Question

find the equation for the least squares regression line of the data described below.
a french car company wants to cut costs by using cheaper low - carbon steel in its car frames. to determine how varying the carbon content will affect strength, company engineers manufactured several car frames.
for each car frame, the engineers noted the percentage of carbon, x, as well as the weight it could support (in kilograms), y.
round your answers to the nearest thousandth.
y = x +

Explanation:

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

Let \(x_i\) be the percent carbon values and \(y_i\) be the weight - supported values.
The mean of \(x\), \(\bar{x}=\frac{1.14 + 1.26+1.29+1.40+1.44+1.76+1.80}{7}=\frac{9.09}{7}\approx1.299\)
The mean of \(y\), \(\bar{y}=\frac{1171 + 1152+1129+1256+1333+1275+1360}{7}=\frac{8676}{7}\approx1239.429\)

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

The formula for the slope \(b=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}\)
\((x_1-\bar{x})(y_1 - \bar{y})=(1.14 - 1.299)(1171-1239.429)=(- 0.159)\times(-68.429)\approx10.880\)
\((x_2-\bar{x})(y_2 - \bar{y})=(1.26 - 1.299)(1152 - 1239.429)=(-0.039)\times(-87.429)\approx3.410\)
\((x_3-\bar{x})(y_3 - \bar{y})=(1.29 - 1.299)(1129-1239.429)=(-0.009)\times(-110.429)\approx0.994\)
\((x_4-\bar{x})(y_4 - \bar{y})=(1.40 - 1.299)(1256 - 1239.429)=(0.101)\times(16.571)\approx1.674\)
\((x_5-\bar{x})(y_5 - \bar{y})=(1.44 - 1.299)(1333 - 1239.429)=(0.141)\times(93.571)\approx13.294\)
\((x_6-\bar{x})(y_6 - \bar{y})=(1.76 - 1.299)(1275 - 1239.429)=(0.461)\times(35.571)\approx16.498\)
\((x_7-\bar{x})(y_7 - \bar{y})=(1.80 - 1.299)(1360 - 1239.429)=(0.501)\times(120.571)\approx60.406\)
\(\sum_{i = 1}^{7}(x_i-\bar{x})(y_i - \bar{y})=10.880+3.410 + 0.994+1.674+13.294+16.498+60.406=107.156\)

\((x_1-\bar{x})^2=(1.14 - 1.299)^2=(-0.159)^2 = 0.025\)
\((x_2-\bar{x})^2=(1.26 - 1.299)^2=(-0.039)^2=0.002\)
\((x_3-\bar{x})^2=(1.29 - 1.299)^2=(-0.009)^2=0.000\)
\((x_4-\bar{x})^2=(1.40 - 1.299)^2=(0.101)^2=0.010\)
\((x_5-\bar{x})^2=(1.44 - 1.299)^2=(0.141)^2=0.020\)
\((x_6-\bar{x})^2=(1.76 - 1.299)^2=(0.461)^2=0.213\)
\((x_7-\bar{x})^2=(1.80 - 1.299)^2=(0.501)^2=0.251\)
\(\sum_{i=1}^{7}(x_i-\bar{x})^2=0.025 + 0.002+0.000+0.010+0.020+0.213+0.251=0.521\)

\(b=\frac{107.156}{0.521}\approx205.674\)

Step3: Calculate the intercept \(a\)

The formula for the intercept \(a=\bar{y}-b\bar{x}\)
\(a = 1239.429-205.674\times1.299\)
\(a=1239.429 - 267.170\)
\(a\approx972.259\)

Answer:

\(y = 205.674x+972.259\)