The following data is given, based on 450 students for marks in Statistics and Economics at a certain examination: Mean marks in Statistics Mean marks in Economics 48 S.D. of marks (Statistics)12 Variance of marks (Econcmics)256 Sum of the products of deviations of marks from their respective mean = 42075 The average marks in Economics of candidates who obtained 50 marks in Statistics is: - 40?

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Solution:

Given data:

Mean marks in Statistics = 48
Mean marks in Economics = ?
S.D. of marks (Statistics) = 12
Variance of marks (Economics) = 256
Sum of the products of deviations of marks from their respective mean = 42075

To find: The average marks in Economics of candidates who obtained 50 marks in Statistics

Finding the Mean marks in Economics:

We know that the sum of the products of deviations of marks from their respective mean is given by:

Σ(xy) = nΣx'y'

Where,
Σ(xy) = Sum of the products of deviations of marks from their respective mean
n = Number of students
Σx'y' = Sum of the products of deviations of marks in Statistics and Economics from their respective means

Substituting the given values, we get:

42075 = 450Σx'y'

Σx'y' = 93.5

Now, we can use the formula for the covariance of Statistics and Economics:

Cov(x,y) = Σ(x-x')(y-y')/(n-1)

Where,
Cov(x,y) = Covariance of Statistics and Economics
x = Marks in Statistics
y = Marks in Economics
x' = Mean marks in Statistics
y' = Mean marks in Economics
n = Number of students

Substituting the given values, we get:

Cov(x,y) = 93.5/(450-1) = 0.209

Now, we can use the formula for the regression line of Economics on Statistics:

y-y' = b(x-x')

Where,
y = Marks in Economics
x = Marks in Statistics
y' = Mean marks in Economics
x' = Mean marks in Statistics
b = Covariance of Statistics and Economics / Variance of Statistics

Substituting the given values, we get:

b = 0.209/144 = 0.00145

y - 48 = 0.00145(x-50)

y = 0.00145x + 47.928

Therefore, the average marks in Economics of candidates who obtained 50 marks in Statistics is:

y = 0.00145(50) + 47.928 = 48.007

Hence, the average marks in Economics of candidates who obtained 50 marks in Statistics is 48.007.

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54.5

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