Question

2. Big TV prices Here are prices and screen sizes (in inches, measured diagonally) for 7 different sizes of one brand of LED HD television.47 Use technology to calculate the equation of the least- squares regression line relating y = price to x = screen size. Screen size Price 60 1000 55 800 50 700 47 600 42 430 39 400 32 300

Answer 1

The general formula for the least-squares regression is:

`y=a+bx_i`

Where

a represents the y-intercept

b represents the slope

To estimate the y-intercept and the slope of the regression line you have to apply the following formulas:

`b=(\Sigma xy-(\Sigma x\Sigma y)/(n))/(\lbrack\Sigma x^2-((\Sigma x)^2)/(n)\rbrack)`
`a=\bar{y}-b\bar{x}`

First, calculate the sums and the means for both variables.

X= screen size

Y= price

n=7

`\begin{gathered} \Sigma x=60+55+50+47+42+39+32 \\ \Sigma x=325 \end{gathered}`
`\begin{gathered} \Sigma x^2=60^2+55^2+50^2+47^2+42^2+39^2+32^2 \\ \Sigma x^2=15643 \end{gathered}`
`\begin{gathered} \bar{x}=(\Sigma x)/(n) \\ \bar{x}=(325)/(7) \\ \bar{x}=46.43 \end{gathered}`
`\begin{gathered} \Sigma y=1000+800+700+600+430+400+300 \\ \Sigma y=4230 \end{gathered}`
`\begin{gathered} \bar{y}=(\Sigma y)/(n) \\ \bar{y}=(4230)/(7) \\ \bar{y}=604.29 \end{gathered}`
`\begin{gathered} \Sigma xy=60\cdot1000+55\cdot800+50\cdot700+47\cdot600+42\cdot430+39\cdot400+32\cdot300 \\ \Sigma xy=210460 \end{gathered}`

Calculate the slope of the line:

`\begin{gathered} b=(\Sigma xy-(\Sigma x\Sigma y)/(n))/(\Sigma x^2-((\Sigma x)^2)/(n)) \\ b=(210460-(325\cdot4230)/(7))/(15643-(325^2)/(7)) \\ b=(210460-196392.85)/(15643-15089.29) \\ b=(14067.75)/(553.71) \\ b=25.41 \end{gathered}`

Once you have calculated the slope, you can calculate the y-intercept:

`\begin{gathered} a=\bar{y}-b\bar{x} \\ a=604.29-25.41\cdot46.43 \\ a=604.29-1179.7863 \\ a=-575.23 \end{gathered}`

The regression line for the price with respect to the screen size is:

`y=-575.23+25.41x`