Unit 5 Business Statistics and Research Methods PYQ’s 9 Sep 2026 Shift 2

The correct answer is – Asking filtering questions

Key Points

• Asking filtering questions
  • Filtering questions are used to determine whether a respondent qualifies for answering subsequent questions.
  • They help in directing respondents to relevant sections of a questionnaire and improve the accuracy of collected data.
  • Example: “Have you used online banking services in the last six months?” followed by related questions only for users.
  • Unlike poorly structured questions, filtering questions are considered a useful questionnaire design technique.

Additional Information

• Common questionnaire design errors
  • Loaded questions
    • Contain biased wording that influences the respondent’s answer.
    • Example: “Why do you support this ineffective policy?”
  • Double-barrelled questions
    • Ask two different issues in a single question, making responses unclear.
    • Example: “Are you satisfied with the price and quality of this product?”
  • Leading questions
    • Suggest a particular answer or influence the respondent’s opinion.
    • Example: “Don’t you agree that this service is excellent?”

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

Key Points

• Kurtosis (A-I): It is a statistical measure used to describe the degree to which scores cluster in the tails or the peak of a frequency distribution. It essentially indicates how ‘flat’ or ‘peaked’ the curve is relative to a normal distribution.
• Dispersion (B-IV): Also known as variability, scatter, or spread, it denotes the extent to which a numerical data set is likely to vary about an average value. Common measures include Standard Deviation and Variance.
• Skewness (C-III): This refers to a distortion or asymmetry that deviates from the symmetrical bell curve, or normal distribution, in a set of data. If the curve is shifted to the left or right, it is said to be skewed.
• Leptokurtic (D-II): This is a specific type of Kurtosis where the distribution has a higher peak and fatter tails than a normal distribution (Mesokurtic). It has a positive excess kurtosis value.

Additional Information

• Types of Kurtosis:
  • Mesokurtic: Distributions with kurtosis medium-peaked (Normal distribution).
  • Platykurtic: Distributions that are flatter and more spread out than a normal distribution.
• Types of Skewness:
  • Positive Skew: The tail on the right side is longer or fatter than the left side.
  • Negative Skew: The tail on the left side is longer or fatter than the right side.

The correct answer is ‘Simplicity’

Key Points

• In statistical inference, an estimator is a rule or formula used to estimate the value of an unknown population parameter based on sample data.
• While an estimator can be simple or complex in its mathematical form, Simplicity is not considered a formal technical property or a criterion for a ‘good’ estimator in statistical theory.
• The quality of an estimator is judged by its mathematical properties such as Unbiasedness, Consistency, Efficiency, and Sufficiency, rather than how easy it is to calculate or understand.

Additional Information

• Consistency: An estimator is said to be consistent if, as the sample size increases, the estimator converges in probability to the true value of the parameter. Mathematically, as n→∞, the estimate becomes increasingly accurate.
• Efficiency: This refers to the variance of the estimator’s sampling distribution. Among all unbiased estimators, the one with the minimum variance is called the most efficient estimator. It utilizes the sample information most effectively.
• Sufficiency: An estimator is sufficient if it utilizes all the information about the population parameter that is contained in the sample. No other estimator calculated from the same sample can provide any additional information about the parameter.
• Unbiasedness: An estimator is unbiased if its expected value is equal to the true population parameter. While not listed in the options, it is the fourth primary property of a good estimator.

The correct answer is ‘Cluster Sampling’

Key Points

• Cluster Sampling:
  • In this method, the entire population is divided into non-overlapping, mutually exclusive groups called clusters.
  • These clusters are often based on geographical areas (e.g., cities, districts, or blocks) or other natural groupings.
  • Unlike stratified sampling where individuals are selected from every group, in cluster sampling, the researcher randomly selects entire clusters. All individuals within the selected clusters are then included in the sample.
  • It is particularly useful when the population is widely dispersed and it is impractical or expensive to create a complete list of all individual elements.

Additional Information

• Simple Random Sampling:
  • Every member of the population has an equal and independent chance of being selected. It typically requires a complete list of the population (sampling frame).
• Systematic Sampling:
  • Members are selected at regular intervals from an ordered list. For example, selecting every 10th person from a directory after a random starting point.
• Convenience Sampling:
  • A non-probability sampling method where samples are chosen based on their accessibility and proximity to the researcher. It is the least rigorous method and often prone to bias.
• Stratified Sampling:
  • The population is divided into subgroups (strata) based on shared characteristics (e.g., age, gender), and a random sample is taken from each subgroup to ensure representation.

The Correct answer is: Only B and E

Key Points

• Statement A: It is a continuous probability distribution:
  • This statement is correct.
  • The Normal Distribution (and its standardized form) deals with continuous variables that can take any value within a range, represented by a smooth curve rather than discrete points.
• Statement B: It has mean value of 1:
  • This statement is incorrect.
  • In a Standardized Normal Distribution (Z-distribution), the mean (μ) is always 0, and the standard deviation (σ) is 1.
• Statement C: The mean of the normal distribution lies at the centre of the normal curve:
  • This statement is correct.
  • The normal distribution is perfectly symmetrical. The highest point of the curve represents the mean, which divides the area under the curve into two equal halves (50% on each side).
• Statement D: The value of mean, median, and mode are alike:
  • This statement is correct.
  • Due to its perfect symmetry and unimodal nature, the Mean, Median, and Mode all coincide at the exact center of the distribution.
• Statement E: It is non-symmetrical probability distribution:
  • This statement is incorrect.
  • The normal distribution is famously symmetrical (bell-shaped). A non-symmetrical distribution is referred to as a skewed distribution.

Additional Information

• Standard Normal Distribution (Z-score):
  • Any normal distribution can be converted into a standard normal distribution using the formula: z=x-μσ.
  • The total area under the curve is exactly 1 (or 100%).
  • The curve is asymptotic to the horizontal axis, meaning it approaches but never actually touches the x-axis.
• Empirical Rule (68-95-99.7 Rule):
  • Approximately 68.2% of the data falls within ±1 standard deviation of the mean.
  • Approximately 95.4% falls within ±2 standard deviations.
  • Approximately 99.7% falls within ±3 standard deviations.

The correct answer is ‘Test-statistic’

Key Points

• Test-statistic:
  • A test-statistic is a numerical value calculated from sample data during a hypothesis test. It is used to determine whether to reject or fail to reject the null hypothesis.
  • Common examples include Z-scores, t-scores, F-statistics, and Chi-square values.
  • Crucially, the test-statistic is a result of the data analysis process after the sample has already been collected; it is not a factor used to determine the size of the sample beforehand.

Additional Information

• Confidence Level:
  • This refers to the degree of certainty that the sample results represent the true population. Common levels are 95% or 99%. A higher confidence level requires a larger sample size to reduce the risk of error.
• Magnitude of Acceptable Error (Margin of Error):
  • This is the precision required for the estimate. It represents the maximum allowable difference between the sample statistic and the actual population parameter. To achieve a smaller margin of error, a larger sample size is necessary.
• Heterogeneity of the Population:
  • This refers to the variability or diversity within the population. If a population is highly heterogeneous (diverse), a larger sample is needed to capture all the different characteristics accurately. Conversely, a homogeneous population requires a smaller sample.
• Sample Size Formula Context:
  • In standard formulas (like Cochran’s formula), sample size (n) is determined by the Z-score (derived from confidence level), the standard deviation (representing heterogeneity), and the margin of error. The test-statistic itself never appears in these predictive formulas.

The correct answer is ‘Two-independent sample t-test’

Key Points

• Mann-Whitney U Test:
  • The Mann-Whitney U test (also known as the Wilcoxon rank-sum test) is a non-parametric statistical test used to determine whether there is a difference between two independent groups.
  • It is specifically designed for ordinal data or continuous data that do not follow a normal distribution.
  • It functions by ranking all the observations from both groups together and then comparing the sum of the ranks for each group.
• Analogue to Two-independent sample t-test:
  • The Two-independent sample t-test is the parametric counterpart to the Mann-Whitney U test.
  • Both tests aim to compare two independent groups to see if they come from the same population.
  • While the t-test compares the means of two groups (assuming normality and interval/ratio scale), the Mann-Whitney U test compares the distributions (medians) of the groups.

Additional Information

• One-sample t-test:
  • Used to determine whether a sample mean is statistically different from a known or hypothesized population mean. Its non-parametric analogue is the One-sample Wilcoxon Signed-Rank Test.
• Paired t-test:
  • Used to compare means from the same group at different times (e.g., before and after treatment). Its non-parametric analogue is the Wilcoxon Signed-Rank Test for matched pairs.
• Chi-square test:
  • A non-parametric test used to determine if there is a significant association between two categorical variables or to check the goodness of fit between observed and expected frequencies.
• Parametric vs. Non-parametric:
  • Parametric tests assume the data follows a specific distribution (usually normal) and involve population parameters like mean and standard deviation.
  • Non-parametric tests are ‘distribution-free’ and are used when data is skewed, has outliers, or is measured on an ordinal scale.

The correct answer is ‘Sample is not truly representing the population’

Key Points

• Sampling Error:
  • Sampling error occurs when the characteristics of a sample do not perfectly match the characteristics of the entire population.
  • It arises specifically because we are observing a subset (sample) rather than the whole population.
  • The statement ‘Sample is not truly representing the population’ is the fundamental definition of a sampling error. This error is inherent in any sampling process and can be reduced by increasing the sample size.
• Non-Sampling Error:
  • Non-sampling errors are errors that occur during data collection or processing, regardless of whether a sample or a complete census is taken.
  • These errors are more serious because they cannot be minimized by simply increasing the sample size.
  • Common types include measurement errors, data entry errors, non-response bias, and investigator bias.

Additional Information

• Non-response bias:
  • This is a non-sampling error that occurs when the individuals who respond to a survey differ significantly from those who do not respond.
• Inefficiency of the field worker:
  • This is a non-sampling error related to the human element of data collection. It includes mistakes made by interviewers in asking questions or recording answers.
• Errors occurred during data compilation:
  • This is a non-sampling error that happens during the processing stage, such as coding, data entry, or tabulation of the collected information.
• Comparison Table:

The correct answer is ‘Correlation’

Key Points

• Correlation:
  • Correlation is a statistical measure that indicates the extent to which two or more variables fluctuate together.
  • It measures the relationship or association between variables, not the spread or scatter of a single variable’s data points.
  • Since it deals with the interdependence between variables rather than the internal variation within a dataset, it is not classified as a method of variation (dispersion).
• Measures of Variation (Dispersion):
  • Variation or Dispersion refers to the extent to which a set of data is clustered or spread out.
  • Common methods to measure variation include Range, Quartile Deviation, Mean Deviation, Standard Deviation, and graphical methods like the Lorenz Curve.

Additional Information

• Standard Deviation:
  • It is the most widely used measure of dispersion. It calculates the square root of the average of the squared deviations from the mean.
  • It provides a precise measurement of how much the individual values in a dataset deviate from the arithmetic mean.
• Lorenz Curve:
  • The Lorenz Curve is a graphical method used to study the degree of dispersion or inequality in a distribution.
  • It is often used in economics to represent income or wealth inequality within a population.
• Range:
  • The Range is the simplest measure of variation. It is the difference between the maximum value and the minimum value in a dataset.
  • While easy to calculate, it is highly sensitive to outliers and does not account for the distribution of values between the extremes.