Correlation Mcqs​

Question 1.
The coefficient of correlation r between x and y when: Cov. (x, y) = -16.5, Var (x) = 2.89, Var (y) = 100 is:
(a) -0.97
(b) 0.97
(c) 0.89
(d) -0.89
Solution:
(a)
Coefficient of correlation
r = cov(x,y)σx⋅σy = −16.52.89√×100√ = −16.51.7×10
= – 0.97

Question 2.
Take 200 and 150 respectively as the assumed mean for X and Y series of 11 values, then dx = X- 200, dy = Y – 150, Σ dx = 13, Σ dx² = 2667, Σ dy = 42, Σ dy² = 6964, Σ dx dy = 3943. The value of r is:
(a) 0.77
(b) 0.98
(c) 0.92
(d) 0.82
Solution :
(c) Formula
r = n∑dxdy−∑dx⋅∑dyn∑d2−(∑dx)2√n∑dy2−(∑dy)2√
r = 11×3943−13×4211×2667−(13)2√11×6964−(42)2√
= 0.92

Question 3.
If the sum of squares of the rank difference in Mathematics and Physics marks of 10 students is 22, then the coefficient of rank correlation is:  
(a) 0.267
(b) 0.867
(c) 0.92
(d) None
Solution:
(b) Coefficient of Rank Correlation
R = 1 – 6∑d2n(n2−1) = 1 – 6×2210(102−1)
= 1 – 6×210×9 = 1315 = 0.867(approx)

Question 4.
The coefficient of correlation between X and Y is 0.6. U and V are two variables defined as U = x−32, V = y−23, then the coefficient of correlation between U and V is :
(a) 0.6
(b) 0.4
(c) 0.8
(d) 1
Solution:
(a) Since correlation coefficient (Karl Pearson’s) is independent of the change of scale and origin,
So ; r(u, v) = r(x, y) = 0.6 [Because X & Y have same sign]
Tricks : See Quicker BMLRS CORRELATION

Question 5.
For 10 pairs of observations, number of concurrent deviations was found to be 4. What is the value of the coefficient of concurrent deviation? 
(a) 0.2−−−√
(b) 1/3
(c) -1/3
(d) –0.2−−−√
Solution:
(c)
Given C = 4, N = 10,
So, n = N – 1 = 10 – 1 = 9
2C – n = 2.4 – 9 = -1 (-ve)
∵ r_c = ± ±(2c−n)n−−−−−−√
∴ r_c = –−(2×4−9)9−−−−−−−√ ⇒ r_c = (−1)3 = –13
Note : Here (2c – n) = – ve, so – ve sign taken at both the places.

Question 6.
In rank correlation, the association need not be linear:
(a) True
(b) False
(c) Partly True
(d) Partly False
Solution:
(a)
In Spearman’s Rank Correlation; the association need not be Linear because it is based on ranks of observations not on actual observations.

Question 7.
The coefficient of rank correlation of marks obtained by 10 students, in English and Economics was found to be 0.5. It was later discovered that the difference in ranks in the two subjects obtained by one student was wrongly taken as 3 instead of 7. The correct coefficient of rank correlation is : 
(a) 0.32
(b) 0.26
(c) 0.49
(d) 0.93
Solution:
(b) Given Incorrect R = 0.5 ; N = 10 ; Incorrect D = 3, Correct D = 7
Coefficient of rank correlation = R = 6∑D2 N( N2−1)
⇒ 0.5 = 1 – 6∑D210(102−1) ⇒ 6∑D2990 = 0.5
⇒ ΣD² = 82.5 [Incorrect because incorrect R = 0.5 is used here]
Corrected value of ΣD² = 82.5 – 3² + 7² = 122.5
Corrected R = 1 – 6×122.5990 = 1 – 735990 = 1 – 0.74 = 0.26

Question 8.
If the sum of square of differences of rank is 50 and number of items is 8 then what is the value of rank correlation coefficient.
(a) 0.59
(b) 0.40
(c) 0.36
(d) 0.63
Solution:
(b) Given That ΣD² = 50, N = 8
R = 1 – 6∑D2N(N2−1) = 1 – 6×508(82−1) = 1 – 6×508×63 = 0.40

Question 9.

If coefficient of correlation between x and y is 0.46. Find coefficient of correlation between x and y2 
(a) 0.46
(b) 0.92
(c) -0.46
(d) -0.92
Solution:
(a) r_xy = 0.46
RULE: – The value of r does not change with respect to the change of origin and scale.
TRICKS : Sign of r will not change because signs of X & Y are same.

Question 10.
The coefficient of correlation is significant if: 
(a) r > 5 P . E
(b) r < 6 P . E
(c) r ≥ 6 × P.E.,
(d) r = 6 P . E
Solution:
(c)
If r ≥ 6 × P.E., then coefficient of correlation is significant and the correlation exists.

Question 11.
If Ranks of two characteristics by two judges are in reverse order then find the value of Spearman rank correlation co-efficient. 
(a) -1
(b) 0
(c) 1
(d) 0.75
Solution:
(a)
Clearly ; In this case, it is perfectly negatively correlated.
So, r = -1

Question 12.
If the rank correlation co-efficient between marks in Management and Mathematics for a group of students is 0.6 and the sum ofthe squares ofthe difference in rank is 66. Then what is the number of students in the group ? 
(a) 9
(b) 10
(c) 11
(d) 12
Solution:
(b) Given, R = 0.6 ; Σ D² = 66
R = 1 – 6∑D2N(N2−1)
TRICKS : Go by choices,
[See QUICKER BMLRS Tricks ]
Use, directly by calculator
For, N = 10, R = 1 – 6×66990 = 0.6 = Given Value
So, N = 10 is correct.

Question 13.
Correlation coefficient between X and Y will be negative when:- 
(a) X and Y are decreasing
(b) X is increasing, Y is decreasing
(c) X and Y are increasing
(d) None of these
Solution:
(b) r is -ve, means X & Y are in reverse order.

Question 14.
If ‘ρ’ is the simple correlation coefficient, the quantity ρ² is known as :
(a) Coefficient of determination
(b) Coefficient of Non-determination
(c) Coefficient of alienation
(d) None of the above.
Solution:
(a)

Question 15.

If the correlation coefficient between X and Y is r, & U = X−510 and V = Y−72 then r_uv is 
(a) r
(b) -r
(c) (r-5)/2
(d) (r-7)/10
Solution:
(a) U = X−510 = X10 – 510 & V = Y2 – 72
TRICKS : Since X & Y have same sign.
So, r_xy = r_uv = r

Question 16.
If the sum of the product of deviations of x and y series from their mean is zero, then the coefficient of correlation will be
(a) 1
(b) -1
(c) 0
(d) None of these
Solution:
(c)
Given ∑(X−X¯¯¯¯)(Y−Y¯¯¯¯) =
Formula, r = ∑(X−X¯¯¯¯)(Y−Y¯¯¯¯)N×σx×σy
= 0N×σx×σy = 0

Question 17.
The covariance between two variables X and Y is 8.4 and their variances are 25 and 36 respectively. Calculate Karl Pearson’s coefficient of correlation between them. 
(a) 0.82
(b) 0.28
(c) 0.01
(d) 0.09
Solution:
(b) Given : Cov. (x, y) = 8.4 σ_x = 25−−√ = 5, σ_y = 36−−√ =
r = Cov.(x,y)σx,σy = 8.45×6 = 0.28

Question 18.
In Spearman’s Correlation Coefficient, the sum of the differences of ranks between two variables shall be 
(a) 0
(b) 1
(c) -1
(d) None of them
Solution:
(a) In Spearman’s correlation coefficient the sum of the differences of ranks between two variable shall be Zero.
Σ D = Σ (R₁ – R₂) = 0

Question 19
The coefficient of correlation between X and Y series is – 0.38 The linear relation between U & Y are 3X + 5U = 3 and -8Y – 7V = 44, what is the coefficient of correlation between U & V? 
(a) 0.38
(b) -0.38
(c) 0.40
(d) None of them
Solution:
(b) Given r_xy = – 0.38
TRICKS : 3X + 5U = 3 and
-8Y – 7V = 44
8Y + 7V = 44
r = – 0.38
For Tricks
Note See QUICKER BMLRS Examples

Question 20
Two variables X and Y are related as 4x + 3y = 7 then correlation between x and y is ___________
(a) Perfect positive
(b) Perfect negative
(c) Zero
(d) None of these
Solution:
(b)
Since, 4x+3y = 7
∴ 3y = -4x + 7; .∴ r = -1
∴ X and are y perfectly negative because x and y have opposite sign.

Question 21.
If r is the karl pearson’s coefficient of correlation in a bivariate distribution the two regression lines are at right angles when __________
(a) r = ±1
(b) r = 0
(c) r = ±∞
(d) None
Solution:
(b) If r = 0 ; Two Regression Lines are perpendicular to each other.

Question 22.
If r = 0.28, Cov. (x, y) = 7.6, V(x) = 9 then σ_y = ______________
(a) 8.75
(b) 9.04
(c) 6.25
(d) None
Solution:
(b) is correct
r = Cov⋅(x;y)σx⋅σy
or; 0.28 = 7.63.σy or σ_y = 9.04

Question 23.
Price and Demand is example for:
(a) No correlation
(b) Positive correlation
(c) Negative correlation
(d) None of these
Solution:
(c)

Question 24.
When each individual gets the exactly opposite rank by the two judges then the rank correlation will be ____________ 
(a) -1
(b) 0
(c) +1
(d) +1/2
Solution:
(a) is correct
Note Opposite Ranks Means r = -1

Question 25.

Correlation coefficient between x and y is 1, then correlation coefficient between x – 2 and (-y/2)+1 is. 
(a) 1
(b) -1
(c) -1/2
(d) 1/2
Solution:
(b) is correct,
∵ r = + 1 (given)
For variables (x – 2) and (−y2+1)
r = -1 (because sign of x & y are opposite)

Question 26.
When r = 1, all the points in a scatter diagram would lie: 
(a) On a straight line directed from lower left to upper right
(b) On a straight line
(c) On a straight line directed from upper left to lower right
(d) Both (a) and (b)
Solution:
(a) is correct.

Question 27.
In case ‘Insurance Companies’ Profits and the No. of claims they have to pay: 
(a) Positive correlation
(b) Negative correlation
(c) No correlation
(d) None of these
Solution:
(b) is correct.

Question 28.
If r = 0.6 then the coefficient of non-determination is __________
(a) 0.4
(b) -0.6
(c) 0.36
(d) 0.64
Solution:
(d)
co-efficient of Non-determination
= 1 – r² = 1 – (0.6)²
= 0.64
= 64%

Question 29.

If the coefficient of correlation between x and y variables is -0.90 then what will be the coefficient of determination 
(a) 0.10
(b) 0.81
(c) 0.94
(d) None
Solution:
(b)
co-eff. of determination = r²
= (-0.90)² = 0.81

Question 30.
If the sum of the squares of Rank differences in the marks of 10 students in two students is 44, then the coefficient of rank correlation is ____________ 
(a) 0.78
(b) 0.73
(c) 0.87
(d) None
Solution:
(b) is correct.
Given, N = 10 & Σ D² = 44
R = 1 – 6∑D2N3−N = 1 – 6×44103−10 = 0.73

Question 31.
Correlation between temperature and power consumption is 
(a) Positive
(b) Negative
(c) Zero
(d) None
Solution:
(a)

Question 32.
Coefficient of correlation between X & Y is 0.6. If both X and Y are multiplied by -1. Then resultant coefficient of correlation is
(a) 0.6
(b) Negative
(c) 1/0.6
(d) None
Solution:
(a) [∵ r does not change with respect to the change of origin and scale]

Question 33
If r = 0.6 then the coefficient of non-determination is : 
(a) 0.4
(b) -0.6
(c) 0.36
(d) 0.64
Solution:
(d)
Co-efficient of Non-determination
= 1 – r² = 1 – (0.6)²
= 0.64

Question 34
If there is a constant increase in the series then the obtained graph is : 
(a) Convex
(b) Concave
(c) Parabola
(d) Straight line from left to right
Solution:
(d)

Question 35

If r = 0.58, correlation coefficient of u = -5x + 3 and v = y + 2 is __________:
(a) 0.58
(b) -0.58
(c) 0.62
(d) None
Solution:
(b)
The value of “r” does not change with respect to the change of origin and scale but sign may change.
So r_uv = – 0.58
[Note : Here; sign of x & y in both eqns. are opposite, so sign of “r” changes ].

Question 36
If the sum of squares of deviations of ranks of 8 students is 50 then the rank correlation coefficient is _______: 
(a) 0.40
(b) 0.45
(c) 0.5
(d) 0.8
Solution:
(a)
Given, N = 8; Σ D² = 50
R = 1 – 6∑D2N3−N = 1 – 6×5083−8 = 0.40

Question 37.
If the plotted points in a scatter diagram are evenly distributed, then the correlation is _________ 
(a) Zero
(b) Negative
(c) Positive
(d) (a) or (b)
Solution:
(a)

Question 38
The covariance between two variables is __________ 
(a) Strictly positive
(b) Strictly negative
(c) Always Zero
(d) Either positive or negative or zero
Solution:
(d)

Question 39
The coefficient of determination is defined by the formula 
(a) r² = 1− unexplained variance  total variance 
(b) r² =  explained variance  total variance 
(c) both (a) and (b)
(d) None
Solution:
(c)

Question 40.
In the method of Concurrent Deviations, only the directions of change (Positive direction/Negative direction) in the variables are taken into account for calculation of 
(a) Coefficient of S.D
(b) Coefficient of regression
(c) Coefficient of correlation
(d) None
Solution:
(c)

Question 41.
Correlation coefficient is ___________ of the units of measurement.
(a) dependent
(b) independent
(c) both
(d) None
Solution:
(b)

Question 42
In case speed of an automobile and the distance required to stop the car after applying brakes correlation is ……………
(a) Positive
(b) Negative
(c) Zero
(d) None
Solution:
(a)

Question 43.
A relationship r² = 1 – 500300 is not possible
(a) True
(b) False
(c) Both
(d) None
Solution:
(a) Given :
r² = 1 – 500300 = −200300 is not possible [∵ r² is always positive]

Question 44.
Rank correlation coefficient lies between 
(a) 0 to 1
(b) -1 to +1 inclusive of these values
(c) -1 to 0
(d) Both
Solution:
(b)

Question 45.
If the correlation coefficient between the variables X and Y is 0.5, then the correlation coefficient between the variables 2x – 4 and 3 – 2y is 
Solution:
(c)
Tricks : See Quicker BMLRS
Chapter: Correlation.
r. = -0.5 (Sign, of X & Y in both are opposite).

ECPGURGAON.COM

Question 46

A.M. of regression coefficient is [1 Mark, June 2019]
(a) Equal to r
(b) Greater than or equal to r
(c) Half of r
(d) None
Solution:
(b).

Question 47
Find the probable error if r = 210√ and N = 36
(a) 0.6745
(b) 0.067
(c) 0.5287
(d) None
Solution:
(b)
r = 210√ , N = 36, P.E = ?
Probable Error P.E = 0.6745 1−r2N√
= 0.06745[1−(210√)236√] = 0.067 (1−410)6

Question 48
Determine Spearman’s rank correlation coefficient from the given data Σ D² = 30, N = 10 [1 Mark, June 2019]
(a) R = 0.82
(b) R = 0.32
(c) R = 0.40
(d) None of these
Solution:
(a)
Here, Σ D² =30, TV = 10
Spearman’s rank correlation
R = 1 – 6∑D2N(N2−1)
= 1 – 6×3010(102−1) = 1 – 180990 = 1 – 211 = 911
= 0.8