CGLE 2024 Tier II Paper II Statistics — Kerala PSC PYQ Practice with Answers

Correct answer (Option D):\nTo find the marginal probability density function (pdf) of X from a given bivariate joint pdf f(x, y), we integrate the joint pdf over the entire support range of the random variable Y.\nFormula: f_X(x) = \int f(x, y) dy\nWhen evaluating the integral within the valid boundaries for the distribution parameters, the integration yields the simplified fraction:\nResult: 3(2x+3) / (4(1+x)⁴)\nOption D matches this resulting algebraic expression perfectly. Therefore, Option D is correct.\n\nWhy others are wrong:\nOption A provides an incorrect denominator scaling factor and lacks the appropriate fourth power. Option B modifies the numerator term to (2x+1) instead of the correct (2x+3). Option C lacks the essential factor of 4 in the denominator layout, leading to an incorrect total area calculation over the distribution support.\n\nStudy tip:\nAlways ensure that a marginal pdf integrates to 1 over its entire support range. Remember that integrating out Y yields the marginal distribution of X, while integrating out X yields the marginal distribution of Y.

Correct answer (Option C):\nFormula: Average Seasonal Ratio = (Sum of trend-adjusted ratios) / (Number of periods)\nGiven values:\nJanuary ratio = 1.1\nFebruary ratio = 0.9\nMarch ratio = 1.0\nStep 1: Calculate the total sum of the ratios.\nSum = 1.1 + 0.9 + 1.0 = 3.0\nStep 2: Divide the sum by the total number of months (3).\nAverage Seasonal Index = 3.0 / 3 = 1.0\nAnswer: 1.0\nOption C is correct.\n\nWhy others are wrong:\nOption A (0.95) is an incorrect arithmetic mean computation. Option B (0.9) merely chooses the lowest individual monthly ratio value instead of taking the quarterly average. Option D (1.1) represents the maximum individual monthly ratio from January instead of computing the average.\n\nStudy tip:\nSeasonal indices are typically computed so that their overall average is equal to 1.0 (or 100%). If the index is above 1.0, it denotes a sales level above the average trend value; if below 1.0, it represents a sales level lower than average.

Correct answer (Option C):\nDuncan's Multiple Range Test (DMRT) is a post-hoc multiple comparison procedure used in analysis of variance (ANOVA) to determine which specific treatment means differ significantly from one another.\nIt calculates a series of specific shortest significant ranges based on the studentized Range Statistic (R).\nThe test compares the sample ranges of treatment means against these calculated critical range thresholds.\nOption C is correct.\n\nWhy others are wrong:\nOption A is an incorrect choice because while the standard error of the mean is used within the standard calculation, the test fundamentally evaluates values based on the studentized range statistic. Option B is wrong because regression coefficients are indicators of line slope, not multiple comparison indicators. Option D is incorrect because the F-statistic identifies global variance significance across groups rather than performing pairwise range tracking.\n\nStudy tip:\nRemember that Duncan's Multiple Range Test does not require a prior significant global F-test, unlike Fisher's LSD test. It protects against Type I errors across sets of ordered averages using multi-stage range values.

Correct answer (Option C):\nIn a three-variable system where all pairwise correlation coefficients are equal to r (i.e., r₁₂ = r₁₃ = r₂₃ = r), the formula for the coefficient of multiple determination is evaluated based on the correlation matrix determinant structural relationships.\nFormula: 1 - R_1.23² = (1 - r₁₂² - r₁₃² - r₂₃² + 2r₁₂r₁₃r₂₃) / (1 - r₂₃²)\nSubstituting r into the equation:\nStep 1: 1 - R_1.23² = (1 - 3r² + 2r³) / (1 - r²)\nStep 2: Factoring the numerator gives (1 - r)²(1 + 2r).\nStep 3: Factoring the denominator gives (1 - r)(1 + r).\nStep 4: Simplifying the fraction yields ((1-r)(1+2r)) / (1+r).\nOption C is correct.\n\nWhy others are wrong:\nOption A uses an incorrect (1-r) divisor term instead of a (1+r) term, reversing the structural cancellation behavior. Option B misses the essential (1-r) factor in the numerator completely. Option D changes the sign within the second factor to (1-2r), which fails to correctly reflect the structural correlation matrix algebra properties.\n\nStudy tip:\nMultiple correlation values always fall inside the interval [0, 1]. When all pairwise correlations are equal to r, the validity requirement for the correlation matrix requires that r must lie within the range (-0.5, 1).

Correct answer (Option B):\nThe free-hand curve method involves drawing a smooth curve through time series data points based on purely visual judgment.\nBecause it relies completely on the investigator's personal interpretation, different analysts can construct completely different trends for identical data sets.\nTherefore, the method is highly subjective and lacks empirical reproducibility.\nOption B is correct.\n\nWhy others are wrong:\nOption A is incorrect because the free-hand curve method is exceptionally fast and requires absolutely no mathematical computations. Option C is wrong because visual curves can follow cyclical behaviors easily based on visual tracking paths. Option D is incorrect because it can be used easily by non-statisticians without requiring formal statistical calculations.\n\nStudy tip:\nWhile simple and highly flexible, the free-hand method should be avoided for formal projections. For objective and mathematically reproducible trend tracking, use regression analysis techniques such as the method of least squares.