Question

2.5 hours, 79
problem 4
the table below gives the height and shoe size of 6 randomly selected students.

height (in)	67	70	73.5	75	78	66
shoe size	8.5	9.5	11	12	13	8

regression equation:
if a student is 73 inches tall what is his or her predicted shoe size?
if a student is 60 inches tall what is his or her predicted shoe size?

Explanation:

Step1: Calculate the regression coefficients

Let \(x\) be the height and \(y\) be the shoe - size. First, calculate the following sums:
Let \(n = 6\).
\(\sum_{i = 1}^{n}x_{i}=67 + 70+73.5 + 75+78+66=429.5\)
\(\sum_{i = 1}^{n}y_{i}=8.5 + 9.5+11+12+13+8=62\)
\(\sum_{i = 1}^{n}x_{i}^{2}=67^{2}+70^{2}+73.5^{2}+75^{2}+78^{2}+66^{2}=67^{2}+4900 + 5402.25+5625+6084+4356=31147.25\)
\(\sum_{i = 1}^{n}x_{i}y_{i}=67\times8.5+70\times9.5 + 73.5\times11+75\times12+78\times13+66\times8\)
\(=569.5+665+808.5+900+1014+528=4485\)

The slope \(b\) of the regression line \(\hat{y}=a + bx\) is given by:

$$b=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}$$

$$b=\frac{6\times4485-429.5\times62}{6\times31147.25-(429.5)^{2}}$$

$$b=\frac{26910 - 26629}{186883.5 - 184470.25}$$

$$b=\frac{281}{2413.25}\approx0.1164$$

The intercept \(a\) is given by:

$$a=\bar{y}-b\bar{x}$$

where \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}=\frac{429.5}{6}\approx71.5833\) and \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}=\frac{62}{6}\approx10.3333\)

$$a = 10.3333-0.1164\times71.5833$$

$$a = 10.3333 - 8.3333=2$$

So the regression equation is \(\hat{y}=2 + 0.1164x\)

Step2: Predict shoe - size for \(x = 73\)

Substitute \(x = 73\) into the regression equation \(\hat{y}=2+0.1164\times73\)
\(\hat{y}=2 + 8.4972=10.4972\approx10.5\)

Step3: Predict shoe - size for \(x = 60\)

Substitute \(x = 60\) into the regression equation \(\hat{y}=2+0.1164\times60\)
\(\hat{y}=2+6.984 = 8.984\approx9\)

Answer:

Regression Equation: \(\hat{y}=2 + 0.1164x\)
If a student is 73 inches tall, the predicted shoe - size is approximately 10.5
If a student is 60 inches tall, the predicted shoe - size is approximately 9