run a regression analysis on the following data set with y as the response variable.\n| x | y |\n| ---- |…

run a regression analysis on the following data set with y as the response variable.\n| x | y |\n| ---- | ---- |\n| 62 | 77.3 |\n| 56.9 | 73.1 |\n| 89.3 | 81.7 |\n| 86.4 | 79.8 |\n| 76 | 77.5 |\n| 79.3 | 80.2 |\n| 81.7 | 79.2 |\n| 67.2 | 78.2 |\nwhat is the predicted value of the response variable when the explanatory variable has value 69.6?\ny = \nround to 1 decimal place.

run a regression analysis on the following data set with y as the response variable.\n| x | y |\n| ---- | ---- |\n| 62 | 77.3 |\n| 56.9 | 73.1 |\n| 89.3 | 81.7 |\n| 86.4 | 79.8 |\n| 76 | 77.5 |\n| 79.3 | 80.2 |\n| 81.7 | 79.2 |\n| 67.2 | 78.2 |\nwhat is the predicted value of the response variable when the explanatory variable has value 69.6?\ny = \nround to 1 decimal place.

Answer

Explanation:

Step1: Calculate sums

Let (n = 8). (\sum_{i = 1}^{n}x_{i}=62 + 56.9+89.3+86.4+76+79.3+81.7+67.2 = 608) (\sum_{i = 1}^{n}y_{i}=77.3+73.1+81.7+79.8+77.5+80.2+79.2+78.2 = 627) (\sum_{i = 1}^{n}x_{i}^{2}=62^{2}+56.9^{2}+89.3^{2}+86.4^{2}+76^{2}+79.3^{2}+81.7^{2}+67.2^{2}) (=3844 + 3237.61+7974.49+7464.96+5776+6288.49+6674.89+4515.84 = 45776.37) (\sum_{i = 1}^{n}x_{i}y_{i}=62\times77.3+56.9\times73.1+89.3\times81.7+86.4\times79.8+76\times77.5+79.3\times80.2+81.7\times79.2+67.2\times78.2) (=4792.6+4159.39+7295.81+6894.72+5890+6359.86+6470.64+5255.04 = 46018.06)

Step2: Calculate slope (b_1)

The formula for the slope (b_1) of the regression - line (y = b_0 + b_1x) is (b_1=\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}}) (n\sum_{i = 1}^{n}x_{i}y_{i}- \sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}=8\times46018.06-608\times627) (=368144.48 - 381216=-13071.52) (n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}=8\times45776.37-608^{2}) (=366210.96 - 369664=-3453.04) (b_1=\frac{-13071.52}{-3453.04}\approx3.786)

Step3: Calculate intercept (b_0)

The formula for the intercept (b_0) is (b_0=\bar{y}-b_1\bar{x}), where (\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}) and (\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}) (\bar{x}=\frac{608}{8}=76) (\bar{y}=\frac{627}{8}=78.375) (b_0 = 78.375-3.786\times76) (=78.375 - 287.736=-209.361)

The regression equation is (y=-209.361 + 3.786x)

Step4: Predict (y) value

When (x = 69.6), (y=-209.361+3.786\times69.6) (y=-209.361 + 263.5056) (y = 54.1446\approx54.1)

Answer:

(54.1)