Unit 5 Business Statistics and Research Methods PYQ’s 6th Dec 2019 Shift 2

The following are the explanation for the correct match:

Posterior probability

1. A posterior probability, in Bayesian statistics, is the revised or updated probability of an event occurring after taking into consideration new information.
2. The posterior probability is calculated by updating the prior probability using Bayes’ theorem.
3. In statistical terms, the posterior probability is the probability of event A occurring given that event B has occurred.
4. This theorem is also known as ‘Inverse probability theorem’ because here moving from the first stage to the second stage, we again find the probabilities (revised) of the events of the first stage i.e. we move inversely.
5. Thus, using this theorem, probabilities can be revised on the basis of having some related new information.

Key Points

We require the following information in order to derive posterior probability:

1. original probability estimates (prior probabilities) of mutually exclusive and collectively exhaustive events
2. conditional probabilities
3. Joint probabilities of prior probability and conditional probability
4. Arbitrary event with probability # 0 and for which conditional probabilities are also known

Thereafter, we can substitute the values in the formula for deriving posterior probability.

Hence, For calculating posterior probabilities (conditional probabilities under statistical dependence) the correct sequence is b) → a) → d) → c)

When two variables are related in such a way that a change in the value of one variable affects the value of another variable, then variables are said to be correlated or there is a correlation between these two variables.

According to the direction of change in variables, there are two types of correlation: 1. Positive Correlation 2. Negative Correlation

Important Points

1. Positive Correlation: Correlation between two variables is said to be positive if the values of the variables deviate in the same direction i.e. if the values of one variable increase (or decrease) then the values of another variable also increase (or decrease). Some examples of positive correlation are correlation between:

1. Heights and weights of a group of persons;
2. Household income and expenditure;
3. Amount of rainfall and yield of crops; and
4. Expenditure on advertising and sales revenue.

In the last example, it is observed that as the expenditure on advertising increases, sales revenue also increases. Thus, the change is in the same direction. Hence the correlation is positive. In the remaining three examples, usually the value of the second variable increases (or decreases) as the value of the first variable increases (or decreases).

2. Negative Correlation: Correlation between two variables is said to be negative if the values of variables deviate in opposite direction i.e. if the values of one variable increase (or decrease) then the values of another variable decrease (or increase). Some examples of negative correlations are correlation between:

1. Volume and pressure of perfect gas;
2. Price and demand for goods;
3. Literacy and poverty in a country; and
4. Time spent on watching TV and marks obtained by students in the examination.

In the first example, pressure decreases as the volume increases or pressure increases as the volume decreases. Thus the change is in opposite direction. Therefore, the correlation between volume and pressure is negative. In the remaining three examples also, the values of the second variable change in the opposite direction of the change in the values of the first variable.

Key Points

Now, since the sum total of the values of two variables ‘X’ and ‘Y’ is equal for all the observations. i.e. x + y = ‘z’ (equal), for ‘z’ to remain same (equal) for all observations, if ‘x’ increases, ‘y’ should decrease and vice-versa. Thus, option 2 is the correct answer.

Regression Coefficients

The regression coefficients are a statistical measure that is used to measure the average functional relationship between variables. In regression analysis, one variable is dependent and the other is independent. Also, it measures the degree of dependence of one variable on the other(s).

Some of the properties of regression coefficient:

1. It is generally denoted by ‘b’.
2. It is expressed in the form of an original unit of data.
3. If two variables are there say x and y, two values of the regression coefficient are obtained. One will be obtained when x is independent and y is dependent and the other when we consider y as independent and x as a dependent. The regression coefficient of y on x is represented by byx and x on y as bxy.
4. Both of the regression coefficients must have the same sign. If byx is positive, bxy will also be positive and it is true for vice versa.
5. If one regression coefficient is greater than unity, then others will be lesser than unity.
6. The geometric mean between the two regression coefficients is equal to the correlation coefficient
7. R=sqrt(byx*bxy)
8. Also, the arithmetic means (am) of both regression coefficients are equal to or greater than the coefficient of correlation.
9. (byx + bxy)/2= equal or greater than r.
10. The regression coefficients are independent of the change of the origin. But, they are not independent of the change of the scale. It means there will be no effect on the regression coefficients if any constant is subtracted from the value of x and y. If x and y are multiplied by any constant, then the regression coefficient will change.
11. The two lines of regression coincide i.e. become identical when r = –1 or 1 or in other words, there is a perfect negative or positive correlation between the two variables under discussion.
12. The coefficient of determination is the square of the correlation (r) between predicted y scores and actual y scores; thus, it ranges from 0 to 1. With linear regression, the coefficient of determination is also equal to the square of the correlation between x and y scores.

Hence, from the above explanation, The regression coefficients are independent of the change of origin but they are not independent of the change of the scale. Therefore, Option 2 is false.

Although every researcher wants to be able to generalize the research findings to the wider population, such a study will require much more time, money and effort. For this reason, we select a sample i.e. a smaller representation of the larger whole (population), so that we can generalize our findings from the sample for the population.

Sampling: The concept of sampling involves selecting a portion (sample) from a bigger group (the sampling population). There are three methods of sampling in research:

1. Random/Probability Sampling
2. Non-random/Non-probability Sampling
3. ‘Mixed’ Sampling

Types of Non-Random/Non-Probability Sampling Designs: These designs do not operate on the principle of randomization rather these are used when the number of elements in the population is either unknown or cannot be individually identified.

1. Accidental Sampling: It is also based upon convenience in accessing the sampling population. People who are unwilling to provide data are simply ignored and the researcher moves to the next person until he/she meets somebody who is willing to be a participant. You stop collecting data when you reach the required number of respondents you decided to have in your sample.
2. Judgment/Purposive Sampling: The primary consideration in purposive sampling is the researcher’s judgment as to who can provide the best information to achieve the objectives of your study. The researcher will only go to those people who in his opinion is likely to have the required information and will be willing to share it.
3. Snowball Sampling: It is also called network or chain referral sampling. To start with, the researcher identifies a small number of respondents having a set of characteristics of interest to the researcher. After collecting the required data from those respondents, the same respondents are asked to identify others having the same characteristics set. E.g., collecting data from drug addicts, rape victims, etc.
4. Quota Sampling: The researcher is guided by some visible characteristics, such as gender or race, of the study population that is of interest to him. The sample is selected from a location that is convenient and easily accessible to the researcher and whenever a person with this visible relevant characteristic is seen that person is asked to participate in the study.
5. Dimensional Sampling: It is an extension of quota sampling where the researcher takes into account several characteristics such as gender, residence, education, etc. and ensures that there is at least one individual in the study representing each of the chosen characteristics.

NOTE:

Random/Probability Sampling: In this type, each element in the population has an equal and independent chance of selection in the sample.

1. Systematic Sampling: Systematic sampling is a type of probability sampling method in which sample members from a larger population are selected according to a random starting point but with a fixed, periodic interval. This interval, called the sampling interval, is calculated by dividing the population size by the desired sample size.
2. Stratified Random Sampling: It combines randomization with stratification. Here, the population is divided into strata, the population within each stratum is homogeneous with respect to the characteristic based on which it is being stratified and such characteristics must be identifiable in the study population (e.g. age, income, sex, etc.).
3. Multi-Stage Sampling: In statistics, multistage sampling is the taking of samples in stages using smaller and smaller sampling units at each stage. Multistage sampling can be a complex form of cluster sampling because it is a type of sampling which involves dividing the population into groups (or clusters).

Therefore, Quota Sampling is the non-random method of selecting samples from a given population.

The Experimental Research Design:

1. The experimental design is a way to carefully plan experiments in advance so that your results are both objective and valid.
2. The terms “Experimental Design” and “Design of Experiments” are used interchangeably and mean the same thing.
3. However, the medical and social sciences tend to use the term “Experimental Design” while engineering, industrial, and computer sciences favor the term “Design of experiments.”

Design of experiments involves:

1. The systematic collection of data
2. A focus on the design itself, rather than the results
3. Planning changes to independent (input) variables and the effect on dependent variables or response variables
4. Ensuring results are valid, easily interpreted, and definitive.

The following are the Experimental research design:

1. Before and after with control group design
2. Factorial design
3. Latin Square design

Important Points

One group before-after design is a type of quasi-experimental design in which the dependent variable is treated once before the intervention is introduced, and then after the implementation of an intervention. A quasi-experimental design is similar to another experimental design, but they lack random assignment, hence they are incomplete experimental designs.

Below are the features of Karl Pearson’s Coefficient of Skewness:

1. Skewness studies or measures the lack of symmetry in the distribution.
2. Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean.

3. The skewness value can be positive, zero, negative, or undefined.
4. If there is a long tail of the distribution towards the higher values or right-hand side i.e., larger spread on the higher side of mode, the skewness is positive or right-handed. In this case, Mean > Median > Mode. Such data is termed ‘elongated bell-shaped data‘.
5. if there is a long tail towards the lower value or left-hand side i.e., larger spread on the lower side of mode, the skewness is negative or left-handed. In such a case Mean < Median < Mode.
6. Karl Pearson’s Coefficient of Skewness is most frequently used for measuring skewness and is based on the first absolute measure.
7. The formula for measuring skewness is Skewness = Mean – Mode / Standard Deviation i.e., the first absolute measure of skewness is divided by the standard deviation.
8. Thus, this value will be free of units of the data.
9. The value of this coefficient would be zero in a symmetrical distribution.
10. If the mean is greater than the mode, the coefficient of skewness would be positive otherwise negative.
11. In practice, the value of this coefficient usually lies between ± 3. If the mode is ill-defined, then using the approximate relationship: Mode = 3 Median – 2 Mean. This formula reduces to Skewness = 3 (Mean – Median) / Standard Deviation.

Therefore, from the above explanation, Option 2 is the correct answer.

Below are the explanation of the above statements:

Key Points

1.Deviation:

• Method of Mean Deviation does not give accurate results.

• The reason is that the mean deviation gives the best results when deviations are taken from the median.
• But the median is not a satisfactory measure when the degree of variability in a series is very high.
• And if we compute mean deviation from mean that is also not desirable because the sum of the deviations from the mean (ignoring signs) is greater than the sum of the deviations from the median (ignoring signs).
• If the mean deviation is computed from a mode that is also not scientific because the value of mode cannot always be determined. Thus, statement A is correct.

2. Standard deviation is independent of the change of origin and but not of scale. Thus, statement B is incorrect.

3. In a symmetrical distribution, the mean deviation equals 4/5 of standard deviation. Thus, statement C is correct.

4 In symmetrical and bell-shaped distribution, quartile deviation is the distance from the median to the lower quartile or to the upper quartile. Thus, statement D is incorrect.

Central Tendency is defined as the tendency of the data to concentrate towards the central or middlemost region of the distribution. In other words, Central Tendency indicates average. One of the requirements of an ideal central tendency measure is that it should be capable of further algebraic treatment.

1. Geometric Mean:

• The geometric mean is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum).
• The geometric mean is defined as the nth root of the product of n numbers It lends itself to algebraic treatment.
• If geometric means of two or more groups are given, the geometric mean of the combined group can be obtained.

2. Median:

• The median of a given series is the value of the variable that divides the series-into two equal parts.
• It is the most central point of a series where half of the items lie above this value and the remaining half lie below this value.
• Median is based on the position of a given observation in a series arranged in an ascending or descending order.
• Therefore, it is called a positional average.
• The Median is not capable of algebraic treatment. That means we cannot have a combined median of two or more groups unless all the items of the groups are known.

3. Arithmetic mean:

• The arithmetic mean is commonly known as mean.
• It is a measure of central tendency because other figures of the data congregate around it.
• The arithmetic mean is obtained by dividing the sum of the values of all observations in the given data set by the number of observations in that set.
• The arithmetic mean is also called a simple Arithmetic Mean.
• It is useful in further statistical analysis. It is used in the computation of other statistical measures like standard deviation, coefficient of variation, coefficient of skewness, etc.

4. Mode:

• The mode is also a measure of central tendency.
• The mode is often considered to be that value of the variate which occurs most frequently. But it is not exactly true for every frequency distribution.
• It is not the center of gravity like mean.
• It is a positional measure similar to the median. It is not capable of further algebraic treatment.

5. Harmonic mean:

• The harmonic mean is a type of numerical average.
• It is calculated by dividing the number of observations by the reciprocal of each number in the series.

• The Harmonic Mean (HM) is defined as the reciprocal of the arithmetic mean of the reciprocal of individual observations. Like the arithmetic mean and geometric mean, it lends itself to further algebraic treatment.
• Amongst the three means, the harmonic mean is the least i.e. Arithmetic Mean ≥ Geometric Mean ≥ Harmonic Mean.

Thus, option 3 is the correct answer.