Question

the data below show the height, in feet, and the foot length, in inches, of ten people. according to the linear model that best fits the data, what is the predicted foot length of a person with a height of 6.2 feet? enter the predicted length, in inches, in the response box.

Explanation:

Step1: Calculate the mean of height ($\bar{x}$) and foot length ($\bar{y}$)

Let \(x\) be height and \(y\) be foot - length.
\(\bar{x}=\frac{5.4 + 5.6+5.7+5.8+6.1+6.1+6.3+6.4+6.8+7.2}{10}=\frac{61.4}{10}=6.14\)
\(\bar{y}=\frac{9.1 + 9.4+9.3+9.6+10.2+10.3+12.1+12.5+12.4+13.3}{10}=\frac{108.2}{10}=10.82\)

Step2: Calculate the slope ($b$)

The formula for the slope \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=(5.4 - 6.14)(9.1 - 10.82)+(5.6 - 6.14)(9.4 - 10.82)+(5.7 - 6.14)(9.3 - 10.82)+(5.8 - 6.14)(9.6 - 10.82)+(6.1 - 6.14)(10.2 - 10.82)+(6.1 - 6.14)(10.3 - 10.82)+(6.3 - 6.14)(12.1 - 10.82)+(6.4 - 6.14)(12.5 - 10.82)+(6.8 - 6.14)(12.4 - 10.82)+(7.2 - 6.14)(13.3 - 10.82)\)
\(=(- 0.74)\times(-1.72)+(-0.54)\times(-1.42)+(-0.44)\times(-1.52)+(-0.34)\times(-1.22)+(-0.04)\times(-0.62)+(-0.04)\times(-0.52)+(0.16)\times(1.28)+(0.26)\times(1.68)+(0.66)\times(1.58)+(1.06)\times(2.48)\)
\(=1.2728 + 0.7668+0.6688+0.4148+0.0248+0.0208+0.2048+0.4368+1.0428+2.6288=7.58\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=(5.4 - 6.14)^{2}+(5.6 - 6.14)^{2}+(5.7 - 6.14)^{2}+(5.8 - 6.14)^{2}+(6.1 - 6.14)^{2}+(6.1 - 6.14)^{2}+(6.3 - 6.14)^{2}+(6.4 - 6.14)^{2}+(6.8 - 6.14)^{2}+(7.2 - 6.14)^{2}\)
\(=(-0.74)^{2}+(-0.54)^{2}+(-0.44)^{2}+(-0.34)^{2}+(-0.04)^{2}+(-0.04)^{2}+(0.16)^{2}+(0.26)^{2}+(0.66)^{2}+(1.06)^{2}\)
\(=0.5476+0.2916+0.1936+0.1156+0.0016+0.0016+0.0256+0.0676+0.4356+1.1236 = 2.8\)
\(b=\frac{7.58}{2.8}\approx2.71\)

Step3: Calculate the y - intercept ($a$)

The formula for the y - intercept \(a=\bar{y}-b\bar{x}\)
\(a = 10.82-2.71\times6.14=10.82 - 16.64= - 5.82\)

Step4: Find the regression equation and predict

The regression equation is \(y = 2.71x-5.82\)
When \(x = 6.2\), \(y=2.71\times6.2-5.82\)
\(y = 16.802-5.82=10.982\approx11.0\)

Answer:

\(11.0\)