Question

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 \(x\) be the percent carbon and \(y\) be the weight supported.
The mean of \(x\), \(\bar{x}=\frac{1.03 + 1.11+1.18+1.23+1.30+1.48+1.69+1.71}{8}=\frac{10.73}{8}=1.34125\)
The mean of \(y\), \(\bar{y}=\frac{1066 + 1186+1123+1102+1208+1265+1194+1301}{8}=\frac{9445}{8}=1180.625\)

Step2: Calculate the slope \(b_1\)

The formula for the slope \(b_1=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}\)
\(\sum_{i = 1}^{8}(x_i-\bar{x})(y_i-\bar{y})=(1.03 - 1.34125)(1066 - 1180.625)+(1.11 - 1.34125)(1186 - 1180.625)+(1.18 - 1.34125)(1123 - 1180.625)+(1.23 - 1.34125)(1102 - 1180.625)+(1.30 - 1.34125)(1208 - 1180.625)+(1.48 - 1.34125)(1265 - 1180.625)+(1.69 - 1.34125)(1194 - 1180.625)+(1.71 - 1.34125)(1301 - 1180.625)\)
\(=(- 0.31125)\times(-114.625)+(-0.23125)\times(5.375)+(-0.16125)\times(-57.625)+(-0.11125)\times(-78.625)+(-0.04125)\times(27.375)+(0.13875)\times(84.375)+(0.34875)\times(13.375)+(0.36875)\times(120.375)\)
\(=35.65917969-1.243164063 + 9.29296875+8.740234375 - 1.137695313+11.703125+4.66796875+44.4140625\)
\(=112.1067823\)

\(\sum_{i = 1}^{8}(x_i-\bar{x})^2=(1.03 - 1.34125)^2+(1.11 - 1.34125)^2+(1.18 - 1.34125)^2+(1.23 - 1.34125)^2+(1.30 - 1.34125)^2+(1.48 - 1.34125)^2+(1.69 - 1.34125)^2+(1.71 - 1.34125)^2\)
\(=(-0.31125)^2+(-0.23125)^2+(-0.16125)^2+(-0.11125)^2+(-0.04125)^2+(0.13875)^2+(0.34875)^2+(0.36875)^2\)
\(=0.0968765625+0.0534765625+0.0260015625+0.0123765625+0.0017015625+0.0192515625+0.1216265625+0.1360765625\)
\(=0.4673882813\)

\(b_1=\frac{112.1067823}{0.4673882813}\approx239.850\)

Step3: Calculate the intercept \(b_0\)

Using the formula \(b_0=\bar{y}-b_1\bar{x}\)
\(b_0 = 1180.625-239.850\times1.34125\)
\(=1180.625 - 321.737\)
\(=858.888\)

Answer:

\(y = 239.850x+858.888\)