Answer to Question #218741 in Statistics and Probability for RAFSAN

Question

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If you want to fit a linear regression line y=a+bx from the above data, then what is the value of 'a'?

Single choice.

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If you want to fit a linear regression line y=a+bx from the above data, then what is the value of 'a'?

Single choice.

Answer 1

A linear regression line has an equation of the form: y = a+ bx

where 'Y' is called the dependent variable or response.

and 'X' is called independent variable or predictors or explanatory variable

'a' is the y-intercept of the line and

'b' is the slope of the line.

From the above data,

$\sum x = 247\\\sum y=486\\\sum xy=20485\\\sum x^2=11409\\\sum y^2=40022\\n=6$

Based on the above table, the following is calculated:

$\bar X=\dfrac{1}{n}\sum X_i=\dfrac{247}{6}=41.167$

$\bar Y= \dfrac{1}{n}\sum Y_i=\dfrac{486}{6}=81\\\ \\SS_{XX}=\sum X_i^2-\dfrac{1}{n}(\sum X_i)^2=11409-\dfrac{(247)^2}{6}=120.83$

$SS_{YY}=\sum Y_i^2-\dfrac{1}{n}(\sum Y_i)^2=40022-\dfrac{(486)^2}{6}=656$

$SS_{XY}=\sum X_iY_i-\dfrac{1}{n}(\sum X_i)(\sum Y_i)=20485 -\dfrac{247\times 486}{6}=478$

Therefore, based on the above calculations, the regression coefficients (the slope b, and the y-intercept a) are obtained as follows:

$b=\dfrac{SS_{XY}}{SS_{XX}}=\dfrac{478}{1240.83}=0.3852\\\ \\a=\bar Y-\bar X\cdot b=81-41.16\times 0.3852=65.14$

Therefore, we find that the regression equation is:

$$y=a+bx\\y=65.14+0.3852x$$