma p_1 q_0}{\Sigma p_0 q_0} \times 100\)
\( = \frac{2040.9}{1610.55} \times 100 \)
\( = 1.2672 \times 100 \)
\( = 126.72 \)

Paasche's index number:

\(I_P = \frac{\Sigma p_1 q_1}{\Sigma p_0 q_1} \times 100\)
\( = \frac{3180.25}{2507.15} \times 100 \)
\( = 1.2685 \times 100 \)
\( = 126.85 \)

Fisher's index number:

\(I_F = \sqrt{I_L \times I_P}\)
\( = \sqrt{126.72 \times 126.85}\)
\( = \sqrt{16074.432}\)
\( = 126.78 \)

🎯 Exam Tip: Clearly label each column in your calculation table. Double-check your sums (\(\Sigma P_1q_0\), \(\Sigma P_0q_0\), \(\Sigma P_1q_1\), \(\Sigma P_0q_1\)) as errors here will affect all three index numbers. Remember to show the full formula for each index before substituting values.

Question 4. Find the Laspeyre's, Paasche's and Fisher's index numbers for the year 2015 with the base year 2014 using the following information:

Here, the base year is 2014.
So, \(P_0\) = Price in 2014, \(q_0\) = Quantity in 2014,
\(P_1\) = Price in 2015, \(q_1\) = Quantity in 2015.

The units for price and quantity are not the same for items A, B, C, and D. We need to convert them to consistent units before calculation.

Explanation:

Item A: The price unit is 20 kg, and the quantity unit is kg.
\( \implies \) In 2014, the price per kg = \( \frac{80}{20} = \text{Rs.}4 \)
\( \implies \) In 2015, the price per kg = \( \frac{120}{20} = \text{Rs.}6 \)

Item B: The price unit is kg, and the quantity unit is gram.
\( \implies \) In 2014, the quantity in kg = \( \frac{2400}{1000} = 2.4 \text{ kg} \)
\( \implies \) In 2015, the quantity in kg = \( \frac{4000}{1000} = 4 \text{ kg} \)
[Note: 1000 gram = 1 kg]

Item C: The price unit is quintal, and the quantity unit is kg.
\( \implies \) In 2014, the price per kg = \( \frac{2000}{100} = \text{Rs.}20 \)
\( \implies \) In 2015, the price per kg = \( \frac{2800}{100} = \text{Rs.}28 \)
[Note: 1 Quintal = 100 kg]

Item D: The price unit is dozen, and the quantity unit is a piece.
\( \implies \) In 2014, the price per piece = \( \frac{48}{12} = \text{Rs.}4 \)
\( \implies \) In 2015, the price per piece = \( \frac{72}{12} = \text{Rs.}6 \)
[Note: 1 Dozen = 12 pieces]

To calculate Laspeyre's, Paasche's, and Fisher's index numbers, we prepare the following table:

Laspeyre's index number:

\(I_L = \frac{\Sigma p_1 q_0}{\Sigma p_0 q_0} \times 100\)
\( = \frac{547.6}{388.0} \times 100 \)
\( = 1.4113 \times 100 \)
\( = 141.13 \)

Paasche's index number:

\(I_P = \frac{\Sigma p_1 q_1}{\Sigma p_0 q_1} \times 100\)
\( = \frac{768.0}{548.0} \times 100 \)
\( = 1.4015 \times 100 \)
\( = 140.15 \)

Fisher's index number:

\(I_F = \sqrt{I_L \times I_P}\)
\( = \sqrt{141.13 \times 140.15}\)
\( = \sqrt{19779.3695}\)
\( = 140.64 \)

🎯 Exam Tip: Unit consistency is critical. Always convert all price and quantity units to a common base (e.g., Rs. per kg, kg, Rs. per piece, pieces) before making any calculations. This is a common source of error if overlooked.

Question 5. Find the ideal index number from the following data for the year 2015:

Here, \(P_0\) = Price in 2014, \(q_0\) = Quantity in 2014,
\(P_1\) = Price in 2015, \(q_1\) = Quantity in 2015.

After ensuring uniform units for price and quantity, we will compute the ideal index number, which is Fisher's index number.

Explanation:

Item A: The unit of price is 20 kg, and the unit of quantity is kg.
\( \implies \) In 2014, the price per kg = \( \frac{120}{20} = \text{Rs.}6 \)
\( \implies \) In 2015, the price per kg = \( \frac{280}{20} = \text{Rs.}14 \)

Item B: The unit of price is 5 dozen, and the unit of quantity is dozen.
\( \implies \) In 2014, the price per dozen = \( \frac{120}{5} = \text{Rs.}24 \)
\( \implies \) In 2015, the price per dozen = \( \frac{140}{5} = \text{Rs.}28 \)
\( \implies \) In 2015, the quantity in dozen = \( \frac{48}{12} = 4 \text{ dozen} \)

Item C: The unit of price is kg, and the unit of quantity is gram.
\( \implies \) In 2014, the quantity in kg = \( \frac{5000}{1000} = 5 \text{ kg} \)

Item D: The unit of price is 5 litre, and the unit of quantity is litre.
\( \implies \) In 2014, the price per litre = \( \frac{52}{5} = \text{Rs.}10.40 \)
\( \implies \) In 2015, the price per litre = \( \frac{58}{5} = \text{Rs.}11.60 \)

To compute Fisher's index number, the table for calculation is prepared as follows:

Fisher's index number:

\(I_F = \sqrt{\frac{\Sigma p_1 q_0}{\Sigma p_0 q_0} \times \frac{\Sigma p_1 q_1}{\Sigma p_0 q_1}} \times 100\)
\( = \sqrt{\frac{438}{308} \times \frac{586}{410}} \times 100\)
\( = \sqrt{1.4221 \times 1.4293} \times 100\)
\( = \sqrt{2.03260853} \times 100\)
\( = 1.4257 \times 100 \)
\( = 142.57 \)

🎯 Exam Tip: Pay close attention to unit conversions for each item. Any mistake in converting units (e.g., kg to grams, dozens to pieces) will propagate errors throughout the calculation. Ensure all terms for Fisher's index are correctly substituted into the formula.

Question 6. Find the Paasche's and Fisher's index numbers for the year 2015 with the base year 2014 using the data given below:

Here, the prices and total expenditures for the items are provided.
Total expenditure for an item = (Price per unit of an item) \(\times\) (Quantity consumed of an item)
\( \implies \) Quantity consumed of an item = \( \frac{\text{Total expenditure of an item}}{\text{Price per unit of an item}} \)

Using the formula above, we will find the quantity consumed for each item. To compute Paasche's and Fisher's index numbers, the calculation table is prepared as follows:

Paasche's index number:

\(I_P = \frac{\Sigma p_1 q_1}{\Sigma p_0 q_1} \times 100\)
\( = \frac{4320}{3750} \times 100 \)
\( = 1.152 \times 100 \)
\( = 115.2 \)

Fisher's index number:

\(I_F = \sqrt{\frac{\Sigma p_1 q_0}{\Sigma p_0 q_0} \times \frac{\Sigma p_1 q_1}{\Sigma p_0 q_1}} \times 100\)
\( = \sqrt{\frac{4120}{3580} \times \frac{4320}{3750}} \times 100\)
\( = \sqrt{1.1508 \times 1.152} \times 100\)
\( = \sqrt{1.3257} \times 100\)
\( = 1.1514 \times 100\)
\( = 115.14 \)

Therefore, Paasche's Index number is 115.2, and Fisher's index number is 115.14.

🎯 Exam Tip: When total expenditure is given, remember to first calculate the quantity for each item for both the base and current years. This step is essential before you can compute the \(\Sigma Pq\) terms required for Paasche's and Fisher's indices.