Question

use the given data to find the equation of the regression line. examine the scatterplot and identify a characteristic of the data that is ignored by the regression line. x 10 9 12 9 10 13 7 4 13 6 5 y 7.19 6.46 12.72 6.85 7.67 0.79 6.05 5.30 0.20 6.17 5.55 create a scatterplot of the data. choose the correct graph below. find the equation of the regression line. y = □ + □x (round the y - intercept two decimal places as needed. round the slope to three decimal places as needed.)

Explanation:

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

Let \(x_1,x_2,\cdots,x_{n}\) and \(y_1,y_2,\cdots,y_{n}\) be the data - points. Here \(n = 11\).
\(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), \(\sum_{i=1}^{11}x_{i}=10 + 9+12 + 9+10+13+7+4+13+6+5=98\), so \(\bar{x}=\frac{98}{11}\approx8.91\).
\(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\), \(\sum_{i = 1}^{11}y_{i}=7.19+6.46+12.72+6.85+7.67+0.79+6.05+5.30+0.20+6.17+5.55 = 65.95\), so \(\bar{y}=\frac{65.95}{11}\approx5.995\).

Step2: Calculate the slope \(b_1\)

The formula for the slope \(b_1=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\).
First, calculate \((x_{i}-\bar{x})(y_{i}-\bar{y})\) and \((x_{i}-\bar{x})^{2}\) for each \(i\):

\(x_{i}\)	\(y_{i}\)	\(x_{i}-\bar{x}\)	\(y_{i}-\bar{y}\)	\((x_{i}-\bar{x})(y_{i}-\bar{y})\)	\((x_{i}-\bar{x})^{2}\)
9	6.46	\(9 - 8.91 = 0.09\)	\(6.46 - 5.995=0.465\)	\(0.09\times0.465 = 0.04185\)	\(0.09^{2}=0.0081\)
12	12.72	\(12 - 8.91 = 3.09\)	\(12.72 - 5.995 = 6.725\)	\(3.09\times6.725=20.78025\)	\(3.09^{2}=9.5481\)
9	6.85	\(9 - 8.91 = 0.09\)	\(6.85 - 5.995 = 0.855\)	\(0.09\times0.855 = 0.07695\)	\(0.09^{2}=0.0081\)
10	7.67	\(10 - 8.91 = 1.09\)	\(7.67 - 5.995 = 1.675\)	\(1.09\times1.675 = 1.82575\)	\(1.09^{2}=1.1881\)
13	0.79	\(13 - 8.91 = 4.09\)	\(0.79 - 5.995=-5.205\)	\(4.09\times(-5.205)=-21.28845\)	\(4.09^{2}=16.7281\)
7	6.05	\(7 - 8.91=-1.91\)	\(6.05 - 5.995 = 0.055\)	\((-1.91)\times0.055=-0.10505\)	\((-1.91)^{2}=3.6481\)
4	5.30	\(4 - 8.91=-4.91\)	\(5.30 - 5.995=-0.695\)	\((-4.91)\times(-0.695)=3.41245\)	\((-4.91)^{2}=24.1081\)
13	0.20	\(13 - 8.91 = 4.09\)	\(0.20 - 5.995=-5.795\)	\(4.09\times(-5.795)=-23.70155\)	\(4.09^{2}=16.7281\)
6	6.17	\(6 - 8.91=-2.91\)	\(6.17 - 5.995 = 0.175\)	\((-2.91)\times0.175=-0.50925\)	\((-2.91)^{2}=8.4681\)
5	5.55	\(5 - 8.91=-3.91\)	\(5.55 - 5.995=-0.445\)	\((-3.91)\times(-0.445)=1.73995\)	\((-3.91)^{2}=15.2881\)

\(\sum_{i = 1}^{11}(x_{i}-\bar{x})(y_{i}-\bar{y})=1.30255+0.04185 + 20.78025+\cdots+1.73995=-16.244\)
\(\sum_{i = 1}^{11}(x_{i}-\bar{x})^{2}=1.1881+0.0081 + 9.5481+\cdots+15.2881=96.2169\)
\(b_1=\frac{-16.244}{96.2169}\approx - 0.169\)

Step3: Calculate the \(y\) - intercept \(b_0\)

The formula for the \(y\) - intercept \(b_0=\bar{y}-b_1\bar{x}\).
\(b_0 = 5.995-(-0.169)\times8.91=5.995 + 1.50579=7.50079\approx7.50\)
The equation of the regression line is \(\hat{y}=7.50-0.169x\)

To identify a characteristic of the data ignored by the regression line, we need to look at the scatter - plot. If there is a non - linear pattern (such as a curve) in the scatter - plot, the regression line (which is a straight line) will ignore this non - linearity.

Answer:

\(\hat{y}=7.50-0.169x\)