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 8.79 6.05 5.38 8.28 6.17 5.55 find the equation of the regression line 𝑦̂=□+□x (round the y - intercept two decimal places as needed. round the slope to three decimal places as needed.)

Explanation:

Step1: Calculate necessary sums

Let \(n = 11\). Calculate \(\sum x\), \(\sum y\), \(\sum x^{2}\), \(\sum xy\).
Let \(x_i\) and \(y_i\) be the individual data - points.
\(\sum_{i = 1}^{n}x_i=10 + 9+12 + 9+10+13+7+4+13+6+5=98\)
\(\sum_{i = 1}^{n}y_i=7.19+6.46+12.72+6.85+7.67+8.79+6.05+5.38+8.28+6.17+5.55 = 79.11\)
\(\sum_{i = 1}^{n}x_{i}^{2}=10^{2}+9^{2}+12^{2}+9^{2}+10^{2}+13^{2}+7^{2}+4^{2}+13^{2}+6^{2}+5^{2}\)
\(=100 + 81+144+81+100+169+49+16+169+36+25 = 970\)
\(\sum_{i = 1}^{n}x_{i}y_{i}=10\times7.19+9\times6.46+12\times12.72+9\times6.85+10\times7.67+13\times8.79+7\times6.05+4\times5.38+13\times8.28+6\times6.17+5\times5.55\)
\(=71.9+58.14+152.64+61.65+76.7+114.27+42.35+21.52+107.64+37.02+27.75 = 771.68\)

Step2: Calculate the slope \(b_1\)

The formula for the slope \(b_1\) of the regression line is \(b_1=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}\)
\(b_1=\frac{11\times771.68 - 98\times79.11}{11\times970-98^{2}}\)
\(=\frac{8488.48-7752.78}{10670 - 9604}\)
\(=\frac{735.7}{1066}\approx0.690\)

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

The formula for the y - intercept \(b_0\) is \(b_0=\bar{y}-b_1\bar{x}\), where \(\bar{x}=\frac{\sum x}{n}\) and \(\bar{y}=\frac{\sum y}{n}\)
\(\bar{x}=\frac{98}{11}\approx8.91\), \(\bar{y}=\frac{79.11}{11}\approx7.19\)
\(b_0 = 7.19-0.690\times8.91\)
\(=7.19 - 6.1479\approx1.04\)

Answer:

\(\hat{y}=1.04 + 0.690x\)