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Introduction to Matrix (2021)

Sep 7, 2024Aug 28, 2021 by Muhammad Imdad Ullah

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.

Firestone	Q1	Q2	Q3	Q4
Tirestone	21	20	3	2
Michigan	5	11	15	24
Copper	6	14	7	28

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;

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.

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

May 13, 2024Aug 26, 2021 by Muhammad Imdad Ullah

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.

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.

Online MCQs Estimation with Answers

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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Measures of Dispersion: Variance (2021)

May 13, 2024Aug 25, 2021 by Muhammad Imdad Ullah

Variance is one of the most important measures of dispersion of a distribution of a random variable. The term variance was introduced by R. A. Fisher in 1918. The variance of a set of observations (data set) is defined as the mean of the squares of deviations of all the observations from their mean. When it is computed for the entire population, the variance is called the population variance, usually denoted by $\sigma^2$, while for sample data, it is called sample variance and denoted by $S^2$ to distinguish between population variance and sample variance. Variance is also denoted by $Var(X)$ when we speak about the variance of a random variable. The symbolic definition of population and sample variance is

$\sigma^2=\frac{\sum (X_i – \mu)^2}{N}; \quad \text{for population data}$

$\sigma^2=\frac{\sum (X_i – \overline{X})^2}{n-1}; \quad \text{for sample data}$

It should be noted that the variance is in the square of units in which the observations are expressed and the variance is a large number compared to the observations themselves. The variance because of its nice mathematical properties, assumes an extremely important role in statistical theory.

Variance can be computed if we have standard deviation as the variance is the square of standard deviation i.e. Variance = (Standard Deviation)$^2$.

Variance can be used to compare dispersion in two or more sets of observations. Variance can never be negative since every term in the variance is the squared quantity, either positive or zero.
To calculate the standard deviation one has to follow these steps:

First, find the mean of the data.
Take the difference of each observation from the mean of the given data set. The sum of these differences should be zero or near zero it may be due to the rounding of numbers.
Square the values obtained in step 1, which should be greater than or equal to zero, i.e. should be a positive quantity.
Sum all the squared quantities obtained in step 2. We call it the sum of squares of differences.
Divide this sum of squares of differences by the total number of observations if we have to calculate population standard deviation ($\sigma$). For sample standard deviation (S) divide the sum of squares of differences by the total number of observations minus one i.e. degree of freedom.
Find the square root of the quantity obtained in step 4. The resultant quantity will be the standard deviation for the given data set.

The major characteristics of the variances are:
a)   All of the observations are used in the calculations
b)   Variance is not unduly influenced by extreme observations
c)   The variance is not in the same units as the observation, the variance is in the square of units in which the observations are expressed.

Consider a scenario: Imagine two groups of students both score an average of 70% on an exam. However, in Group A, most scores are clustered around 70%, while in Group B, scores are spread out widely. The measure of spread (like standard deviation or variance) helps distinguish these scenarios, providing a more nuanced understanding of student performance.

By understanding how spread out (scatterness of) the data points are from the average value (mean), standard deviation offers valuable insights in various practical scenarios. It allows for data-driven decision making in quality control, investment analysis, scientific research, and other fields.

Important Statistical Inference Quiz 5

Sep 11, 2024Aug 23, 2021 by Muhammad Imdad Ullah

MCQs from the Statistical Inference Quiz cover the topics of estimation and hypothesis testing for the preparation of exams and different statistical job tests in the government/semi-government or private organization sectors. These Quizzes are also helpful in getting admission to other colleges and Universities. The Estimation Statistical Inference Quiz will help the learner understand the related concepts and enhance their knowledge.

Please go to Important Statistical Inference Quiz 5 to view the test

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 Estimation of parameters and testing of the hypothesis.

Statistical Inference Quiz

The following statistics are unbiased estimators
A statistic is an unbiased estimator of a parameter if:
Which one of the following is a biased estimator?
For $n$ paired number of observations, the degrees of freedom for the Paired Sample t-test will be
If t-distribution for two independent samples $n_1=n_2=n$, then the degrees of freedom will be
If $1-\alpha=0.90$ then value of $Z_{\frac{\alpha}{2}}$ is
If the population standard deviation ($\sigma$) is known and the sample size ($n$) is less than or equal to or more than 30, the confidence interval for the population mean ($\mu$) will be
If the population standard deviation ($\sigma$) is unknown and the sample size ($n$) is greater than 30, the confidence interval for the population mean $\mu$ is
If the population standard deviation $\sigma$ is unknown and the sample size $n$ is less than or equal to 30, the confidence interval for the population mean $\mu$ is
Suppose the 90% confidence Interval for population mean $\mu$ is -24.3 cents to 64.3 cents, the sample mean $\overline{X}$ is
A 95% confidence interval for a population proportion is 32.4% to 47.6%, and the value of the sample proportion $\hat{p}$ is
For a normal population with a known population standard deviations $\sigma_1$ and $\sigma_2$, the confidence interval estimate for the difference between two population means $(\mu_1-\mu_2)$ is
If $n_1, n_2\le 30$ the confidence interval estimate for the difference of two population means ($\mu_1-\mu_2$) when population standard deviations $\sigma_1, \sigma_2$ are unknown but equal in case of pooled variates is:
In the case of paired observations (for a small sample $n\le 30$), the confidence interval estimate for the difference of two populations means $\mu_1-\mu_2=\mu_d$ is
For a large sample, the confidence interval estimate for the difference between two population proportions $p_1-p_2$ is
Each of the following increases the width of a confidence interval except
‘Statistic’ is an estimator, and its computed value(s) is called
Confidence lists for mean when population SD is known
Mean and median are both estimators of population mean _________.
What does it mean when someone calculates a 95% confidence interval?

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Introduction to Matrix

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