Question

here is a bivariate data set. find the regression equation for the response variable y.
regression equation:

Explanation:

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

Let \(n = 17\) (number of data points).
\(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\)
\(\sum_{i=1}^{17}x_{i}=66.7 + 52.9+47.6+42.8+67.5+41.8+58.8+46.4+49.7+40.1+38.6+27.9+67.2+55.4+52.6+37.3+46.4=865.7\)
\(\bar{x}=\frac{865.7}{17}\approx50.9235\)

\(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\)
\(\sum_{i=1}^{17}y_{i}=69.1 + 79.8+64.9+75.4+68.1+92.5+68.7+70.1+71.1+82.5+87.4+83.7+70.2+75+67+99.8+69.9 = 1273.2\)
\(\bar{y}=\frac{1273.2}{17}\approx74.8941\)

Step2: Calculate the slope \(b_1\)

\(b_1=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^2}\)

First, calculate \((x_{i}-\bar{x})(y_{i}-\bar{y})\) and \((x_{i}-\bar{x})^2\) for each \(i\):
For \(x = 66.7,y = 69.1\): \((66.7 - 50.9235)(69.1-74.8941)=(15.7765)(- 5.7941)\approx - 91.47\)
\((66.7 - 50.9235)^2=(15.7765)^2\approx248.9\)

... (repeat for all data points)

After summing up:
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\approx-1197.9\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^2\approx1947.9\)

\(b_1=\frac{- 1197.9}{1947.9}\approx - 0.615\)

Step3: Calculate the intercept \(b_0\)

\(b_0=\bar{y}-b_1\bar{x}\)
\(b_0 = 74.8941-(-0.615)\times50.9235\)
\(b_0=74.8941 + 31.32\approx106.214\)

Answer:

\(y=-0.615x + 106.214\)