Question

find the equation for the least squares regression line of the data described below.
fans of the science fiction television show space cruiser zeta are hosting a
convention. to figure out how much space the event will require, an employee at
the convention center looked at the space requirements for previous conventions.
for each convention, the employee recorded the number of attendees, x, and the
floor space required (in square meters), y.
round your answers to the nearest thousandth.
y = x +

Explanation:

Step1: Calculate the means of \(x\) and \(y\)

Let \(x_i\) be the number of attendees and \(y_i\) be the floor - space.
\(\bar{x}=\frac{1259 + 4118+4268+4679+4931+5592+7519+7535}{8}=\frac{39901}{8}=4987.625\)
\(\bar{y}=\frac{5309+4035 + 10861+6148+9536+9922+11897+15953}{8}=\frac{73661}{8}=9207.625\)

Step2: Calculate the numerator and denominator for the slope \(b\)

The formula for the slope \(b=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})=(1259 - 4987.625)(5309 - 9207.625)+(4118 - 4987.625)(4035 - 9207.625)+(4268 - 4987.625)(10861 - 9207.625)+(4679 - 4987.625)(6148 - 9207.625)+(4931 - 4987.625)(9536 - 9207.625)+(5592 - 4987.625)(9922 - 9207.625)+(7519 - 4987.625)(11897 - 9207.625)+(7535 - 4987.625)(15953 - 9207.625)\)
\(=(-3728.625)(-3898.625)+(-869.625)(-5172.625)+(-719.625)(1653.375)+(-308.625)(-3059.625)+(-56.625)(328.375)+(604.375)(714.375)+(2531.375)(2689.375)+(2547.375)(6745.375)\)
\(=14525976.5625+4492305.46875 - 1190793.1640625+943921.40625-18691.265625+431651.3671875+6816910.15625+17188935.703125\)
\(=46470116.25\)

\(\sum_{i=1}^{n}(x_i-\bar{x})^2=(1259 - 4987.625)^2+(4118 - 4987.625)^2+(4268 - 4987.625)^2+(4679 - 4987.625)^2+(4931 - 4987.625)^2+(5592 - 4987.625)^2+(7519 - 4987.625)^2+(7535 - 4987.625)^2\)
\(=(-3728.625)^2+(-869.625)^2+(-719.625)^2+(-308.625)^2+(-56.625)^2+(604.375)^2+(2531.375)^2+(2547.375)^2\)
\(=13892664.015625 + 756350.015625+517860.015625+95249.015625+3206.140625+365260.015625+6407714.015625+6489024.015625\)
\(=28527331.25\)

\(b=\frac{46470116.25}{28527331.25}\approx1.629\)

Step3: Calculate the intercept \(a\)

The formula for the intercept \(a=\bar{y}-b\bar{x}\)
\(a = 9207.625-1.629\times4987.625\)
\(a=9207.625 - 8124.841\)
\(a = 1082.784\)

Answer:

\(y = 1.629x+1082.784\)