Assignment Code: MCO-022/ASST/TMA/2022-2023

Course Code: MCO-22

Assignment Name: Quantitative Analysis and Managerial Application

Year: 2022-2023

Verification Status: Verified by Professor

1. (a) Discuss some criteria where the probability associated with the associated outcome is not known. (10)

Ans) Indicators when probability is unknown.

Criterion of Pessimism: As the name implies, the decision-making process is pessimistic, presuming that, regardless of whatever option is selected, the worst outcome associated with each option will actually happen. Maximizing the minimal reward is a sensible criterion for decision-making in such a situation.

Criterion of Optimism: In a variant of (a), in addition to the maximum of the lowest payoff, the maximum of the maximum payoff is computed. A decision would require complete optimism. It is suggested that the d.m. calculate the greatest and minimum payoffs for each alternative and then weigh them using his coefficient of optimism to determine the expected benefit for each option. Then, choose the choice with the largest anticipated payout. The range of the Coefficient of Optimism is 0 to 1. It demonstrates how much the d.m. favours the greatest return above the smallest payoff.

Criterion of Regret: The criteria are based on the possibility that the final decision made on an alternative and the actual result after the decision has been made will not match. When it matches, a regret of 0 is produced.

The following are the effects:

Therefore, if option 1 is selected and a depressive episode actually occurs, there is cause for regret because option 2 would have only resulted in a loss of six compared to ten, so regret equals ten minus six minus four. Similar to this, if alternative 2 has been selected and there isn't genuinely a depression, a regret of 40-20 = takes place. If alternative1 is selected and depression is subsequently discovered, there would be no regret. The regret matrix is thus discovered:

Now, a pessimistic stance is adopted, and the decision-making standard is the minimization of maximum regret. The greatest regret is determined for each option, and then the option with the lowest value of the maximum regret is selected. As a result, our d.m. would have gone with option 1.

Subjectivists' Criterion: In this situation, the outcomes are thought to be equally likely, and the EMV is employed to make the decision. This is referred to as the subjectivist position.

The four criteria listed above are the most well-known. It should be clear by now that choosing the ultimate criterion is entirely subjective. Each offers a different justification, but the d.m. is free to select any based on his personal preferences.

(b) Explain the purpose and methods of classification of data giving suitable examples. (10)

Ans)

Purpose of Classification of Data

1. It helps in presenting the mass of data in a concise and simple form.

2. It divides the mass of data on the basis of similarities and resemblances so as to enable comparison.

3. It is a process of presenting raw data in a systematic manner enabling us to draw meaningful conclusions.

4. It provides a basis for tabulation and analysis of data.

5. It provides us a meaningful pattern in the data and enables us to identify the possible characteristics in the data.

Methods of Classification

1) Classification According to Attributes

An attribute is a qualitative characteristic which cannot be expressed numerically. Only the presence or absence of an attribute can be known. For example. intelligence, religion, caste, sex, etc., are attributes. You cannot quantify these characteristics. When classification is to be done on the basis of attributes, groups are differentiated either by the presence or absence of the attribute (e.g., male and female) or by its differing qualities.

The qualities of an attribute can easily be differentiated by means of some natural line of demarcation. Based on this natural difference, we can determine the group into which a particular item is placed. For instance, if we select colour of hair as the basis of classification, there will be a group of brown haired people and another group of black haired people.

There are two types of classification based on attributes.

a) Simple Classification : In simple classification the data is classified on the basis of only one attribute. The data classified on the basis of sex will be an example of simple classification.

b) Manifold Classification: In this classification the data is classified on the basis of more than one attribute. For example, the data relating to the number of students in a university can be classified on the basis of their sex and marital.

2) Classification According to Variables

Variables refer to quantifiable characteristics of data and can be expressed numerically. Examples of variable are wages, age, height, weight, marks, distance etc. All these variables can be expressed in quantitative terms. In this form of classification, the data is shown in the form of a frequency distribution. A frequency distribution is a tabular Presentation that generally organises data into classes and shows the number of observations (frequencies) falling into each of these classes. Based on the number of variables used, there are three categories of frequency distribution: 1) uni-variate frequency distribution, bi-variate frequency distribution, and  Multi-variate frequency distribution 

a) Uni-variate Frequency Distribution : The frequency distribution with one variable is called a uni-variate frequency distribution. For example, the students in a class may be classified on the basis of marks obtained by them.

b) Bi-variate Frequency Distribution: The frequency distribution with two variable is called bi-variate frequency distribution. If a frequency distribution shows two variables it is known as bi-variate frequency distribution.

c) Multi-variate Frequency Distribution: The frequency & distribution with more than two variables is called multivariate frequency distribution. For example, the students in a class may be classified on the basis of marks, age and sex

2. (a) What do you understand by time series analysis? How would you go about conducting such an analysis for forecasting the sales of a product in your firm? (10)

Ans) Time series analysis is a specific way of analysing a sequence of data points collected over an interval of time. In time series analysis, analysts record data points at consistent intervals over a set period of time rather than just recording the data points intermittently or randomly. However, this type of analysis is not merely the act of collecting data over time.

What sets time series data apart from other data is that the analysis can show how variables change over time. In other words, time is a crucial variable because it shows how the data adjusts over the course of the data points as well as the final results. It provides an additional source of information and a set order of dependencies between the data.

Time series analysis typically requires a large number of data points to ensure consistency and reliability. An extensive data set ensures you have a representative sample size and that analysis can cut through noisy data. It also ensures that any trends or patterns discovered are not outliers and can account for seasonal variance. Additionally, time series data can be used for forecasting—predicting future data based on historical data.

Deseasonalising the Time Series

Prior to anything else, moving averages and original variable to moving average ratios must be computed.

Fitting a Trend Line

Deseasonalizing the data is followed by developing the trend line. Here, we'll employ the least-squares method. Calculations are significantly simplified if the origin is chosen in the middle of the data with the appropriate scale. To fit a straight line to the deseasonalized sales using the formula Y = a + bX,

Identifying Cyclical Variation

Deseasonalized variation measurements are used to determine the cyclical component.

represents the ratio of actual deseasonalized sales to the trend line,

value that the trend line predicts.

Forecasting with the Decomposed Components of the Time Series

Suppose that the management of the Engineering firm is interested in estimating the sales for the second and third quarters of 1988. The estimates of the deseasohalised sales can be obtained by using the trend line

Y = 6.3 + 0.04(23)

= 7.22 (2nd Quarter 1988)

and Y = 6.3 + 0.04 (25)

= 7.30 (3rd Quarter 1988)

The second and third quarters of these estimates will now need to be seasonalized. Here's how to go about it:

For 1988 2nd quarter seasonalised sales estimate = 7.22 x 1.013 = 7.31

For 1988 3rd quarter seasonalised sales estimate = 7.30 x 1.56 = 8.44

As a result, the Engineering firm's projected sales for the second and third quarters of 1988 are Rs. 7.31 lakh and Rs. 8.44 lakh, respectively, based on the analysis shown above.

These estimations were generated by factoring in seasonal variations as well as trend. Uneven and cyclical components have not been considered. The method for cyclical variations only aids in the analysis of previous behaviour; it does not aid in the forecasting of future behaviour. Moreover, it is challenging to measure random or irregular fluctuations.

(b) What is the practical utility of the central limit theorem in applied statistics? (10)

Ans) If x1, x2, … , xn are n random variables which are independent and having the same distribution with mean μ. and standard deviation 𝜎, then if n → ∞, the limiting distribution of the standardised mean z =  is the standard normal distribution.

In practice, if the sample size is sufficiently large, we need not know the population distribution because the central limit theorem assures us that the distribution of x can be approximated by a normal distribution. A sample size larger than 30 is generally considered to be large enough for this purposes.

Many practical samples are of size higher than 30. In all these cases, we know that the sampling distribution of the mean can be approximated by a normal distribution with an expected value equal to the population mean and a variance which is equal to the population variance divided by the sample size n.

Practical Application

The Central Limit Theorem (CLT) is one of the most popular theorems in statistics and it’s very useful in real world problems. In this article we’ll see why the Central Limit Theorem is so useful and how to apply it. In a lot of situations where you use statistics, the ultimate goal is to identify the characteristics of a population.

Central Limit Theorem is an approximation you can use when the population you’re studying is so big, it would take a long time to gather data about each individual that’s part of it.

Population: Population is the group of individuals that you are studying. And even though they are referred to as individuals, the elements that make a population don’t need to be people. If you’re a regional manager at a grocery chain and you’re trying to be more efficient at re-stocking the seltzer water section every week in every store, so you sell as much seltzer as possible and avoid ending up with a lot of unsold inventory, all the cases of seltzer sold in that particular store represent the population. If you’re a poultry farmer and want to put in an order for chicken feed, you’ll need to know how many pounds of grain your hens typically eat. So here, the chickens are your population.

Studying the population is hard: Depending on the problem you’re solving; it will be extremely hard to gather data for the entire population. If a company like Coca-Cola wants to know if their US customers will like the new product they are developing, they can’t send an army of researchers to talk to every single person in the US. Well, they probably could, but it would be very expensive and would take a long time to collect all the data.

That’s why companies do user studies with several groups of people that represent of their product’s audience, their population, so they can gather data and determine if it’s worth moving forward with product development. All of this, without talking to the entire population. So, in statistical terms, you’re going to collect samples from your population, and by combining the information from the samples you can draw conclusions about your population.

A good sample must be

1. Representative of the population,

2. Big enough to draw conclusions from, which in statistics is a sample size greater or equal to 30.

3. Picked at random, so you’re not biased towards certain characteristics in the population.

Representative samples: A representative sample must showcase all the different characteristics of the population. If you want to know who is more likely to win the Super Bowl and decide to poll the US population, i.e., take a sample from the US population, you need to make sure to talk to people from:

1. All the different states about who they think is going to win.

2. Different age groups and different genders.

And only include in your study the people that have interest in sports or in the event itself otherwise, they will not be part of the population that is interested in what you’re studying.

3. Briefly comment on the following: (4×5)

a) “Measuring variability is of great importance to advanced statistical analysis.”

Ans) Variability means ‘Scatter’ or ‘Spread.’ Thus, measures of variability refer to the scatter or spread of scores around their central tendency. The measures of variability indicate how the distribution scatter above and below the central tender.

Suppose there are two groups. In one group there are 50 boys and in another group 50 girls. A test is administered to both these groups. The mean score of boys and is 54.4 and girls is we compare the mean score of both the groups, we find that that there is no difference in the performance of the two groups. But suppose the boys’ scores are found to range from 20 to 80 and the girls’ scores range from 40 to 60.

This difference in range shows that the boys are more variable, because they cover more territory than the girls. If the group contains individuals of widely differing capacities, scores will be scattered from high to low, the range will be relatively wide and variability becomes large.

Measures of Variability

The Range: Range is the difference between in a series. It is the most general measure of spread or scatter. It is a measure of variability of the varieties or observation among themselves and does not given an idea about the spread of the observations around some central value.

The Quartile Deviation (Q): It is based upon the interval containing the middle fifty percent of cases in a given distribution. One quarter means 1/4th of something, when a scale is divided in to four equal parts. “The quartile deviation or Q is the one-half the scale distance between the 75t and 25th percentiles in a frequency distribution.”

The Average Deviation (A.D.): Average deviation s arithmetic mean of the deviations of a series computed from some measure of central tendency. So average deviation is the mean of the deviations taken from their mean (Sometimes from Median and Mode.)

The Standard Deviation (SD): Standard deviation is a measure of spread or dispersion. It is root mean squared deviation.” It is commonly used in experimental research as it is the most stable index of variability. Symbolically it is wrote as σ.

b) “Opinion polls are a very common method of gaining knowledge about consumer tastes.”

Ans) Opinion polls are a very popular way to learn about customer preferences, reactions to new products, leader or persona popularity, reactions to election results, or the likely candidate for prime minister after the upcoming elections. Any opinion survey should focus on two things above all else. The information sought is first, followed by the target demographic from which the information is sought. When creating the ideal mechanism for conducting the opinion survey, each of these considerations must be taken into account.

It is possible to conduct opinion surveys using:

1. Individual interviews

2. Distributing questionnaires.

3. Group meetings.

4. Symposia, conferences, and seminars.

The demographic, the required type of information, and the available budget all have a significant role in the approach chosen. For instance, a properly constructed questionnaire could be mailed to the individuals in question if data from a very large number of people needs to be gathered. A good questionnaire design is a significant undertaking in and of itself. It is important to avoid asking questions that are unclear.

The responses should ideally be brief one-word statements or the selection of an appropriate response from a list of many options. This makes it simple for respondents to complete the questionnaire and for analysts to analyse it.

As an illustration, the conclusion could be stated as follows:

1. 80 percent of people voiced their opinions. A.

2. 10% of people voiced an opinion B.

3. 5% of people stated an opinion C.

4. 5% of people were undecided.

Similar to the forecasting of product demand, it is typical to determine the sales forecast by combining the views of local salespeople. A rating for each salesperson or an adjustment for environmental uncertainties might be used to change the forecast. Expert comments are used to guide decisions regarding future R&D and new technologies.

c) “Probability theory provides us with the ways and means to attain the formal and precise expressions for uncertainties involved in different situations.”

Ans) In order to address the three various contexts in which probability measurements are typically required, three distinct probabilistic techniques have developed. In this section, we first examine the methods using illustrations of several kinds of experiments. The idea of probability is defined using the axioms that are common to different techniques after which they are discussed.

Think about the following scenarios, each of which features a different type of experiment. Within these experiments, the events in which we are most interested are also provided.

Draw a number from one of nine numbers in situation one (say 1 to 9).

Event: The number 4 appears on every draw.

Experiment 2: administering a certain medication.

Event: It takes the medicine ten minutes to put someone to sleep.

Experiment 3: Starting up a solar power facility.

Event: The factory proves to be a profitable endeavour.

If the experiment is conducted fairly, the first situation is defined by the fact that each of the nine participants has an equal chance of occurring on any draw. As a result, all of the numbers have an equal chance of appearing in any given draw. The Classical Approach to probability theory was born out of situations like these, which are characterised by the presence of "equally likely" clients. The definition of probability in the classical approach is:

the total number of outcomes, including those that were favourable to the event.

The total number of outcomes in a draw of the numbers 1 to 9 is therefore 9, as any one of the numbers may occur, if we designate the event that "a 4 comes out in a draw" as A and the probability of the event as P (A). Only the number 4 happens.

1 of these 9 possibilities. Thus P(A) =

We do not examine the second circumstance. We discover that we cannot predict that the drug will be equally effective for every person if we attempt to apply the aforementioned concept of probability to the second trial. Furthermore, we are unsure of the total number of test subjects. This suggests that we should have information from the past on those who received the medication and how many of them passed out within ten minutes. In the absence of historical data, we must conduct an experiment in which we deliver the medicine to a set of participants and observe how they respond to it. Here, we're presuming that experimenting is risk-free, i.e., the medication has no negative side effects.

In situations like these, probability is computed using the Relative Frequency Approach. According to this method, the likelihood that an event will occur is determined by the ratio of the number of times the event occurs to the total number of trials. The event is denoted by B, and the probability of the event is denoted by P(B).

that is, a measure of the probability will be the relative frequency of successes. However, this calculation assumes that either an experiment can be conducted with such businesses or that historical information on such ventures would be available. Since a solar power plant is a relatively recent development using cutting-edge technology, previous experiences are not readily available. Like drug testing, experimentation is not an option due to the significant expense and time commitment. The Subjective Approach to Probability is the sole option in these circumstances. With this strategy, we attempt to draw the probability from our personal experiences. To access this, we can bring in any information. We might investigate how the commissioning authority performed in other recent and related technologies in the aforementioned scenario. You should take note that because this evaluation is entirely subjective, it will differ from person to person.

d) “The primary purpose of correlation is to establish an association between any two random variables.”

Ans) Creating an association between any two random variables is the main goal of correlation. Although the existence of causality inherently requires the presence of connection, the opposite is also true. Only the presence or absence of a relationship between variables can be determined statistically. The existence or absence of causation is solely a matter of logic. For instance, there is evidence to suggest that spending more money on clothing of greater quality is correlated with income levels. However, one must be cautious of fictitious or absurd association that might be discovered between completely unrelated variables by pure accident.

Regression analysis's useful independent variables are first chosen using correlation analysis. For instance, a building company might recognise elements such as

1. Population.

2. Employment in construction.

3. It believes that the building permits given last year will have an impact on its sales this year.

By calculating the correlation coefficient of each pair of variables from the provided historical data, these and other factors that may be discovered could be examined for mutual correlation (this type of study is easily completed by utilising a suitable programme on a computer). Only variables with a strong association to the annual sales could be chosen for a regression model.

A wide range of measurable variables are attempted to be resolved in terms of a relatively small number of new Categories, known as factors, using correlation in factor analysis.

The following three applications of the findings might be helpful:

1. To uncover the hidden or latent elements that control how the seen data are related.

2. To reveal connections between data that were previously hidden through analysis.

3. When data evaluated on several rating scales needs to be brought together, to offer a classification scheme.

With the aid of time series models, forecasting is a significant additional use of correlation. Before constructing an acceptable forecasting model utilising historical data (which is frequently a time series of the variable of interest accessible at equal time intervals), one must determine the trend, seasonality, and random pattern in the data. Plots showing auto-correlation for different time lags and the idea of auto-correlation both aid in determining the underlying process' nature.

4. Write short notes on the following: (4×5)

a) Geometric Mean

Ans) In Mathematics, the Geometric Mean (GM) is the average value or mean which signifies the central tendency of the set of numbers by finding the product of their values. Basically, we multiply the numbers altogether and take the nth root of the multiplied numbers, where n is the total number of data values.

In other words, the geometric mean is defined as the nth root of the product of n numbers. It is noted that the geometric mean is different from the arithmetic mean. Because, in arithmetic mean, we add the data values and then divide it by the total number of values. But in geometric mean, we multiply the given data values and then take the root with the radical index for the total number of data values. For example, if we have two data, take the square root, or if we have three data, then take the cube root, or else if we have four data values, then take the 4th root, and so on.

Geometric Mean Formula

The formula to calculate the geometric mean is given below:

The Geometric Mean (G.M) of a series containing n observations is the nth root of the product of the values.

Consider, if x1, x2 …. Xn are the observation, then the G.M is defined as:

where the notation's typical meaning is present. Geometric mean is very helpful when creating index numbers. When giving big weights to tiny values of observations and small weights to large values of observations, it is an average that is most appropriate. This average can be used to estimate population growth.

b) Binomial Distribution

Ans) Binomial distribution is a probability distribution used in statistics that summarizes the likelihood that a value will take one of two independent values under a given set of parameters or assumptions. The underlying assumptions of binomial distribution are that there is only one outcome for each trial, that each trial has the same probability of success, and that each trial is mutually exclusive, or independent of one another.

To start, the “binomial” in binomial distribution means two terms. We’re interested not just in the number of successes, nor just the number of attempts, but in both. Each is useless to us without the other.

Binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as normal distribution. This is because binomial distribution only counts two states, typically represented as 1 (for a success) or 0 (for a failure) given a number of trials in the data. Binomial distribution thus represents the probability for x successes in n trials, given a success probability p for each trial.

Binomial distribution summarizes the number of trials, or observations when each trial has the same probability of attaining one particular value. Binomial distribution determines the probability of observing a specified number of successful outcomes in a specified number of trials.

Analysing Binomial Distribution

The expected value, or mean, of a binomial distribution is calculated by multiplying the number of trials (n) by the probability of successes (p), or n × p.

For example:

The expected value of the number of heads in 100 trials of heads or tails is 50, or (100 × 0.5). Another common example of binomial distribution is by estimating the chances of success for a free-throw shooter in basketball, where 1 = a basket made and 0 = a miss.

The binomial distribution formula is calculated as:

P(x:n,p) = nCx x px(1-p)n-x

where:

1. n is the number of trials (occurrences)

2. x is the number of successful trials

3. p is probability of success in a single trial

4. nCx is the combination of n and x. A combination is the number of ways to choose a sample of x elements from a set of n distinct objects where order does not matter and replacements are not allowed. Note that nCx=n!/(r!(n−r)!), where ! is factorial (so, 4! = 4 × 3 × 2 × 1).

The mean of the binomial distribution is np, and the variance of the binomial distribution is np (1 − p). When p = 0.5, the distribution is symmetric around the mean. When p > 0.5, the distribution is skewed to the left. When p < 0.5, the distribution is skewed to the right.

The binomial distribution is the sum of a series of multiple independent and identically distributed Bernoulli trials. In a Bernoulli trial, the experiment is said to be random and can only have two possible outcomes: success or failure.

For instance, flipping a coin is considered to be a Bernoulli trial; each trial can only take one of two values (heads or tails), each success has the same probability (the probability of flipping a head is 0.5), and the results of one trial do not influence the results of another. Bernoulli distribution is a special case of binomial distribution where the number of trials n = 1.

c) Rank correlation

Ans) The Spearman’s Correlation Coefficient, represented by ρ or by rR, is a nonparametric measure of the strength and direction of the association that exists between two ranked variables. It determines the degree to which a relationship is monotonic, i.e., whether there is a monotonic component of the association between two continuous or ordered variables.

Monotonicity is “less restrictive” than that of a linear relationship. Although monotonicity is not actually a requirement of Spearman’s correlation, it will not be meaningful to pursue Spearman’s correlation to determine the strength and direction of a monotonic relationship if we already know the relationship between the two variables is not monotonic.

On the other hand, if, for example, the relationship appears linear (assessed via scatterplot) one would run a Pearson’s correlation because this will measure the strength and direction of any linear relationship.

Spearman Ranking of the Data

We must rank the data under consideration before proceeding with the Spearman’s Rank Correlation evaluation. This is necessary because we need to compare whether on increasing one variable, the other follows a monotonic relation (increases or decreases regularly) with respect to it or not.

Thus, at every level, we need to compare the values of the two variables. The method of ranking assigns such ‘levels’ to each value in the dataset so that we can easily compare it.

Assign number 1 to n (the number of data points) corresponding to the variable values in the order highest to lowest.

In the case of two or more values being identical, assign to them the arithmetic mean of the ranks that they would have otherwise occupied.

For example, Selling Price values given: 28.2, 32.8, 19.4, 22.5, 20.0, 22.5 The corresponding ranks are: 2, 1, 5, 3.5, 4, 3.5 The highest value 32.8 is given rank 1, 28.2 is given rank 2,…. Two values are identical (22.5) and in this case, the arithmetic means of ranks that they would have otherwise occupied (3+4/2) has to be taken.

The Formula for Spearman Rank Correlation

where n is the number of data points of the two variables and di is the difference in the ranks of the ith element of each random variable considered. The Spearman correlation coefficient, ρ, can take values from +1 to -1.

A ρ of +1 indicates a perfect association of ranks

A ρ of zero indicates no association between ranks and

ρ of -1 indicates a perfect negative association of ranks.

The closer ρ is to zero, the weaker the association between the ranks.

d) Consideration in the choice of a forecasting method

Ans) Whatever the system used to generate forecasts, it is important to keep an eye on the results to make sure the gap between expected and actual demand figures stays within a set range that is acceptable for random variation.

The system creates a forecast based on historical data, which can be adjusted based on managerial discretion and knowledge. When new data is made available, the forecast is compared with it, and the error is watched or tracked to determine how well the forecast generation system is working.

A valuable statistical tool for tracking and confirming a forecasting system's accuracy is the moving chart.

The control chart is simple to create and keep up with. Let's say there are n periods of data available. When D, which is the actual demand for period t, and Ft, which is the prediction for time t,

1. If the demand in the past has been statistically steady.

2. Whether the current demand is consistent with historical patterns.

3. The control chart demonstrates how to modify the forecasting approach if the demand pattern has altered.

It demonstrates that the changes are caused by chance reasons and that the underlying mechanism of forecast production is acceptable as long as the displayed error points continue to stay within the control boundaries. When a situation spirals out of control, there is cause to doubt the accuracy of the prediction generation system, which needs to be updated to account for these modifications.

5. Distinguish between the following: (4×5)

a) Census and Sampling methods of data collections

Ans) The paramount differences between census and sampling are discussed in detail in the given below points:

1. The census is a systematic method that collects and records the data about the members of the population. The sampling is defined as the subset of the population selected to represent the entire group, in all its characteristics.

2. The census is alternately known as a complete enumeration survey method. In contrast, sampling is also known as a partial enumeration survey method.

3. In the census, each and every unit of population is researched. On the contrary, only a handful of items is selected from the population for research.

4. Census is a very time-consuming method of survey, whereas, in the case of sampling, the survey does not take much time.

5. The census method requires high capital investment as it involves the research and collection of all the values of the population. Unlike sampling which is a comparatively economical method.

6. The results drawn by conducting a census is accurate and reliable while there are chances of errors in the results drawn from the sample.

7. The size of the sample determines the probability of errors in the outcome, i.e., the larger the size of population the less are the chances of errors and the smaller the size; the higher are the chances of errors. This is not possible with census as all the items are taken into consideration.

8. Census is best suited for the population of heterogeneous nature. As opposed to sampling which is appropriate for homogeneous nature.

b) One-tailed and Two-tailed tests of variance

Ans) The fundamental differences between one-tailed and two-tailed test, is explained below in points:

1. One-tailed test, as the name suggest is the statistical hypothesis test, in which the alternative hypothesis has a single end. On the other hand, two-tailed test implies the hypothesis test; wherein the alternative hypothesis has dual ends.

2. In the one-tailed test, the alternative hypothesis is represented directionally. Conversely, the two-tailed test is a non-directional hypothesis test.

3. In a one-tailed test, the region of rejection is either on the left or right of the sampling distribution. On the contrary, the region of rejection is on both the sides of the sampling distribution.

4. A one-tailed test is used to ascertain if there is any relationship between variables in a single direction, i.e., left or right. As against this, the two-tailed test is used to identify whether or not there is any relationship between variables in either direction.

5. In a one-tailed test, the test parameter calculated is more or less than the critical value. Unlike, two-tailed test, the result obtained is within or outside critical value.

6. When an alternative hypothesis has ‘≠’ sign, then a two-tailed test is performed. In contrast, when an alternative hypothesis has ‘> or <‘ sign, then one-tailed test is carried out.

c) Linear Regression and Non-Linear Regression

Ans) The difference between linear and nonlinear regression models isn’t as straightforward as it sounds. You’d think that linear equations produce straight lines and nonlinear equations model curvature. Unfortunately, that’s not correct. Both types of models can fit curves to your data—so that’s not the defining characteristic. The difference between nonlinear and linear is the “non.” OK, that sounds like a joke, but, honestly, that’s the easiest way to understand the difference. First, we will define what linear regression is, and then everything else must be nonlinear regression.

Linear Regression Equations

A linear regression model follows a very particular form. In statistics, a regression model is linear when all terms in the model are one of the following:

1. The constant

2. A parameter multiplied by an independent variable (IV)

Then, you build the equation by only adding the terms together. These rules limit the form to just one type:

Dependent variable = constant + parameter * IV + … + parameter * IV

Y = +  + +      

Statisticians say that this type of regression equation is linear in the parameters. However, it is possible to model curvature with this type of model. While the function must be linear in the parameters, you can raise an independent variable by an exponent to fit a curve. For example, if you square an independent variable, the model can follow a U-shaped curve.

Y= +  +

While the independent variable is squared, the model is still linear in the parameters. Linear models can also contain log terms and inverse terms to follow different kinds of curves and yet continue to be linear in the parameters.

Nonlinear Regression Equations

It is showed how linear regression models have one basic configuration. Now, we’ll focus on the “non” in nonlinear! If a regression equation doesn’t follow the rules for a linear model, then it must be a nonlinear model. It’s that simple! A nonlinear model is literally not linear.

The added flexibility opens the door to a huge number of possible forms. Consequently, nonlinear regression can fit an enormous variety of curves. However, because there are so many candidates, you may need to conduct some research to determine which functional form provides the best fit for your data.

The defining characteristic for both types of models are the functional forms. If you can focus on the form that represents a linear model, it’s easy enough to remember that anything else must be a nonlinear.

d) Type I and Type II Errors

Ans) The points given below are substantial so far as the differences between type I and type II error is concerned:

1. Type I error is an error that takes place when the outcome is a rejection of null hypothesis which is, in fact, true. Type II error occurs when the sample results in the acceptance of null hypothesis, which is actually false.

2. Type I error or otherwise known as false positives, in essence, the positive result is equivalent to the refusal of the null hypothesis. In contrast, Type II error is also known as false negatives, i.e., negative result, leads to the acceptance of the null hypothesis.

3. When the null hypothesis is true but mistakenly rejected, it is type I error. As against this, when the null hypothesis is false but erroneously accepted, it is type II error.

4. Type I error tends to assert something that is not really present, i.e., it is a false hit. On the contrary, type II error fails in identifying something, that is present, i.e., it is a miss.

5. The probability of committing type I error is the sample as the level of significance. Conversely, the likelihood of committing type II error is same as the power of the test.

6. Greek letter ‘α’ indicates type I error. Unlike, type II error which is denoted by Greek letter ‘β.’