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SOLVED ASSIGNMENTS FOR JUNE & DEC TEE 2026

MCOM 2^ND SEMESTER

TUTOR MARKED ASSIGNMENT

COURSE CODE : MCO-022

COURSE TITLE : QUANTITATIVE ANALYSIS & MANAGERIAL APPLICATION

ASSIGNMENT CODE : MCO-022/TMA/2025-2026

 

1. a) What do you understand by the term correlation? Explain how the study of correlation helps in forecasting demand of a product.

Introduction

In the field of economics, business, and statistics, decision-making often requires understanding how two or more variables are related to each other. For instance, businesses want to know how sales are affected by advertising expenditure, how demand is influenced by price changes, or how consumer income impacts the consumption of luxury goods. The concept of correlation provides a systematic way to measure such relationships. Particularly in demand forecasting, correlation acts as a guiding tool, helping organizations to predict future sales or demand by analyzing how demand is associated with other measurable factors.

Meaning of Correlation

Correlation is a statistical technique used to study the degree of relationship between two or more variables. It tells us whether variables move together, in the same direction, or in opposite directions, and how strongly they are related.

·        If both variables move in the same direction, we call it a positive correlation.
Example: As consumer income rises, demand for smartphones increases.

·        If both variables move in opposite directions, it is a negative correlation.
Example: As the price of tea rises, the demand for tea decreases.

·        If there is no systematic relationship, then there is zero correlation.
Example: The number of hours of sunshine in a city and the sales of laptops may show no correlation.

The correlation coefficient, usually represented by ‘r’, ranges from –1 to +1.

·        +1 shows perfect positive correlation.

·        –1 shows perfect negative correlation.

·        0 shows no correlation at all.

Types of Correlation

1.     Positive Correlation – Both variables increase or decrease together. E.g., more advertising expenditure → more sales.

2.     Negative Correlation – One variable increases while the other decreases. E.g., higher price of petrol → lower demand for petrol.

3.     Linear and Non-Linear Correlation – If the change is proportional, it is linear; if not, it is non-linear.

4.     Simple and Multiple Correlation – Simple correlation studies relation between two variables, while multiple correlation deals with more than two factors at the same time (e.g., demand depending on income, price, and advertising together).

Correlation and Demand Forecasting

Forecasting demand is essential for businesses because it helps in production planning, inventory control, marketing strategies, and financial management. Correlation analysis allows businesses to identify and measure how demand is influenced by various economic, social, and business factors. Some key applications include:

1. Price-Demand Relationship

One of the most fundamental applications of correlation is to measure the relationship between price and demand. Generally, demand and price are negatively correlated: when prices rise, demand falls. For example, if the price of cold drinks increases by 20%, sales might fall by 10%. By studying past data of price and sales, firms can forecast how future changes in price will impact demand.

2. Income and Demand

Income is another strong determinant of demand. For normal and luxury goods, demand increases with rising income, showing positive correlation. For inferior goods, demand falls when income rises, showing negative correlation. For instance, demand for branded clothing and cars is positively correlated with rising income levels in urban India. Businesses can use this relationship to forecast demand based on income projections.

3. Advertising and Sales

Advertising and promotional activities are usually positively correlated with demand. A company can analyze how much additional sales are generated per unit of advertising expenditure. For example, if past records show that a 10% increase in ad budget leads to a 5% increase in sales, firms can use this correlation to predict sales for future campaigns.

4. Substitute and Complementary Goods

Demand for a product is often correlated with the demand or price of related goods. Complementary goods (like petrol and cars, or printers and cartridges) show positive correlation, while substitute goods (like coffee and tea, or butter and margarine) show negative correlation. Studying these correlations helps businesses anticipate changes in demand when related product markets fluctuate.

5. Seasonal and Climatic Correlation

Some products show strong correlation with seasonal or climatic variables. For example, sales of air conditioners are positively correlated with temperature rise, while sales of umbrellas are positively correlated with rainfall. By analyzing these correlations, companies can forecast seasonal demand and plan inventories accordingly.

6. Macroeconomic Variables

Demand may also correlate with GDP growth, inflation, or interest rates. For example, the demand for automobiles and consumer durables is strongly correlated with GDP growth. Similarly, demand for housing loans may be negatively correlated with interest rates. Firms can forecast future demand by studying how demand responds to changes in such macroeconomic variables.

Illustration

Suppose a company selling smartphones studies past data and finds that whenever average consumer income rises by 10%, its sales increase by 8%. This indicates a strong positive correlation between income and demand. If government reports project a 15% increase in average income next year, the company can forecast a likely 12% rise in demand for its products, allowing it to adjust production and marketing strategies in advance.

Significance of Correlation in Forecasting

·        Helps in identifying demand determinants.

·        Provides a scientific basis for demand forecasting.

·        Reduces uncertainty in business planning.

·        Improves accuracy of regression and econometric forecasting models.

·        Enables businesses to respond proactively to market changes.

 

Conclusion

Correlation, therefore, is not just a statistical measure but a practical tool for business forecasting. By studying how demand is correlated with price, income, advertising, seasonal changes, and macroeconomic variables, firms can predict future demand more accurately. This aids in efficient production planning, effective marketing, better financial control, and long-term strategic decision-making. In a competitive environment, businesses that successfully use correlation analysis in demand forecasting are better equipped to meet customer needs, minimize risks, and maximize profitability.

 

b) What are ogives? Discuss the method of constructing ogives with the help of an example.

b) What are Ogives? Discuss the Method of Constructing Ogives with the Help of an Example.

Meaning of Ogives:
An ogive is a type of cumulative frequency graph used in statistics to represent how frequencies accumulate over different class intervals. It provides a visual picture of the cumulative distribution of data and helps identify the number or proportion of observations that fall below or above a specific value. In other words, ogives show how the total frequency builds up progressively across class boundaries.

There are two types of ogives:

1. Less than Ogive
2. More than Ogive

Both curves are plotted on a graph using cumulative frequencies but in opposite directions, and when drawn together, they often intersect at a point that represents the median of the data.

 

1. Less than Ogive

The less than ogive is drawn by plotting the upper class boundaries of each class interval against the cumulative frequencies. It shows the total number of observations less than or equal to a particular value.

Steps to Construct a Less than Ogive:

1. Prepare the cumulative frequency table by adding frequencies successively from top to bottom.
2. Determine upper class boundaries for each class interval.
3. Plot points where the upper class boundary is on the X-axis and corresponding cumulative frequency is on the Y-axis.
4. Join the points smoothly with a freehand curve or straight lines to obtain the ogive.

 

2. More than Ogive

The more than ogive is drawn by plotting the lower class boundaries against cumulative frequencies starting from the bottom. It represents the total number of observations greater than or equal to a particular value.

Steps to Construct a More than Ogive:

1. Prepare a cumulative frequency table starting from the bottom (subtracting frequencies successively).
2. Determine lower class boundaries of each class interval.
3. Plot points where the lower boundary is on the X-axis and cumulative frequency on the Y-axis.
4. Join the points smoothly to obtain the more than ogive.

 

Example:

Let’s consider the following data showing marks of 50 students:

Marks (Class Interval)	Frequency
0 – 10	5
10 – 20	8
20 – 30	12
30 – 40	10
40 – 50	9
50 – 60	6

(a) For Less than Ogive:

Marks (Less than)	Cumulative Frequency
10	5
20	13
30	25
40	35
50	44
60	50

Plot these points (10,5), (20,13), (30,25), (40,35), (50,44), (60,50) and join them smoothly to get the less than ogive.

 

(b) For More than Ogive:

Marks (More than)	Cumulative Frequency
0	50
10	45
20	37
30	25
40	15
50	6

Plot these points (0,50), (10,45), (20,37), (30,25), (40,15), (50,6) and join them to get the more than ogive.

 

Interpretation:

• The intersection of the two ogives gives the median value of the distribution.
• The shape of the ogive helps in understanding how data is distributed — whether concentrated in a specific range or spread widely.

 

Conclusion:

Ogives are powerful graphical tools for visualizing cumulative frequency data. They make it easier to estimate medians, percentiles, and the proportion of observations below or above certain values. By constructing both types of ogives and analyzing their intersection, one can derive valuable statistical insights about the distribution of a dataset.

 

2. a) Describe the four states of decision environment in managerial applications. Which is the most prevalent state? Give reasons with examples.

Introduction:
Decision-making is one of the most essential managerial functions. Every manager continuously makes choices about planning, organizing, directing, or controlling business operations. However, decisions are never made in isolation—they are influenced by the surrounding decision environment, which refers to the conditions under which decisions are taken. The nature of information available, the predictability of outcomes, and the degree of uncertainty define the type of decision environment.

Scholars generally classify decision environments into four states:

1. Certainty
2. Risk
3. Uncertainty
4. Ambiguity

 

1. Decision-Making Under Certainty

In this environment, the manager has complete and accurate information about all possible alternatives, outcomes, and their consequences. The results of each decision are known in advance, so there is little to no ambiguity.

Example:
When a manager invests in a government bond with a fixed interest rate, the return is certain. Similarly, calculating production costs using known input prices falls under this environment.

Managerial Implication:
Since all variables are known, decision-making under certainty involves simple analytical tools like break-even analysis or linear programming. It is rare in real business scenarios because complete information is seldom available.

 

2. Decision-Making Under Risk

In this environment, the manager knows the probable outcomes of various alternatives, though not with complete certainty. Decisions are based on probability estimates, often derived from past data or statistical analysis.

Example:
A company launching a new product may use market research data to estimate the probability of success or failure. While the exact result isn’t known, managers can predict outcomes such as a 70% chance of high sales and 30% chance of low sales.

Managerial Implication:
This environment requires tools like expected value analysis, decision trees, and risk assessment models. Managers often prefer this setting because it balances analytical reasoning with real-world unpredictability.

 

3. Decision-Making Under Uncertainty

Under uncertainty, the manager lacks adequate information to determine probabilities or predict outcomes. Both internal and external variables are unpredictable, and historical data may be unreliable.

Example:
When a company decides to enter a completely new foreign market with no prior experience or reliable market data, it operates under uncertainty. External factors such as political instability or sudden regulatory changes make prediction difficult.

Managerial Implication:
Here, managers rely on judgment, intuition, and experience rather than mathematical models. Scenario planning and sensitivity analysis are often used to anticipate possible outcomes.

 

4. Decision-Making Under Ambiguity

Ambiguity represents the most complex decision environment. In this situation, managers are unclear not only about the outcomes but also about the nature of the problem itself. The objectives, variables, and even decision criteria are undefined or changing.

Example:
When a company explores adopting emerging technologies like quantum computing or metaverse marketing, the business model, outcomes, and impacts are not yet clear. Managers face ambiguous conditions requiring creativity and experimentation.

Managerial Implication:
Decision-making under ambiguity demands flexibility, strategic thinking, and continuous learning. Managers often use design thinking, pilot projects, or adaptive leadership strategies to navigate ambiguity.

 

Most Prevalent State: Decision-Making Under Risk

In contemporary managerial practice, decision-making under risk is the most prevalent environment. Modern managers typically operate with partial but quantifiable information—enough to estimate probabilities but not complete certainty.

Reasons:

1. Businesses operate in dynamic markets with measurable risks such as price fluctuations, demand variation, or competition.
2. Advanced data analytics and forecasting tools allow managers to assess risk probabilities more effectively.
3. Managers often face time constraints that prevent gathering complete data, making risk-based decisions more practical.

Example:
Banks evaluating loan applications, investors predicting stock returns, or marketing managers planning new campaigns—all rely on probabilistic models rather than complete certainty.

 

Conclusion:

Decision-making environments range from certainty to ambiguity, depending on the availability of information and predictability of outcomes. While decisions under certainty are rare and those under ambiguity are complex, most managerial choices fall under conditions of risk, where outcomes are uncertain but measurable. Effective managers, therefore, must learn to evaluate and manage risk strategically to make sound, evidence-based decisions in a competitive business environment.

 

b) Define matrices. Give examples of some special matrices. How would you represent the data of a transportation problem in the matrix form?

3. a) What do you understand by decision theory? What are the key issues in decision theory? Explain decision tree approach for managerial applications.

a) Decision Theory: Meaning, Key Issues, and Decision Tree Approach

Meaning of Decision Theory

Decision theory is a systematic and logical approach used to select the best course of action among several alternatives under given conditions. It helps managers make rational choices when outcomes are uncertain. Decision theory combines concepts from economics, mathematics, statistics, and psychology to guide the process of choosing the most suitable decision based on the available data and probable outcomes.

In simple terms, decision theory is the study of how decisions are made and how they should be made. It provides a structured framework to analyze various options, assess associated risks, and select the optimal strategy that aligns with organizational objectives.

 

Key Elements of Decision Theory

1.     Decision Alternatives – The possible courses of action available to a decision-maker (e.g., launching a new product, expanding to a new market).

2.     States of Nature – The possible external conditions or future scenarios that affect outcomes but are beyond the control of the decision-maker (e.g., market demand, competition, government policy).

3.     Payoffs – The expected results or outcomes (profit, cost, revenue) associated with each combination of an alternative and a state of nature.

4.     Probabilities – The likelihood of each state of nature occurring.

5.     Decision Criteria – Rules or techniques used to choose the best alternative, depending on the level of uncertainty (e.g., Maximax, Maximin, Minimax Regret, Expected Monetary Value).

 

Key Issues in Decision Theory

Decision theory addresses several critical issues that affect managerial decision-making:

1.     Uncertainty and Risk – Decisions often have uncertain outcomes due to incomplete information. Managers must assess the degree of risk and uncertainty.

2.     Information Availability – The quality and quantity of available data affect the reliability of decisions.

3.     Human Judgment and Bias – Managers’ personal biases, overconfidence, and perception may distort decision-making.

4.     Complexity of Alternatives – Multiple conflicting objectives and interdependent alternatives make decision-making complex.

5.     Changing Environment – Economic, technological, and social changes influence decision outcomes.

6.     Optimization of Objectives – The goal is to maximize profit, minimize cost, or achieve an optimal balance between conflicting objectives.

7.     Ethical and Social Considerations – Decisions must align with ethical norms and corporate responsibility.

 

Decision Tree Approach in Managerial Applications

The decision tree is a graphical method used to represent complex decision problems in a simplified visual form. It helps managers identify possible alternatives, outcomes, and their respective payoffs systematically.

Steps in Constructing a Decision Tree

1.     Define the Decision Problem – Identify the main decision point (e.g., whether to launch a new product).

2.     Draw Decision Nodes – Represented by a square (□), showing where a decision must be made.

3.     Add Chance Nodes – Represented by circles (○), showing points of uncertainty where various outcomes may occur.

4.     List Alternatives and Probabilities – For each chance node, write possible outcomes and their associated probabilities.

5.     Calculate Payoffs – Assign monetary or utility values for each possible outcome.

6.     Compute Expected Values (EV) – Multiply each payoff by its probability and sum them for each alternative.

7.     Select the Best Alternative – Choose the decision with the highest expected value.

Conclusion

Decision theory serves as a scientific guide for managerial problem-solving under uncertainty. Among its tools, the decision tree is one of the most practical and visual methods for analyzing risk and evaluating multiple alternatives. It enhances managerial judgment, minimizes subjectivity, and supports rational, data-driven decision-making in business environments.

 

b) Explain forecasting methods for long, medium and short-term decisions.

b) Forecasting Methods for Long, Medium, and Short-Term Decisions

Meaning of Forecasting

Forecasting is the process of estimating future events or trends based on the analysis of past and present data. In managerial decision-making, forecasting plays a vital role as it provides a scientific basis for planning and controlling business activities. Managers use forecasting to predict sales, production levels, prices, demand, and market trends to make informed short, medium, and long-term decisions.

Forecasting methods can broadly be classified based on the time horizon they cover—short-term, medium-term, and long-term forecasting—each serving different managerial purposes and requiring distinct techniques.

 

1. Short-Term Forecasting

Time Horizon: Usually covers a period of up to one year.
Purpose: To make operational decisions such as inventory control, production scheduling, manpower planning, and short-term sales targets.

Common Methods:

1.     Moving Average Method:
This method calculates the average of a fixed number of recent data points to predict the next value. It smooths out fluctuations and is effective for stable demand patterns.
Example: A three-month moving average forecast for month 4 = (Month1 + Month2 + Month3) ÷ 3.

2.     Exponential Smoothing:
Assigns higher weights to recent observations, making forecasts more responsive to recent changes. It is widely used in inventory and production planning.

3.     Trend Projection Method:
Based on fitting a straight line (Y = a + bX) to past data to project future values. Suitable when a consistent upward or downward trend exists in demand.

4.     Regression Analysis:
Establishes a relationship between dependent and independent variables (e.g., sales and advertisement expenditure). It helps in short-term forecasting when influencing factors are measurable.

5.     Naïve Method:
Assumes that the demand in the next period will be equal to the demand in the current period. Useful for very short-term forecasts where demand patterns are stable.

Applications:
Used for daily or weekly production scheduling, sales budgeting, cash flow planning, and workforce allocation.

 

2. Medium-Term Forecasting

Time Horizon: Covers one to three years.
Purpose: Helps in tactical decisions such as resource allocation, capacity planning, budgeting, and pricing strategies.

Common Methods:

1.     Trend Analysis with Seasonality:
Combines trend projection with seasonal indices to forecast periodic demand variations, useful for industries affected by seasons (e.g., FMCG, clothing).

2.     Econometric Models:
These use statistical techniques to establish relationships between economic variables (e.g., GDP, interest rate, and sales). Such models help in planning for medium-term market changes.

3.     Delphi Technique:
Involves obtaining forecasts through structured expert opinions. The process continues until a consensus is reached, making it valuable for new product forecasting or uncertain environments.

4.     Scenario Analysis:
Considers multiple potential future scenarios (e.g., economic boom, recession) and prepares forecasts for each. Managers can plan flexible strategies accordingly.

5.     Causal Methods:
Identify cause-and-effect relationships among variables such as advertising expenditure, pricing, and consumer demand to estimate medium-term performance.

Applications:
Useful for annual budgeting, manpower training programs, marketing campaigns, and expansion planning.

 

3. Long-Term Forecasting

Time Horizon: Generally covers a period of five years or more.
Purpose: Assists in strategic decisions like investment in new projects, technology upgradation, capacity expansion, and diversification.

Common Methods:

1.     Trend Extrapolation:
Extends long-term trends in historical data into the future. Effective when major structural changes are not expected in the economy.

2.     Econometric and Simulation Models:
Combine multiple variables and simulate future outcomes under different assumptions. They are used for forecasting macroeconomic indicators and long-term market behavior.

3.     Technology Forecasting:
Predicts future advancements in technology using methods such as patent analysis, expert opinions, and Delphi studies.

4.     Input-Output Analysis:
Examines interrelationships between various sectors of the economy to predict future demand for products or resources.

5.     Strategic Scenarios and Environmental Scanning:
Considers political, social, technological, and economic factors to anticipate long-term market directions and business risks.

Applications:
Used for corporate strategy formulation, infrastructure investment, R&D planning, and long-term financial forecasting.

 

Conclusion

Forecasting is an indispensable tool for effective management at all levels. Short-term forecasts ensure operational efficiency, medium-term forecasts support tactical planning, and long-term forecasts guide strategic growth. By selecting suitable methods for each time horizon, managers can minimize uncertainty, allocate resources effectively, and align business decisions with future opportunities and risks.

 

4. Write short notes on the following:

a) Binomial Distribution

b) Consideration in the choice of a forecasting method

c) Testing the Goodness of Fit

d) Forecast Control

a) Binomial Distribution

The Binomial Distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes — success or failure.
It is widely used in business, statistics, and management for decision-making under uncertainty.

Key Characteristics:

1.     Each trial has two outcomes — success (p) and failure (q = 1 – p).

2.     The number of trials (n) is fixed.

3.     The probability of success remains constant in all trials.

4.     All trials are independent of each other.

Probability Formula:

b) Considerations in the Choice of a Forecasting Method

Choosing an appropriate forecasting method depends on various factors, as no single technique fits all situations. Managers must consider both the nature of the data and the business context before selecting a method.

Key Considerations:

1. Time Horizon:
   Short-term forecasts use methods like moving averages or exponential smoothing, whereas long-term forecasts require trend analysis or econometric models.
2. Data Availability and Quality:
   The choice depends on how much historical data exists and how reliable it is. For example, regression analysis requires consistent quantitative data.
3. Nature of Demand:
   If demand shows seasonal patterns, a seasonal model is needed. If it is stable, simpler methods suffice.
4. Cost and Resources:
   Complex statistical methods require skilled personnel and software, which may not be cost-effective for small firms.
5. Desired Accuracy:
   High accuracy may justify the use of sophisticated methods like econometric models, while simpler techniques may suffice for routine forecasting.
6. Managerial Judgment:
   When data is insufficient, qualitative techniques such as the Delphi method or market surveys are more appropriate.
7. External Factors:
   Economic conditions, government policies, and technological changes influence the choice of method.

Conclusion:
The best forecasting method balances accuracy, cost, and practicality while fitting the decision-making context.

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c) Testing the Goodness of Fit

Goodness of fit tests are statistical methods used to determine how well a theoretical probability distribution fits observed data. It helps assess whether sample data follows a specified distribution like normal, binomial, or Poisson.

Common Method: Chi-Square Test

 

 

Example:
Used to test whether the number of defective items in a batch follows a Poisson distribution.

Applications:
Quality control, market research, risk assessment, and hypothesis testing.

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d) Forecast Control

Forecast Control refers to the process of continuously monitoring forecasting results to ensure accuracy and reliability. Forecasts are based on assumptions and data, both of which can change over time; hence, controlling forecasts ensures that deviations are detected early and corrective measures are applied.

Key Elements of Forecast Control:

1. Establish Forecast Standards:
   Define acceptable levels of accuracy and identify performance indicators like Mean Absolute Deviation (MAD) or Mean Squared Error (MSE).
2. Measurement of Forecast Errors:
   Calculate deviations between actual and forecasted values to measure accuracy.
3. Analysis of Errors:
   Identify whether deviations are random or systematic (bias). Systematic errors require adjustments in the forecasting model.
4. Feedback Mechanism:
   Provide corrective feedback to improve the forecasting process or model.
5. Updating the Forecast:
   Revise forecasts periodically based on new information and trends.
6. Responsibility Assignment:
   Assign clear responsibility for monitoring and revising forecasts.
7. Use of Control Charts:
   Tools such as tracking signals or control charts can visually display whether forecast errors remain within acceptable limits.

Conclusion:
Forecast control ensures that forecasting remains a dynamic and adaptive process, helping managers maintain accuracy, anticipate changes, and make better-informed business decisions.

 

5. Distinguish between the following:

a) Pilot testing and Pre-testing

b) Null Hypothesis and Alternate Hypothesis

c) Mean and Median

d) Probability and Non-Probability Sampling

a) Pilot Testing and Pre-testing

Basis of Difference	Pilot Testing	Pre-testing
Meaning	Pilot testing refers to conducting a small-scale preliminary study to evaluate the feasibility, time, cost, and process of the main research project.	Pre-testing is the process of testing a research instrument (like a questionnaire or survey) on a small sample to check clarity, wording, and respondent understanding.
Purpose	To test the overall design and procedure of the study before launching it fully.	To ensure that the questions, format, and language of the research tool are appropriate and understandable.
Scope	Broader in scope—it examines logistics, sampling, data collection, and analysis.	Narrower—it focuses only on the tool or questionnaire.
Focus	Focuses on the entire research process.	Focuses on the quality and reliability of questions.
Outcome	Helps identify operational problems and improve the research design.	Helps refine and modify the research instrument for clarity and accuracy.
Timing	Conducted before the main study as a rehearsal of the research.	Conducted before the finalization of the questionnaire or schedule.
Example	Conducting a small trial run of a market survey to check response time and logistics.	Testing a 10-question customer satisfaction survey on 5–10 respondents before the main survey.

Summary:
Pre-testing focuses on questionnaire improvement, while pilot testing focuses on the entire research process.

 

b) Null Hypothesis and Alternate Hypothesis

Basis of Difference	Null Hypothesis (H₀)	Alternate Hypothesis (H₁ or Ha)
Meaning	A null hypothesis states that there is no significant relationship or difference between variables.	The alternate hypothesis states that there is a significant relationship or difference between variables.
Purpose	Serves as a starting point for statistical testing and is assumed true until proven otherwise.	Represents what the researcher aims to prove through evidence.
Symbol	Denoted as H₀	Denoted as H₁ or Ha
Nature	Conservative and default assumption.	Contradictory to the null hypothesis.
Decision Rule	Accepted if there is not enough evidence to reject it.	Accepted when sufficient evidence exists against H₀.
Example	H₀: There is no difference in average sales before and after training.	H₁: There is a difference in average sales before and after training.
Result Interpretation	Acceptance of H₀ means observed differences are due to chance.	Acceptance of H₁ means differences are statistically significant.

Summary:
Null hypothesis implies no effect, while alternate hypothesis indicates presence of an effect.

 

c) Mean and Median

Basis of Difference	Mean	Median
Meaning	The mean is the arithmetic average of all data values.	The median is the middle value that divides the data into two equal halves.
Formula	Mean (𝑥̄) = ΣX / N	Median = Middle value after arranging data in ascending or descending order.
Type of Measure	It is a measure of central tendency based on all observations.	It is a positional average, not affected by extreme values.
Effect of Extreme Values	Greatly affected by extreme or outlier values.	Not affected by extreme values.
Usefulness	Suitable for data with relatively uniform distribution.	Suitable for skewed or unequal distributions.
Example	For 5, 10, 15, 20 → Mean = (5+10+15+20)/4 = 12.5	For 5, 10, 15, 20 → Median = (10+15)/2 = 12.5
Application	Used in economics, accounting, and performance analysis.	Used in income distribution and demographic studies.

Summary:
Mean gives the average, while median gives the central position in a data set.

 

d) Probability and Non-Probability Sampling

Basis of Difference	Probability Sampling	Non-Probability Sampling
Meaning	Every element of the population has a known and equal chance of being selected.	The selection of elements is based on researcher’s judgment or convenience, not random.
Basis of Selection	Random selection method.	Non-random, based on subjective choice.
Bias	Less biased as every unit has equal opportunity.	More prone to bias and sampling errors.
Representativeness	Produces a representative sample of the population.	May not represent the entire population accurately.
Statistical Inference	Allows use of statistical tests and generalization to the population.	Does not allow reliable generalization beyond the sample.
Examples	Simple random sampling, stratified sampling, cluster sampling.	Convenience sampling, judgmental sampling, quota sampling.
Usefulness	Suitable for quantitative and large-scale studies.	Suitable for exploratory or qualitative research.
Cost and Time	More costly and time-consuming.	Less expensive and faster to conduct.

Summary:
Probability sampling ensures objectivity and generalization, while non-probability sampling provides flexibility and practicality for exploratory studies.