bivariate data for the quantitative variables x and y are given in the table below. these data are plotted…

bivariate data for the quantitative variables x and y are given in the table below. these data are plotted in the scatter plot shown next to the table. in the scatter plot, sketch an approximation of the least - squares regression line for the data.\n\n|x|y|\n|----|----|\n|1.6|3.7|\n|3.7|3.7|\n|8.6|7.0|\n|5.2|7.0|\n|9.2|8.2|\n|4.3|4.4|\n|10.3|7.9|\n|2.1|3.9|\n|7.2|7.0|\n|8.2|8.2|\n|4.8|5.1|\n|6.5|5.9|\n|9.7|7.8|\n|2.5|2.9|\n|6.8|5.7|\n|5.9|7.8|\n|2.8|3.9|\n|3.8|5.4|

bivariate data for the quantitative variables x and y are given in the table below. these data are plotted in the scatter plot shown next to the table. in the scatter plot, sketch an approximation of the least - squares regression line for the data.\n\n|x|y|\n|----|----|\n|1.6|3.7|\n|3.7|3.7|\n|8.6|7.0|\n|5.2|7.0|\n|9.2|8.2|\n|4.3|4.4|\n|10.3|7.9|\n|2.1|3.9|\n|7.2|7.0|\n|8.2|8.2|\n|4.8|5.1|\n|6.5|5.9|\n|9.7|7.8|\n|2.5|2.9|\n|6.8|5.7|\n|5.9|7.8|\n|2.8|3.9|\n|3.8|5.4|

Answer

Explanation:

Step1: Calculate means of x and y

Let (n = 18). (\bar{x}=\frac{1}{n}\sum_{i = 1}^{n}x_{i}=\frac{1.6 + 3.7+8.6+5.2+9.2+4.3+10.3+2.1+7.2+8.2+4.8+6.5+9.7+2.5+6.8+5.9+2.8+3.8}{18}) (\bar{x}=\frac{100.7}{18}\approx5.594) (\bar{y}=\frac{1}{n}\sum_{i = 1}^{n}y_{i}=\frac{3.7 + 3.7+7.0+7.0+8.2+4.4+7.9+3.9+7.0+8.2+5.1+5.9+7.8+2.9+5.7+7.8+3.9+5.4}{18}) (\bar{y}=\frac{107.6}{18}\approx5.978)

Step2: Calculate 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) and sum them up. (\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})) and (\sum_{i = 1}^{n}(x_{i}-\bar{x})^2) values are obtained by: For example, for (x_1 = 1.6,y_1 = 3.7): ((1.6 - 5.594)(3.7- 5.978)) and ((1.6 - 5.594)^2) After calculating all 18 - pairs and summing: (\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\approx97.97) (\sum_{i = 1}^{n}(x_{i}-\bar{x})^2\approx109.17) (b_1=\frac{97.97}{109.17}\approx0.897)

Step3: Calculate intercept (b_0)

(b_0=\bar{y}-b_1\bar{x}) (b_0 = 5.978-0.897\times5.594) (b_0=5.978 - 5.018\approx0.96) The least - squares regression line is (y = b_0 + b_1x=0.96+0.897x) To sketch the line on the scatter - plot:

  1. Locate the point ((\bar{x},\bar{y})\approx(5.594,5.978)) on the scatter - plot.
  2. Use the slope (b_1 = 0.897\approx0.9). From the point ((\bar{x},\bar{y})), for a 1 - unit increase in (x), the (y) - value increases by approximately (0.9) units. Draw a straight line passing through points estimated using the slope and the point ((\bar{x},\bar{y})).

Answer:

Sketch the line (y = 0.96+0.897x) on the scatter - plot using the steps above.