Index Numbers An Introduction

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The index numbers are numbers that “measure a relative change in a variable or an average relative change in a group of related variables concerning a base”. An index number indicates the level of certain phenomena at some given period in comparison with the level of the same phenomena at some reference period. The index numbers are usually constructed for economic variables such as price, quantity, wage, unemployment, investment, cost of living, etc.

Index numbers are free from units of measurement because they show relative changes. For ease of understanding, index numbers are expressed in percentages. To construct an index number at least two periods are required and a period that is economically stable and has no major crisis caused by wars, diseases, strikes, food shortage, etc. known as the normal period is selected as a base. Index numbers of wholesale prices and consumer prices, etc. are published by the Federal Bureau of Statistics and State Bank of Pakistan.

Uses/ Need of Index Numbers

There are many uses for index numbers but the most important are:

  • Many economic plans and Government policies depend on index numbers, for example, to control rising prices of government imports from other countries or give subsidies (financial support) to the manufacturer.
  • The Price index number is used to know the purchasing ability of money at different periods and places.
  • The quantity index number is used to know the changes in the quantities produced, consumed, sold, purchased, imported or exported, etc.
  • Consumer price index numbers are used to know people’s standards of living and the goods and services used by them.
  • Index numbers are used to forecast future economic trends
  • Cyclical (long-term movements, which are in the form of oscillation) and seasonal (short-term movements, which are linked with the seasons or movements that repeat themselves within a fixed period) movements are measured by index number.
Index Numbers

Shortcomings of Index Number

Index numbers can not be used freely due to the following shortcomings:

  • An improper base period gives misleading results. Base periods must be free from all types of crises caused by wars, diseases, strikes, food shortages, etc. If such a period is not available then the average of some or all the periods is selected as the base.
  • Selection of favorite commodities is difficult because the use of services and commodities by individuals varies with the locality of people, social customs, standard of living, occupation, ideas of saving, courage of investment, and sources of income, etc.
  • The quality of a product cannot be observed at each point, that is, ball-to-ball commentary is difficult. For example, if we want to view the quality of cloth at each thread before purchasing it becomes impossible.
  • Index number gives a rough measure of relative changes because sampling error or error of measurement may occur at the stages of gathering data or the base period may be improper or the number of commodities may be less than required. According to Dr. Arriving Fisher, the accuracy of index numbers may be increased by increasing the number of commodities.
  • Different methods of index numbers usually give different results.
  • Prices vary from place to place according to the idea of profit of investors, expenditures on transportation, and awareness about the psychology of buyers, hence their collection is difficult.

Examples of Important Index Numbers:

  • Consumer Price Index (CPI): Tracks changes in the prices of goods and services purchased by consumers.
  • Producer Price Index (PPI): Measures the average change in prices received by domestic producers for their output.
  • Wholesale Price Index (WPI): Tracks the price changes of goods traded in wholesale markets.
  • Industrial Production Index (IPI): Measures the volume of physical production in the industrial sector.

In conclusion, index numbers are a powerful tool for summarizing complex economic information and identifying trends. They play a vital role in economic analysis, decision-making, and understanding changes in our world over time.

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Introduction to Matrix (2021)

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This post is about some basic introduction to matrix.

Matrices are everywhere. If you have used a spreadsheet program such as MS Excel, or Lotus, written a table (such as in Ms-Word), or even have used mathematical or statistical software such as Mathematica, Matlab, Minitab, SAS, SPSS, Eviews, etc., you have used a matrix. Let us start with the Introduction to matrix.

Introduction to Matrix

Matrices make the presentation of numbers clearer and make calculations easier to program. For example, the matrix is given below about the sale of tires in a particular store given by quarter and make of tires.

FirestoneQ1Q2Q3Q4
Tirestone212032
Michigan5111524
Copper614728

It is called a matrix, as information is stored in a particular order and different computations can also be performed. For example, if you want to know how many Michigan tires were sold in Quarter 3, you can go along the row ‘Michigan’ and column ‘Q3’ and find that it is 15.

Similarly, the total number of sales of ‘Michigan’ tiers can also be found by adding all the elements from Q1 to Q4 in the Michigan row. It sums to 55. So, a matrix is a rectangular array of elements. The elements of a matrix can be symbolic expressions or numbers. Matrix $[A]$ is denoted by;

Introduction to Matrix

Row $i$ of the matrix $[A]$ has $n$ elements and is $[a_{i1}, a_{i2}, cdots, a_{1n}] and column of $[A]$ has $m$ elements and is $begin{bmatrix}a_{1j}\ a_{2j} \ vdots\ a_{mj}end{bmatrix}$.

The size (order) of any matrix is defined by the number of rows and columns in the matrix. If a matrix $[A]$ has $m$ rows and $n$ columns, the size of the matrix is denoted by $(mtimes n)$. The matrix $[A]$ can also be denoted by $[A]_{mtimes n}$ to show that $[A]$ is a matrix that has $m$ rows and $n$ columns in it.

Each entry in the matrix is called the element or entry of the matrix and is denoted by $a_{ij}$, where $i$ represents the row number and $j$ is the column number of the matrix element.

Matrix of tire Sale

The above-arranged information about sales and types of tires can be denoted by the matrix $[A]$, that is, the matrix has 3 rows and 4 columns. So, the order (size) of the matrix is 3 x 4. Note that element $a_{23}$ indicates the sales of tires in ‘Michigan’ in quarter 3 (Q3). That is all about Introduction to Matrix.

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Important MCQs Estimation 6

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MCQs Estimation Quiz covers the topics of Estimation for the preparation of exams and different statistical job tests in Government/ Semi-Government or Private Organization sectors. These quizzes help get admission to different colleges and Universities. The MCQs Estimation Quiz will help the learner to understand the related concepts and enhance the knowledge too.

MCQs about statistical inference covering the topics estimation, estimator, point estimate, interval estimate, properties of a good estimator, unbiasedness, efficiency, sufficiency, Large sample, and sample estimation.

1. A quantity obtained by applying a certain rule or formula is known as

 
 
 
 

2. Crammer-Rao inequality is valid in the case of:

 
 
 
 

3. An estimator $T_n$ is said to be a sufficient statistic for a parameter function $\tau(\theta)$ if it contains all the information which is contained in the:

 
 
 
 

4. Sample median as an estimator of the population mean is always

 
 
 
 

5. Since $E(X)=$______. $X$ is said to be an unbiased estimator of the population mean.

 
 
 
 

6. By the method of moments, one can estimate:

 
 
 
 

7. If $\alpha=0.10$ and $n=15$ then $t_{\frac{\alpha}{2}}$ will be

 
 
 
 

8. Criteria to check a point estimator to be good are

 
 
 
 

9. The consistency of an estimator can be checked by comparing

 
 
 
 

10. Roa-Blackwell Theorem enables us to obtain minimum variance unbiased estimator through:

 
 
 
 

11. If the sample average $\overline{x}$ is an estimate of the population mean $\mu$, then $\overline{x}$ is:

 
 
 
 

12. The estimation and testing of the hypothesis are the main branches of ________.

 
 
 
 

13. ________ is an estimate expressed by a single value.

 
 
 
 

14. Sampling error decreases by ______ the sample size.

 
 
 
 

15. An Estimator $\hat{T}$ is an unbiased estimator of the population parameter $T$ if

 
 
 
 

16. For an estimator to be consistent, the unbiasedness of the estimator is

 
 
 
 

17. If $n_1=16$, $n_2=9$ and $\alpha=0.01$, then $t_{\frac{\alpha}{2}}$ will be

 
 
 
 

18. The sample proportion $\hat{p}$ is ______ estimator

 
 
 
 

19. A _____ is the specific value of the statistics used to estimate the population parameter.

 
 
 
 

20. _______ is the value of a sample statistic.

 
 
 
 

Statistical inference is a branch of statistics in which we conclude (make wise decisions) about the population parameter by making use of sample information. Statistical inference can be further divided into the Estimation of parameters and testing of the hypothesis.

Estimation is a way of finding the unknown value of the population parameter from the sample information by using an estimator (a statistical formula) to estimate the parameter. One can estimate the population parameter by using two approaches (I) Point Estimation and (ii) Interval Estimation.
In point Estimation, a single numerical value is computed for each parameter, while in interval estimation a set of values (interval) for the parameter is constructed. The width of the confidence interval depends on the sample size and confidence coefficient. However, it can be decreased by increasing the sample size. The estimator is a formula used to estimate the population parameter by making use of sample information.

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  • Criteria to check a point estimator to be good are
  • By the method of moments, one can estimate:
  • If the sample average $\overline{x}$ is an estimate of the population mean $\mu$, then $\overline{x}$ is:
  • A quantity obtained by applying a certain rule or formula is known as
  • The consistency of an estimator can be checked by comparing
  • For an estimator to be consistent, the unbiasedness of the estimator is
  • Sample median as an estimator of the population mean is always
  • Roa-Blackwell Theorem enables us to obtain minimum variance unbiased estimator through:
  • An estimator $T_n$ is said to be a sufficient statistic for a parameter function $\tau(\theta)$ if it contains all the information which is contained in the:
  • Crammer-Rao inequality is valid in the case of:
  • If $n_1=16$, $n_2=9$ and $\alpha=0.01$, then $t_{\frac{\alpha}{2}}$ will be
  • If $\alpha=0.10$ and $n=15$ then $t_{\frac{\alpha}{2}}$ will be
  • A _ is the specific value of the statistics used to estimate the population parameter.
  • An Estimator $\hat{T}$ is an unbiased estimator of the population parameter $T$ if
  • Since $E(X)=$_______. $X$ is said to be an unbiased estimator of the population mean.
  • The sample proportion $\hat{p}$ is ________ estimator Sampling error decreases by the sample size.
  • __________ is the value of a sample statistic.
  • __________ is an estimate expressed by a single value.
  • The estimation and testing of the hypothesis are the main branches of ________.
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