Question

Multiple Choice Questions : Given the following data pairs (x, y), find the regression. equation. (1, 1.24), (2, 5.23), (3, 7.24), (4, 7.60), (5, 9.97), (6, 14.31), (7, 13.99), (8, 14.88), (9, 18.04), (10, 20.70), calculate the correlation coefficient

Answer 1

Regression equation X on Y:

x=0.50y-0.16.

Regression equation for Y on X:

y=2.01x + 0.27

Correlation coefficient is 1.00249688

Given:

data pairs (x, y)

equation. (1, 1.24), (2, 5.23), (3, 7.24), (4, 7.60), (5, 9.97), (6, 14.31), (7, 13.99), (8, 14.88), (9, 18.04), (10, 20.70)

To Find:

Regression Equation

Correlation equation

Solution:

Formula for regression equation X on Y

(x- x̄) = r∑x/∑y(y-ȳ)

For the table

x X= (x- x̄) X square    Y    Y=( y-ȳ)     Y square XY

1 -4.5       20.25 1.24     -10.08     101.60     48.6

2 -3.5       12.25   5.23      -6.09     37.08     21.31

3 -2.5       6.25         7.24       -4.08     16.64     10.2

4 -1.5       2.25          7.60        -3.72     13.84     5.58

5 -0.5       0.25         9.97        -1.35     1.82     0.67

6 0.5       0.25         14.31 2.99    8.94     1.49

7 1.5       2.25         13.99 2.67     7.12     4

8 2.5      6.25         14.88 3.56    12.67     8.9

9 3.5      12.25         18.04 6.72     45.15     23.52

10 4.5     20.25         20.70 9.38     87.98      42.21

∑x=55  ∑ x square=82.5 ∑y=113.2  ∑ y square=332.84 ∑xy=166.48

Attached the table in word for reference.

Now we need to find the Regression equation for X on Y

And the formula is

(x- x̄)= r∑x/∑y( y-ȳ)

Now this can be further equated to

(x- x̄) =∑xy/∑ y square( y-ȳ)

That is

Now , insert the values as per the derivation

So it will be

x-5.5= 166.48/332.84 ( y-11.32)

First , divide 166.48/332.84( 0.50)

Now , x-5.5=0.50( y-11.32)

x-5.5=0.50y-5.66

x=0.50y-(-5.66+5.5)

x=0.50y-0.16.

Regression equation for Y on X

Formula is:

(y-ȳ)= r∑y/∑x(y-ȳ)

Now, this can be evaluated to:

( y-ȳ) =∑xy/∑x square (x- x̄)

Now, we need to insert the values as per the table.

y-11.32=166.48/82.5(x-5.5)

y-11.32=2.01(x-5.5)

y-11.32=2.01x-11.05

y=2.01x + 0.27

Now, the formula for correlation coefficient is

Square root of ∑xy/∑ y square * ∑xy/∑x square

Which is square root of 0.50*2.10

It becomes square root of 1.005

1.00249688

• The Regression Equation is the algebraic expression of the regression lines. It is used to predict the values of the dependent variable from the given values of independent variables.
• A correlation coefficient is a measurement of the statistical relationship (correlation), between two variables.

Therefore,

Regression equation X on Y:

x=0.50y-0.16.

Regression equation for Y on X:

y=2.01x + 0.27

Correlation coefficient is 1.00249688

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