TGPSC MPSO / ASO · Paper II · Topic 5
TGPSC MPSO / ASO Correlation & Regression questions with answers
19 questions from TGPSC ASO Paper II Mock 1 with answers and TGPSC ASO Paper II Challenge 1 with answers. Try each one, then tap Show answer.
5.1Karl Pearson's & Spearman's correlation, probable errorNotesPractise this chapter · Set 1 free
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Find Spearman's rank correlation coefficient, corrected for ties (to three decimals):
X: 53, 40, 40, 58, 60, 50, 40
Y: 63, 41, 50, 66, 64, 59, 64
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Answer A. 0.625
Ranks X: 3, 6, 6, 2, 1, 4, 6; Y: 4, 7, 6, 1, 2.5, 5, 2.5. Σd² = 18.5. Ties: 40 three times in X, 3(3² − 1)/12 = 2; 64 twice in Y, 2(2² − 1)/12 = 0.5. R = 1 − 6 × 21/(7 × 48) = 0.625.
Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)
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The coefficient of correlation between X and Y is 0.6. If U = (X − 5)/2 and V = (Y − 10)/(−3), what is the coefficient of correlation between U and V?
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Answer C. −0.6
r is unaffected by change of origin and by change of scale in magnitude, but its sign reverses when the two scale factors have opposite signs. Here h = 2 and k = −3, so r(U, V) = −0.6.
Source: S.P. Gupta, Statistical Methods, Correlation Analysis
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If byx = 1.6 and bxy = 0.4, what is the ratio σy/σx?
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Answer D. 2
byx/bxy = (r σy/σx)/(r σx/σy) = σy²/σx² = 4, so σy/σx = 2. (Also r = √(1.6 × 0.4) = 0.8.)
Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)
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If Cov(X, Y) = 6, Var(X) = 9 and the coefficient of correlation is 0.5, what is Var(Y)?
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Answer D. 16
σx = 3. r = Cov/(σx σy) ⇒ σy = Cov/(r σx) = 6/(0.5 × 3) = 4, so Var(Y) = 16.
Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)
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Assertion (A): If X takes values symmetric about zero and Y = X², the coefficient of correlation between X and Y is zero.
Reason (R): Karl Pearson's coefficient of correlation measures only the linear relationship between two variables.
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Answer A. Both (A) and (R) are true and (R) is the correct explanation of (A)
Y depends fully on X but not linearly; the positive and negative X values cancel in Cov(X, Y). Since r captures only linear association, r = 0, so R explains A.
Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)
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Assertion (A): The coefficient of correlation between the monthly incomes of 50 families measured in rupees and their expenditures measured in thousands of rupees is the same as when both are measured in rupees.
Reason (R): Karl Pearson's coefficient of correlation is independent of change of scale.
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Answer A. Both (A) and (R) are true and (R) is the correct explanation of (A)
Dividing expenditure by 1,000 is a change of scale by a positive factor, which leaves r unchanged. So A is true because of R.
Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)
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Consider the following statements about Spearman's rank correlation with tied ranks:
1. Items with equal values are each given the average of the ranks they would otherwise occupy.
2. For every group of m tied items, m(m² − 1)/12 is added to Σd².
Which of the statements given above is/are correct?
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Answer C. Both 1 and 2
Tied values share the mean of their ranks (e.g. two items tying for 3rd and 4th each get 3.5), and each tie group of size m adds m(m² − 1)/12 to Σd².
Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)
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Which of the following is NOT a property of Karl Pearson's coefficient of correlation?
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Answer C. It changes when the scale of measurement of X is changed by a positive factor
r is bounded by ±1, symmetric in X and Y, equal to ±√(byx bxy), and independent of change of origin and scale.
Source: Medha AI, based on BRAOU/APSCHE degree textbooks (ASO Paper II)
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Consider the following statements about Karl Pearson's coefficient of correlation r:
1. r is unchanged if a constant is subtracted from every value of X.
2. r is unchanged if every value of Y is divided by a positive constant.
Which of the statements given above is/are correct?
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Answer C. Both 1 and 2
r is independent of change of origin (statement 1) and of change of scale by a positive factor (statement 2).
Source: S.P. Gupta, Statistical Methods, Correlation Analysis
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Consider the following statements:
I. If U = 3 − 2X and V = 5 − 4Y, then r(U, V) = r(X, Y).
II. If every value of X is multiplied by −3 and Y is left unchanged, the correlation coefficient changes sign but not size.
Which of the statements given above is/are correct?
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Answer C. Both I and II are correct
Both statements are correct. The correlation coefficient is unaffected by a change of origin, and a change of scale affects only its sign: r(U, V) = (bd/|bd|) r(X, Y) for U = a + bX, V = c + dY. In statement I, b = −2 and d = −4 have the same sign, so r(U, V) = r(X, Y). In statement II, only X is multiplied by a negative number, so r becomes −r.
Source: Standard method (BRAOU Sem5 Correlation, Regression & Sampling, correlation)
5.1Regression lines & regression coefficientsNotesPractise this chapter · Set 1 free
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The regression line of Y on X is y = 0.5x + 4 and the regression line of X on Y is x = 0.5y + 2.5. Which of the following is correct?
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Answer A. x̄ = 6, ȳ = 7, r = 0.5
Substituting y: x = 0.5(0.5x + 4) + 2.5, so 0.75x = 4.5, x̄ = 6 and ȳ = 7. r = √(0.5 × 0.5) = 0.5, positive like the coefficients.
Source: Standard result (BRAOU Sem5 Correlation, Regression & Sampling, Unit 4 Simple Linear Regression)
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The regression coefficient of weight (Y, in kg) on age (X, in years) is 0.4. If weight is recorded in grams instead, the new regression coefficient is
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Answer D. 400
Every Y is multiplied by 1000, so byx (units of Y per unit of X) is also multiplied by 1000: 0.4 × 1000 = 400 g per year.
Source: Standard result (BRAOU Sem5 Correlation, Regression & Sampling, Unit 4 Simple Linear Regression)
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Assertion (A): The correlation coefficient is independent of change of scale.
Reason (R): Regression coefficients are independent of change of scale.
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Answer C. A is true, but R is false
r is unchanged by a positive change of scale, but regression coefficients change with scale (they are independent of origin only).
Source: Standard definition (BRAOU Sem5 Correlation, Regression & Sampling, Unit 4 Simple Linear Regression)
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If r = 0 for a data set, consider the following statements:
1. The two regression lines are perpendicular.
2. byx = bxy = 0.
3. The estimate of Y is ȳ for every value of X.
4. The standard error of estimate of Y on X is 0.
Which of the statements given above are correct?
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Answer D. 1, 2 and 3 only
With r = 0 both coefficients are 0, the lines are y = ȳ and x = x̄ (perpendicular), and ŷ = ȳ. Then Syx = σy, not 0.
Source: Standard definition (BRAOU Sem5 Correlation, Regression & Sampling, Unit 4 Simple Linear Regression)
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Consider the following statements:
1. 1 − r² is called the coefficient of determination.
2. The coefficient of determination always lies between 0 and 1.
Which of the statements given above is/are correct?
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Answer B. 2 only
r² is the coefficient of determination and lies between 0 and 1. 1 − r² is the coefficient of non-determination.
Source: Standard definition (BRAOU Sem5 Correlation, Regression & Sampling, Unit 4 Simple Linear Regression)
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Which of the following pairs is NOT correctly matched?
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Answer D. Charles Spearman : regression coefficient
Spearman is known for the rank correlation coefficient, not for regression coefficients. Galton coined 'regression', Legendre gave the least squares principle and Karl Pearson the product moment coefficient.
Source: Standard definition (BRAOU Sem5 Correlation, Regression & Sampling, Unit 4 Simple Linear Regression)
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Consider the following statements about the least squares regression line:
1. It passes through every observed point.
2. It is called the line of best fit.
Which of the statements given above is/are correct?
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Answer B. 2 only
The least squares line makes the sum of squared deviations a minimum, so it is the line of best fit. It passes through all points only when r = ±1.
Source: Standard definition (BRAOU Sem5 Correlation, Regression & Sampling, Unit 4 Simple Linear Regression)
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The two regression lines of a bivariate distribution are 3x + 2y = 26 and 6x + y = 31. What are the means of X and Y and the correlation coefficient?
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Answer A. x̄ = 4, ȳ = 7, r = −0.5
x̄ = 4, ȳ = 7 and r = −0.5. Both lines pass through (x̄, ȳ), and solving them together gives x = 4, y = 7. Taking the first as Y on X gives byx = −3/2, and the second as X on Y gives bxy = −1/6; their product 1/4 is at most 1, so this choice is valid (the other choice gives a product of 4, which is impossible). r = −√(1/4) = −0.5, negative because both coefficients are negative.
Source: Standard method (BRAOU Sem5 Correlation, Regression & Sampling, regression)
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For 5 pairs of observations, Σx = 20, Σy = 40, Σxy = 180 and Σx² = 100. Using the regression line of Y on X, estimate Y when X = 10.
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Answer D. 14
The estimate is 14. byx = (nΣxy − ΣxΣy)/(nΣx² − (Σx)²) = (900 − 800)/(500 − 400) = 1. With x̄ = 4 and ȳ = 8, the line is y − 8 = 1 × (x − 4), that is y = x + 4. At x = 10, y = 14.
Source: Standard method (BRAOU Sem5 Correlation, Regression & Sampling, regression)
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