It was reported somewhere that children whenever plays the game in computer, they used the computer very roughly which may reduce the lifetime of a computer. The random access memory (RAM) of a computer also plays a crucial role on the lifetime of a computer. A researcher wanted to examine how the lifetime of a personal computer which is used by children is affected by the time (in hours) spends by the children per day to play games and the available random access memory (RAM) measured in megabytes (MB) of a used computer. The data is provided in following table.

Lifetime (years) 5 1 7 2 3 4 6
Play time (hours)/day 2 8 1 5 6 3 2
RAM in MB) 8 2 6 3 2 4 7

Identify which one is dependent variable? Solve this problem using multiple linear regression model and provide problem specific interpretations based on the regression model developed.

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Let, Lifetime = y

Playtime = x1

Ram = x2

Here,

The lifetime (y) is dependent variable

y x1 x2 yx1 yx2 x1x2 y2 x12 x22
5 2 8 10 40 16 25 4 64
1 8 2 8 2 16 1 64 4
7 1 6 7 42 6 49 1 36
2 5 3 10 6 15 4 25 9
3 6 2 18 6 12 9 36 4
4 3 4 12 16 12 16 9 16
6 2 7 12 42 14 36 4 49

Σy = 28, Σx1 = 27, Σx2 = 32, Σyx1 = 77, Σyx2 = 154, Σx1x2 = 91, Σy2 = 140, Σx12 = 143, Σx22 = 182

n= 7

Let the regression equation of y on x1, x2 be

y = b0 + b1x1 + b2x2   —— (i)

where b0, b1, b2 are regression coefficients.

By least square method, The normal equations to the above equations are

Σy = nb0 + b1Σx1 + b2Σx2

28 = 7b0 + 27b1 + 32b2        ——–(ii)

Σyx1 = b0Σx1 + b1Σx12 + b2Σx1Σx2

77 = 27b0 + 143b1 + 91b2               ————–(iii)

Σyx2 = b0Σx2 + b2Σx1Σx2 + b2Σx22

154 = 32b0 + 91b1 + 182b2            —————-(iv)

Solving the equation (ii) , (iii) and (iv) we get the value of regression coefficient.

b0 = 6.96, b1 = -0.79 and b2 = 0.015

Therefore, The final regression model is:-

ŷ = 6.96 + (-0.79)b1 + 0.015b2

Here, b0 = 6.96 is the lifetime when playtime and RAM are considered to be zero.

b1 = -0.79 means the lifetime decrease by 0.79 per unit change in playtime keeping RAM constant.

b2 = 0.015 means the lifetime increase by 0.015 per unit change in RAM keeping playtime constant.

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