Unit 5 Business Statistics and Research Methods PYQ’s 27th June 2025

The correct answer is – 19.2

Key Points

• Harmonic Mean (HM) is calculated using the formula:
  • HM = (2 × a × b) / (a + b), where a and b are the two numbers.
• We are given:
  • Arithmetic Mean (AM) = 30, which means: (a + b)/2 = 30.
  • Geometric Mean (GM) = 24, which means: √(a × b) = 24.
• Solving these equations:
  • a + b = 60 (from AM).
  • a × b = 576 (from GM).
• Substitute the values of a + b and a × b into the HM formula:
  • HM = (2 × 576) / 60.
  • HM = 1152 / 60 = 19.2.

Additional Information

• Arithmetic Mean (AM):
  • It is the sum of two numbers divided by 2: AM = (a + b)/2.
  • AM is always greater than or equal to the Geometric Mean (GM) for non-negative numbers.
• Geometric Mean (GM):
  • It is the square root of the product of two numbers: GM = √(a × b).
  • GM is always less than or equal to the Arithmetic Mean (AM).
• Harmonic Mean (HM):
  • It is calculated as: HM = (2 × a × b) / (a + b).
  • For non-negative numbers, HM ≤ GM ≤ AM.
• Relationship between AM, GM, and HM:
  • The inequality AM ≥ GM ≥ HM holds for any two positive numbers.
  • This relationship is known as the AM-GM-HM inequality.

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

Key Points

• Matching LIST-I with LIST-II
  • Prefatory Information (A) corresponds to Executive Summary (I) because it provides a concise overview of the report before diving into details.
  • Introductory Information (B) matches with Problem Definition (III) as it sets the context and outlines the key issue being addressed.
  • Main Research Body (C) aligns with Research Design (IV), which outlines the methodology and findings of the research.
  • Supplementary Information (D) is linked to Glossary (II), which provides additional details such as definitions or explanations for key terms.
• The correct answer option is 4 based on the logical alignment of the components in LIST-I with their respective contents in LIST-II.

Additional Information

• Prefatory Information
  • Includes elements like the title page, acknowledgments, and executive summary.
  • Designed to offer a high-level overview for readers who may not read the entire report.
• Introductory Information
  • Defines the problem statement and the objectives of the research.
  • Provides clarity on the scope and purpose of the report.
• Main Research Body
  • Focuses on the methodology, including research design, data collection, and analysis.
  • Forms the core of the report, presenting detailed findings and interpretations.
• Supplementary Information
  • Includes materials like appendices, glossaries, and references.
  • Supports the main content by providing additional context or clarifications.

The correct answer is – B, A, D, C, E

Key Points

• Research Design Process
  • The sequence of steps in the research design process helps ensure that the study is systematic, logical, and reliable.
  • The correct sequence is:

• B: Research approach selection – It is the first step where the researcher decides the overall approach (qualitative, quantitative, or mixed methods).
• A: Measurement technique selection – The researcher identifies and selects appropriate techniques or tools for measuring variables.
• D: Sample design selection – The process of selecting the sample group that represents the population under study.
• C: Data collection technique selection – The researcher determines how data will be collected (e.g., surveys, interviews, experiments).
• E: Data analysis method selection – Finally, the researcher decides on the methods and tools for analyzing the collected data.
• Following this sequence ensures that the research is conducted in a structured and logical manner.

Additional Information

• Details of Each Step
  • Research approach selection (B)
    • This step determines the overall strategy for the research.
    • Approaches include qualitative (exploratory, narrative-based), quantitative (statistical, numerical), or mixed methods.
  • Measurement technique selection (A)
    • Involves identifying scales, tools, or instruments to measure the variables.
    • Examples include questionnaires, rating scales, or standardized tests.
  • Sample design selection (D)
    • Defines the sampling method (e.g., random, stratified, cluster sampling) and size.
    • Ensures that the sample is representative of the target population.
  • Data collection technique selection (C)
    • Includes methods like surveys, interviews, focus groups, or observational techniques.
    • The choice depends on the research question and approach.
  • Data analysis method selection (E)
    • Involves selecting methods such as statistical analysis, thematic analysis, or content analysis.
    • Software tools like SPSS, Excel, or NVivo may be used to assist in analysis.
• Importance of Following the Correct Sequence
  • Ensures logical consistency and minimizes errors in the research process.
  • Improves the validity and reliability of research outcomes.

The correct answer is – B and C Only

Key Points

• Standard deviation is independent of change of origin but not of scale
  • When a constant is added or subtracted from all data points (change of origin), the standard deviation remains unchanged.
  • However, when all data points are multiplied or divided by a constant (change of scale), the standard deviation is affected proportionally.
• The standard deviation of the first n natural numbers is sqrt((n² – 1)/12)
  • The formula for the standard deviation of the first n natural numbers is derived using the properties of summation and variance.
  • It is mathematically proven that the standard deviation is sqrt((n² – 1)/12).
• Statement A is incorrect
  • Statement A says that “Standard deviation is independent of both origin and scale,” which is false, as standard deviation is affected by changes in scale.

Additional Information

• Standard deviation and its properties:
  • Variance is the square of the standard deviation, which measures the spread of data points around the mean.
  • Standard deviation is always non-negative, as it represents the distance from the mean.
  • It is a robust measure for comparing variability in datasets with similar scales.
• Mean deviation and its relationship with standard deviation:
  • For moderately skewed distributions, the standard deviation is approximately 1.25 times the mean deviation.
  • This approximation depends on the shape of the distribution and may not hold for highly skewed datasets.
• Formulas for standard deviation:
  • For ungrouped data: SD = sqrt(Σ(xi – x̄)² / n)
  • For grouped data: SD = sqrt(Σf(xi – x̄)² / Σf)
  • For the first n natural numbers: SD = sqrt((n² – 1)/12)

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

Key Points

• Karl Pearson’s coefficient of skewness
  • Uses the formula: (Mean – Mode) / Standard Deviation.
  • It measures the degree of skewness based on the relationship between mean and mode.
  • Applicable for datasets where mode is well-defined.
• Bowley’s coefficient of skewness
  • Uses the formula: (Q3 + Q1 – 2 Median) / (Q3 – Q1).
  • Relies on quartiles and median, making it robust for skewness in distributions.
  • Useful for measuring asymmetry in datasets with extreme values.
• Kelly’s coefficient of skewness
  • Uses the formula: (P90 + P10 – 2 Median) / (P90 – P10).
  • Based on deciles (P90, P10) and median, providing a measure of skewness for distributions.
• Moment coefficient of skewness
  • Uses the formula: (μ₃)² / (μ₁)³, where μ₃ and μ₁ are central moments.
  • Measures skewness using moments, offering a mathematical approach.
  • Applicable for continuous distributions.

Additional Information

• Skewness
  • Skewness quantifies the asymmetry of a distribution.
  • A positive skew indicates a distribution with a longer tail on the right, while a negative skew indicates a longer tail on the left.
  • Common measures include relative measures (e.g., Pearson, Bowley) and moment measures.
• Central moments
  • Used to calculate skewness and kurtosis.
  • Third central moment (μ₃) captures asymmetry, while first moment (μ₁) represents mean.
• Percentile-based measures
  • Kelly’s coefficient utilizes specific percentiles like P90 and P10 for robust skewness measurement.
  • Bowley’s coefficient focuses on quartiles, ensuring resilience against outliers.

The correct answer is – D, C, B, A

Key Points

• Defining target population
  • The first step in the sampling design process is identifying the target population, which refers to the group of individuals or units relevant to the study.
  • It ensures the sample is representative of the population being studied.
• Selection of sampling frame
  • Once the target population is defined, a sampling frame is established. This is a list or database of elements from which the sample will be drawn.
  • It is critical that the sampling frame accurately represents the target population.
• Selecting a sampling technique
  • In this step, the researcher chooses an appropriate sampling technique such as random sampling, stratified sampling, or cluster sampling, depending on the study objectives and constraints.
  • The choice of technique influences the validity and reliability of the results.
• Determine the sample size
  • Finally, the sample size is determined based on statistical considerations like margin of error, confidence level, and variability in the population.
  • A well-calculated sample size ensures that the study findings are statistically significant and generalizable.

Additional Information

• Types of Sampling Techniques
  • Probability Sampling
    • Includes methods like simple random sampling, stratified sampling, and cluster sampling.
    • Ensures that every element in the population has an equal chance of being selected.
  • Non-Probability Sampling
    • Includes methods like convenience sampling, quota sampling, and judgmental sampling.
    • Does not guarantee equal representation but may be used in exploratory studies.
• Importance of Sampling Design
  • A robust sampling design minimizes bias and maximizes the reliability and validity of study results.
  • It is crucial in ensuring that study findings are representative of the population and generalizable to broader contexts.

The correct answer is – 2

Key Points

• Poisson Distribution
  • A Poisson random variable X is characterized by its mean (λ) and variance, both of which are equal to λ.
  • Given that P(X = 1) = 4P(X = 2), we can use the formula for probabilities in Poisson distribution:
    • P(X = k) = (λ^k * e^-λ) / k!
    • Substituting k = 1 and k = 2 into this equation, we get:
      • P(X = 1) = λ * e^-λ
      • P(X = 2) = (λ² * e^-λ) / 2
    • Equating the given condition P(X = 1) = 4P(X = 2), we solve for λ:
      • λ * e^-λ = 4 * (λ² * e^-λ) / 2
      • λ = 2
  • Since the variance of a Poisson distribution is equal to its mean, the variance of X is 2.

Additional Information

• Properties of Poisson Distribution
  • The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space, provided the events occur independently.
  • The mean and variance of the Poisson distribution are both equal to λ.
  • It is defined for non-negative integers, i.e., X ≥ 0.
• Formula for Poisson Distribution
  • P(X = k) = (λ^k * e^-λ) / k!
  • Where:
    • λ: mean number of occurrences.
    • k: number of occurrences.
    • e: Euler’s constant (~2.718).
• Applications of Poisson Distribution
  • Modeling rare events such as the number of calls received by a call center in an hour.
  • Estimating the number of accidents occurring at a traffic junction in a day.
  • Predicting the number of typing errors in a book manuscript.

The correct answer is – Measurement Questions

Key Points

• Measurement Questions
  • These are the actual questions asked to respondents during the process of data collection.
  • They are designed to collect specific data or responses that align with the research objectives.
  • Examples include survey questions, interview questions, or questionnaire items.
  • The focus is on obtaining quantifiable or clearly defined responses from participants.

Additional Information

• Management Questions
  • These address the strategic concerns of managers or decision-makers.
  • They often focus on organizational goals and how the research findings will help achieve them.
  • For example: “What factors influence customer satisfaction in our product line?”
• Research Questions
  • These are the central questions a researcher seeks to answer through their study.
  • They frame the objectives of the research and guide the overall investigation.
  • For example: “What is the relationship between employee engagement and productivity?”
• Investigative Questions
  • These are detailed questions that help break down the research problem into smaller, manageable components.
  • They often guide the development of measurement questions.
  • For example: “What are the key drivers of customer loyalty?”

The correct answer is – A, D and E Only

Key Points

• Property A: The coefficient of correlation and the two regression coefficients have the same signs
  • The coefficient of correlation (denoted as r) and the regression coefficients (b_yx and b_xy) are directly related.
  • If r is positive, both regression coefficients are positive; if r is negative, both regression coefficients are negative.
  • This ensures consistency in the direction of the relationship between variables.
• Property D: If one of the regression coefficients is greater than unity, the other must be less than unity
  • The regression coefficients are calculated based on the standard deviations of the variables.
  • If b_yx > 1, then b_xy < 1, ensuring the product of the two regression coefficients equals the square of the correlation coefficient (r²).
  • This property maintains the mathematical relationship between the coefficients and the correlation.
• Property E: The regression coefficients are independent of the change of origin but not of scale
  • Regression coefficients remain unaffected by shifting the origin (e.g., adding or subtracting a constant).
  • However, they are affected by changes in scale (e.g., multiplication or division by a constant).
  • This distinction is important for understanding how data transformations impact regression analysis.

Additional Information

• Regression Coefficient Properties
  • The regression coefficients b_yx and b_xy are measures of the relationship between two variables, where one is dependent and the other is independent.
  • They help predict the value of one variable based on the value of another.
• Correlation Coefficient and Regression Coefficients
  • The correlation coefficient (r) represents the strength and direction of the linear relationship between two variables.
  • The product of the two regression coefficients is equal to r², which represents the proportion of variance explained.
• Impact of Change in Origin and Scale
  • A change in origin (shifting the data) does not affect the regression coefficients.
  • A change in scale (rescaling the data) alters the regression coefficients proportionally.

The correct answer is – Criterion validity

Key Points

• Criterion validity
  • It refers to the extent to which the results of a selection procedure, such as a test, are statistically correlated with job performance or other relevant outcomes.
  • It ensures that the selection tool is effective in predicting an individual’s performance in a specific role or activity.
  • For example, a test designed to predict sales performance will have high criterion validity if scores on the test strongly correlate with actual sales performance.
  • It is widely used in employment testing, academic assessments, and other contexts where predictive accuracy is essential.

Additional Information

• Types of Validity
  • Test validity
    • It is a broad term that encompasses all types of validity, including criterion validity, content validity, and construct validity.
    • It ensures that a test measures what it is intended to measure.
  • Content validity
    • Refers to how well a test represents the entire domain of the subject being measured.
    • For example, a math test with questions from all relevant topics has high content validity.
  • Construct validity
    • Measures how well a test or tool assesses the theoretical construct it is designed to measure.
    • For instance, a test for intelligence should align with established theories of intelligence.
• Importance of Criterion Validity
  • Helps organizations make data-driven decisions in hiring, training, and performance evaluation.
  • Improves the reliability of assessments and reduces the risk of biased decision-making.
  • Widely applied in areas such as psychological testing, academic assessments, and job recruitment processes.

The correct answer is – t test

Key Points

• Paired t-test
  • The paired t-test is a statistical method used to compare the means of two related groups.
  • It is specifically applied when the samples are dependent, such as measurements taken before and after an intervention on the same subjects.
  • This test assumes that the data is normally distributed and the paired differences are continuous.
  • It tests the null hypothesis that the mean difference between paired observations is zero.
• Statistical Application
  • Used in scenarios like analyzing pre-test and post-test scores in educational research or measuring the effect of a treatment in clinical trials.
  • Key formula: t = (Mean difference) / (Standard Error of Difference).

Additional Information

• Alternative Tests
  • Z test
    • Used for large samples (>30) and assumes the population standard deviation is known.
    • Not suitable for paired samples, as it applies to independent samples and population proportions.
  • Chi-Square test
    • A non-parametric test used to analyze categorical data.
    • Cannot be used for comparing means or paired samples.
  • Mann-Whitney test
    • A non-parametric test used for independent samples to compare medians.
    • It is not appropriate for paired samples.
• Assumptions of Paired t-test
  • Data is continuous and measured on an interval or ratio scale.
  • Paired differences are normally distributed.
  • Observations within each pair are dependent, but pairs are independent.