Explanation

• Fisher-price index = √(Laspeyres × Paasche's index)

⇒ Fisher-price index = √[∑(p₁qo/∑p₀qo) ×(∑(p1q₁/∑p0q₁)] × 100

Where,,p₁ = Price for the current year, p_o = Price for the base year, q₁ = Quantity for the current year, q_o = Quantity for the base year

• By exchanging p₀ with p₁ in the formula for the Fisher-price index 

⇒ Fisher's ideal index = P 01 × P10 = √[∑(p1qo/∑p0qo) ×(∑(p1q1/∑p0q1)] × √[∑(p₀qo/∑p₁qo) ×(∑(p0q1/∑p1q1)] 

⇒ P 01 × P10 = 1

⇒ Fisher's ideal index satisfies the time-reversal test.

• Also, since the weight used while calculating the Fisher-price index, the units of the weight are mentioned for both the current as well as the base year so except, for the weighted aggregate method all the other methods like the Fisher-price index number method, Laspeyres index number method, Paasche's index number method satisfy the Unit Test.
• So, the correct answer is option 3.

Concept:

• Time-series: It is the recording of various numerical data points for the given variable in a successive order such that it can be used for forecasting or some other analysis.
• It is useful in analyzing how certain variables like stocks, prices, economy, or other variables change over a period of time.  
• There are four components of the time series mentioned below:
• Secular trend component or trend series: Long-term, regular and smooth variation of the series observed over a long period of time. The trend can have little ups and downs, but it will continue its long-term pattern. Also, the trend can be linear or non-linear. One of the common examples is inflation since it continues to go up with the increase in time.
• Seasonal component: When a regular pattern of variability is seen in the data within one year period is shown over the years, it can be classified as a seasonal component. For example, electricity consumption over a year shows seasonal variation.
• Cyclical component: It is the component having continuous upward or downward movement in a time series but the period of the cycle is greater than a year. These variations are not as regular and predictable as seasonal components. For example, a business cycle has to pass through 4 different phases named as, prosperity, recession, depression, and recovery one by one. 
• Irregular component: This component is caused by unforeseen circumstances like natural calamities, and floods, and is unpredictable. For example, the effect of lockdown due to Covid-19 on the economic activities. 

Explanation: 

Given:

• We are looking at a generation of electricity from solar panels, which depends on the amount of sunlight received on the panels.
• The amount of sunlight will vary over the year due to seasonal variation.
• Hence, this is a seasonal component. 
• Hence the correct answer is option (2).

Concept:

Weighted Aggregative Method

• Constructing a weighted aggregate index number provides a better and more accurate comparison when the data items have variations of weights.
• Since the index number does not depend on the units in which the prices are quoted, we shall weigh prices by quantities and price relatives by values
• In such a case, the use of a weighted index number allows greater importance to be attached to some items.
• Moreover, the Information also includes factors such as quantity sold or quantity consumed for each item.
• In this method, appropriate weights are assigned to different commodities to make them comparable and thus compatible for summation.
• The advantage of this method of the computing index number is that the allotment of weights enables the commodities of greater importance to have more impact on the index number.
• A weighted aggregative index is constructed as follows,
• \(I = \frac{\Sigma p_1w}{\Sigma p_0w}\times 100\;\;\;\;\; \ldots \left( 1 \right)\)

Where, I = Weighted aggregative Index, w = The quantity of usage for commodity, p₀ = Unit price for the base period, p₁ = Unit price of a commodity for the current period

Explanation:

 ​Referring to the definition of Weighted Aggregative index number, the correct answer is option 1​​

Concept:

• Population: It is a grouping of distinct individuals grouped together or a pool of individuals for which conclusions need to be drawn.
• Sample: It is a small subset taken from a larger dataset that draws some information or makes inferences about a larger dataset.
• For example, a study is required to calculate the average height of men in India. 
• The height of each man in India needs to be measured and then its average should be calculated to complete the study in an ideal scenario.
• However, since the number of men in India is huge, in reality, it is not possible to measure the height of every man in India due to time and resource constraints.
• Hence, a small section of the population, say the group of 50,000 men is selected randomly such that the group exhibits behavior very close to the total men population in India.
• The group of 50,000 men is called the sample and all men in India will represent the population for this example.    

Explanation: 

Given:

• Study: Estimation of percentage of unmarried women in Bihar.
• Here the inferences have to be made about the women in the state of Bihar.
• The population is represented by the women in Bihar.
• Hence the correct answer is the option (3).

Concept:

• if we want to construct an index to be based on the price change for a group of items such as housing, food, medical care cost, stock market, transportation etc., we use an aggregated index. The purpose of the aggregated index is to measure the collective change in a group of items.

Using a simple aggregated method, the index number,

Price index number,

\(I = \frac{\Sigma p_1}{\Sigma p_0}\times 100\;\;\;\;\; \ldots \left( 1 \right)\)

Σp1 = The sum of unit prices for the current period

Σp₀ = The sum of unit prices for the base period

Calculation:

Given:  Σp1 = 165, Σp0 = 150

• Using equation 1,

\(I = \frac{165}{150}\times 100 =110\)

• From the above example, we can conclude that the price index number for the year 2020 has only increased by 10% over the period of 2016 to 2020.
• So, the correct answer will be option 2.

Concept:

Fisher-Price Index = (Laspeyres Price Index × Passche's Price Index)^0.5

Calculation:

The laspeyres price index of a commodity is 208

Passche's index of the same commodity is 52

Fisher-Price Index = (208 × 52)^0.5 = 104

the value of the Fisher index number will be 104

Laspeyre's index number = If we use base period  quantities(q_o) as the weight in the general weighted aggregative index formula 

L = (∑p_nq_o/∑p_oq_o) × 100

p_n =  current year price

p_o = Price at base year

q_o = Quantity at base price

Paasche's index = If we use current year quantities(q_n) as weights in the general aggregative formula we get paasche's formula

P = (∑(p_nq_n/∑p_oq_o) × 100

Concept:

• To address the concerns shared for the simple aggregative method, let us construct a sample average of relatives.
• In this method, we shall convert the price of each commodity into the percentage of the base period.
• To construct the index number, we shall calculate the average of all such relatives because the index number calculated from relatives will remain the same regardless of the units of each commodity.
• Simple average of the relative index,

\(I = \frac{\Sigma \frac{p_1}{p_0}}{{N}}× 100\;\;\;\;\; \ldots \left( 1 \right)\)

\(\Sigma \frac{p_1}{p_0}\) = The sum of the relative index, N = Number of the Commodities

Calculation:

Given:  Referring to the table,

• Using equation 1,

\(\Rightarrow I = \frac{105 +75+ 100 +100}{4} = 95\)

• So, the correct answer will be option 2.

Laspeyres Method:

1. The Laspeyres Price Index is a weighted aggregate price index, where the weights are determined by quantities in the base period.

2. The formula for constructing the index is:

\(P_{01}=\frac{∑_{p_1q_0}}{∑_{p_0q_0}}\times100\)

Steps for calculating Laspeyres Price Index:

Step 1: Multiply the current year prices of various commodities with base year weights and obtain \(\sum_{p_1q_0}\).

Step 2: Multiply the base year prices of various commodities with base year weights and obtain \(\sum_{p_0q_0}\)

Step 3: Divide \(\sum_{p_1q_0}\) by \(\sum_{p_0q_0}\) and multiply the quotient by 100. This gives us the price index.

Laspeyres Index attempts to answer the question "What is the change in the aggregate value of the base period list of goods when valued at given period prices?" 

Concept:

• Suppose, from a population, we select multiple random samples and analyze their properties. For example, we measure the mean of different samples.  
• Central Limit Theorem (CLT) for the above example states that the distribution of all the means of different samples will form a normal distribution curve as the sample size becomes larger, assuming all sizes of the samples as the same.
• The population distribution can be of any shape, normal distribution or not, the CLT can be applied to any population.
• A sample size of 30 or more is considered to be enough to hold CLT.

Explanation:

• The mean of the average quantity over all the samples will be equal to the mean of the population when samples are sufficiently large and the sample size is more than or equal to 30.  
• Hence the required mean will be 60 kg.
• Hence, the correct answer is an option (3).

Concept:

• Representative Sample: It is a sample that accurately represents its population. It should be an unbiased reflection of the characteristic of the population.
• Evaluation of representativeness depends on the scope of study and the information of the population. 
• For example, if most of the doctor population in a region are women (say 80%), then a sample containing a list of all male doctors may not be a representative sample of the larger population. 
• The cause of lack of representation is sampling error or bias. For example, a study requires conducting a survey on the number of sleep hours an average citizen gets in India, but the participants chosen were only surveying volunteers belonging to metro cities like Delhi, Mumbai, Gurgaon, etc. 

Explanation:

• There are a few ways to avoid or reduce sampling errors while creating a sample from a population:
  • Random sampling from the population irrespective of any factors. This will minimize the selection bias that may occur.
  • Increasing the size of the samples since larger samples tend to be more representative of the population.
  • Stratification protocol, for example, if the population contains 80% of women, then the sample should be formed in such a way that it contains 80% of women. 
• Implementing all these procedures will lead to the formation of a representative sample.

Additional Information

• Population: It is a grouping of distinct individuals grouped together or a pool of individuals for which conclusions need to be drawn.
• Sample: It is a small subset taken from a larger dataset that draws some information or makes inferences about a larger dataset.
• For example, a study is required to calculate the average height of men in India. 
• The height of each man in India needs to be measured and then its average should be calculated to complete the study in an ideal scenario.
• However, since the number of men in India is huge, in reality, it is not possible to measure the height of every man in India due to time and resource constraints.
• Hence, a small section of the population, say the group of 50,000 men is selected randomly such that the group exhibits behavior very close to the total men population in India.
• The group of 50,000 men is called the sample and all men in India will represent the population for this example.