Unit 5 Business Statistics and Research Methods PYQ’s 3rd January 2026 Shift 1

The correct answer is – 37

Key Points

• Understanding leap years
  • A leap year consists of 366 days, meaning it has 52 complete weeks and 2 extra days.
  • The two extra days can be any pair of consecutive days of the week, such as Sunday and Monday, Monday and Tuesday, and so on.
• 53 Sundays or 53 Mondays
  • For a day to appear 53 times in a leap year, it must occur on one of the two extra days.
  • In a leap year, there are 7 possible combinations of the two extra days:
    • Sunday & Monday
    • Monday & Tuesday
    • Tuesday & Wednesday
    • Wednesday & Thursday
    • Thursday & Friday
    • Friday & Saturday
    • Saturday & Sunday
  • Out of these combinations, 3 combinations include either Sunday or Monday as one of the extra days:
    • Sunday & Monday
    • Saturday & Sunday
    • Monday & Tuesday
• Probability calculation
  • The probability of selecting a combination with either 53 Sundays or 53 Mondays is:
  • Probability = Favorable outcomesTotal outcomes = 37

Additional Information

• Leap year identification
  • A leap year occurs every 4 years, except for years divisible by 100 but not divisible by 400.
  • For example, 2000 and 2024 are leap years, but 1900 is not.
• Day distribution in a leap year
  • Each week has 7 days, so a leap year has 52 full weeks (52 × 7 = 364 days).
  • The remaining 2 days determine the days that will occur 53 times in that year.
• Probability basics
  • Probability is defined as the ratio of favorable outcomes to total possible outcomes.
  • In this question, the total outcomes are 7 (possible combinations of 2 extra days), and favorable outcomes are 3 (combinations including Sunday or Monday).

The correct answer is – A-II, B-I, C-IV, D-III

Key Points

• Type I Error (A-II)
  • Occurs when a true null hypothesis is incorrectly rejected.
  • Also known as a false positive error.
  • Denoted by the symbol α (alpha).
• Type II Error (B-I)
  • Occurs when a false null hypothesis is not rejected.
  • Also known as a false negative error.
  • Denoted by the symbol β (beta).
• Parametric Tests (C-IV)
  • Used for analyzing data from interval or ratio scales.
  • Examples include t-tests and ANOVA.
  • Assume that the data follows a normal distribution.
• Nonparametric Tests (D-III)
  • Used for data measured on nominal or ordinal scales.
  • Examples include chi-square tests and Mann-Whitney U tests.
  • Do not assume a specific distribution for the data.

Additional Information

• Hypothesis Testing
  • Aims to determine whether there is enough evidence to reject a null hypothesis.
  • The null hypothesis is a statement of no effect or no difference.
• Error Types
  • Type I Error (α)
    • The likelihood of rejecting a true null hypothesis.
    • Controlled by setting a significance level, typically 0.05 or 5%.
  • Type II Error (β)
    • The likelihood of failing to reject a false null hypothesis.
    • Inversely related to the statistical power of a test (1 – β).
• Parametric vs. Nonparametric Tests
  • Parametric Tests
    • Require assumptions about the population distribution (e.g., normality).
    • Provide greater statistical power when assumptions are met.
  • Nonparametric Tests
    • Do not rely on strict distributional assumptions.
    • Useful for small sample sizes or non-normal data.

The correct answer is – A-IV, B-III, C-I, D-II

Key Points

• Probable Error of Correlation Coefficient
  • Formula: 0.6745 × 1 – r²√(n).
  • Used to test the reliability of correlation.
  • Hence matched with IV.
• Standard Error of Correlation Coefficient
  • Formula: 1 – r²√(n).
  • Measures sampling variability of the correlation coefficient.
  • Hence matched with III.
• Coefficient of Determination
  • Represents Explained Variation / Total Variation.
  • Numerically equal to r².
  • Hence matched with I.
• Coefficient of Alienation
  • Represents Unexplained Variation / Total Variation.
  • Equal to 1 − r².
  • Hence matched with II.

Additional Information

• Correlation Coefficient (r)
  • Ranges between −1 and +1.
  • Shows degree and direction of linear relationship.
• Relationship among Measures
  • Coefficient of Determination + Coefficient of Alienation = 1.
  • i.e., r² + (1 − r²) = 1.
• Use of Probable Error
  • If correlation > 6 × Probable Error, correlation is considered significant.
  • Helps judge statistical significance of correlation.
• Sample Size Effect
  • Larger n (sample size) reduces both standard error and probable error.

The correct answer is – 54

Key Points

• Quartile Deviation
  • The quartile deviation is defined as half the difference between the third quartile (Q₃) and the first quartile (Q₁).
  • Formula: Quartile Deviation = (Q₃ – Q₁) / 2.
• Symmetrical Distribution
  • In a symmetrical distribution, the median is equidistant from the first quartile and the third quartile.
  • As per the property of symmetry: Median = Q₁ + Quartile Deviation.
• Calculation
  • Given data: Q₁ = 48, Quartile Deviation = 6.
  • Using the formula Median = Q₁ + Quartile Deviation:
  • Median = 48 + 6 = 54.

Additional Information

• Median in Symmetrical Distributions
  • In a symmetrical distribution, the median divides the data into two equal halves.
  • The positions of Q₁, Median, and Q₃ maintain a consistent relationship.
  • The formula Median = Q₁ + Quartile Deviation is valid only for symmetrical data.
• Quartiles
  • Q₁: The value below which 25% of the data falls.
  • Q₃: The value below which 75% of the data falls.
  • Quartiles divide the data into four equal parts, each representing 25% of the distribution.
• Other Measures of Central Tendency
  • Mean: The arithmetic average of the data.
  • Mode: The most frequently occurring value in the data.
  • In symmetrical distributions, Mean = Median = Mode.

The correct answer is – 0.043

Key Points

• Karl Pearson’s Coefficient of Skewness Formula
  • The formula is given as: Skewness = 3 × (Mean – Median) / Standard Deviation.
• Substitute the given values
  • Mean = 6.82
  • Median = 6.8
  • Standard Deviation = 1.4
• Step-by-step Calculation
  • Skewness = 3 × (6.82 – 6.8) / 1.4.
  • Skewness = 3 × 0.02 / 1.4.
  • Skewness = 0.043.
• Thus, the correct answer is 0.043.

Additional Information

• Understanding Skewness
  • It measures the asymmetry of a data distribution.
  • If the mean is greater than the median, the distribution is positively skewed.
  • If the mean is less than the median, the distribution is negatively skewed.
• Karl Pearson’s Coefficient of Skewness
  • It is a measure of skewness that uses the mean, median, and standard deviation of the distribution.
  • The formula is simple and widely used for quick estimation of skewness.
• Significance in Statistics
  • Skewness helps in understanding whether a distribution is symmetrical or not.
  • It is important for determining the shape of data in probability and data analysis.

The correct answer is – B, A, D, C, E

Key Points

• Problem definition (B) is the first step as it involves clearly identifying the research problem or objective.
• Developing an approach to the problem (A) comes next, where a framework for solving the problem is established through literature review or theoretical understanding.
• Research design formulation (D) is the third step where the methodology and techniques for data collection and analysis are planned.
• Data Collection, Preparation, and Analysis (C) follows, which involves gathering, cleaning, and analyzing the data as per the research design.
• Report writing and presentation (E) is the final step where the results are documented and presented effectively.

Additional Information

• Problem Definition
  • It forms the foundation of the research process by identifying the research objective or issue to address.
  • Common tools include stakeholder interviews and exploratory studies.
• Approach Development
  • This step includes framing hypotheses and identifying variables to be studied.
  • It often involves extensive literature review and theoretical modeling.
• Research Design
  • Includes determining sample size, data collection methods (qualitative or quantitative), and statistical techniques.
  • Ensures reliability and validity of the research outcomes.
• Data Collection and Analysis
  • Data is collected through surveys, experiments, or observations and is then cleaned for errors or inconsistencies.
  • Various statistical methods are used to draw meaningful insights.
• Report Writing
  • Findings are documented in a structured format for stakeholders.
  • Includes visual aids like graphs and tables for better comprehension.

The correct answer is – Prefatory Information

Key Points

• Executive Summary
  • The Executive Summary is a brief overview of the entire research report.
  • It provides a concise summary of the objectives, methods, key findings, and conclusions.
  • This section is often included in the Prefatory Information to give readers an overview before delving into the detailed content.
• Position of Prefatory Information
  • The Prefatory Information section is the first part of a research report, preceding the main content.
  • It typically includes components like the title page, acknowledgments, table of contents, and the Executive Summary.

Additional Information

• Components of a Research Report
  • Prefatory Information:
    • Includes the Executive Summary, acknowledgments, and table of contents.
  • Introductory Information:
    • Outlines the background, objectives, and the scope of the research.
  • Main Research Body:
    • Includes detailed sections such as the literature review, methodology, and data analysis.
  • Supplementary Information:
    • Contains appendices, references, and other supporting documents.
• Purpose of the Executive Summary
  • The Executive Summary aims to provide readers with a quick understanding of the research without needing to read the entire report.
  • It is essential for decision-makers who may not have time to review the entire document.

The correct answer is – A, C and E Only

Key Points

• The normal curve approaches the x-axis asymptotically
  • The curve never touches the x-axis but gets infinitely close as it moves away from the mean.
  • This property ensures that the normal distribution covers the entire range of possible values.
• Irrespective of the shape, the total area under the curve is always 1
  • The total probability represented by the normal distribution equals 1, making it a valid probability distribution.
  • This property ensures that the sum of probabilities over all possible values is conserved.
• The normal distribution is unimodal
  • The curve has a single peak, which corresponds to the mean of the distribution.
  • Unimodal distributions are key for modeling symmetric data around a central value.

Additional Information

• Points of inflection
  • For a normal distribution, the points of inflection occur at μ ± σ, not μ ± 2σ.
  • These are the points where the curve changes from concave to convex or vice versa.
• Area under the curve
  • About 68.27% of the area lies within μ ± σ, not μ ± 2σ.
  • Within μ ± 2σ, approximately 95.45% of the total area is covered.

The correct answer is – (A) is correct but (R) is not correct

Key Points

• Filter Questions
  • Filter questions are designed to assess the familiarity or prior experience of respondents with the topic of research.
  • They help to filter out responses from individuals who are not relevant to a specific part of the questionnaire.
  • For example, a filter question might ask, “Have you ever used this product?” before asking detailed questions about the product.
• Assertion (A) is Correct
  • The statement that filter questions measure the familiarity of respondents is accurate and aligns with the purpose of such questions in research.
• Reason (R) is Incorrect
  • Leading questions do not capture multiple pieces of information at once. Instead, they are questions that are phrased in a way to suggest a particular answer or influence the respondent’s response.
  • For example, a leading question might be, “Don’t you think this product is great?”
  • Questions that capture multiple pieces of information are known as double-barreled questions, not leading questions.

Additional Information

• Types of Questions in Questionnaires
  • Filter Questions
    • Used to determine if a respondent qualifies to answer subsequent questions.
    • Ensures that the questionnaire is relevant to the respondent.
  • Leading Questions
    • Designed in a way that subtly guides respondents to a particular answer.
    • Example: “Don’t you agree that this policy is beneficial?”
  • Double-Barreled Questions
    • Attempt to ask about two topics in a single question, leading to confusion or incomplete responses.
    • Example: “Do you like the product’s design and price?”
• Importance of Question Design
  • Well-designed questions ensure accurate data collection and prevent bias.
  • Avoiding leading and double-barreled questions enhances the validity of research findings.

The correct answer is – D, C, B, A

Key Points

• Sampling design process involves a series of logical steps to ensure accurate representation of the target population.
• Step 1: Target population definitions
  • Clearly define the group of individuals or entities to be studied.
  • This step ensures the research focuses on the relevant population.
• Step 2: Selection of sampling frame
  • Choose a list or database that represents the target population.
  • The sampling frame must be precise to avoid bias or errors.
• Step 3: Select a sampling technique
  • Decide on the appropriate method, such as random sampling, stratified sampling, etc.
  • The technique depends on the nature of the research and objectives.
• Step 4: Determine the sample size
  • Calculate the number of participants required for reliable results.
  • The sample size must balance precision with practical considerations like cost and time.

Additional Information

• Importance of sampling design
  • Ensures representativeness of the sample to generalize findings to the population.
  • Minimizes bias and increases the reliability of research results.
• Sampling techniques
  • Probability sampling: Every individual has an equal chance of selection (e.g., random sampling).
  • Non-probability sampling: Selection based on subjective judgment or convenience (e.g., convenience sampling).
• Factors affecting sample size
  • Desired level of accuracy and confidence.
  • Variability in the population.
  • Available resources such as time and budget.