find the equation for the least squares regression line of the data described below. a team of forestry…

find the equation for the least squares regression line of the data described below. a team of forestry specialists from the hartman lumber company visited a section of the forest that was going to be logged for timber. they took measurements to evaluate the health and growth of trees in the area. the team counted the number of rings on several trees to find their ages (in years), x. they also measured the diameter of these trees (in meters), y. age (in years) diameter (in meters) 36 0.9 50 1.1 52 1.7 54 2.4 63 2.8 round your answers to the nearest thousandth. y = x +

find the equation for the least squares regression line of the data described below. a team of forestry specialists from the hartman lumber company visited a section of the forest that was going to be logged for timber. they took measurements to evaluate the health and growth of trees in the area. the team counted the number of rings on several trees to find their ages (in years), x. they also measured the diameter of these trees (in meters), y. age (in years) diameter (in meters) 36 0.9 50 1.1 52 1.7 54 2.4 63 2.8 round your answers to the nearest thousandth. y = x +

Answer

Explanation:

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

Let (x_1 = 36,x_2=50,x_3 = 52,x_4=54,x_5 = 63) and (y_1 = 0.9,y_2=1.1,y_3 = 1.7,y_4=2.4,y_5 = 2.8). The number of data - points (n = 5). The mean of (x), (\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{36 + 50+52+54+63}{5}=\frac{255}{5}=51). The mean of (y), (\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{0.9 + 1.1+1.7+2.4+2.8}{5}=\frac{8.9}{5}=1.78).

Step2: Calculate the numerator and denominator for the slope (b_1)

The numerator (S_{xy}=\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})) ((x_1-\bar{x})(y_1 - \bar{y})=(36 - 51)(0.9-1.78)=(-15)\times(-0.88) = 13.2) ((x_2-\bar{x})(y_2 - \bar{y})=(50 - 51)(1.1 - 1.78)=(-1)\times(-0.68)=0.68) ((x_3-\bar{x})(y_3 - \bar{y})=(52 - 51)(1.7 - 1.78)=1\times(-0.08)=-0.08) ((x_4-\bar{x})(y_4 - \bar{y})=(54 - 51)(2.4 - 1.78)=3\times0.62 = 1.86) ((x_5-\bar{x})(y_5 - \bar{y})=(63 - 51)(2.8 - 1.78)=12\times1.02 = 12.24) (S_{xy}=13.2 + 0.68-0.08+1.86+12.24=27.9) The denominator (S_{xx}=\sum_{i = 1}^{n}(x_i-\bar{x})^2) ((x_1-\bar{x})^2=(36 - 51)^2=(-15)^2 = 225) ((x_2-\bar{x})^2=(50 - 51)^2=(-1)^2 = 1) ((x_3-\bar{x})^2=(52 - 51)^2=1^2 = 1) ((x_4-\bar{x})^2=(54 - 51)^2=3^2 = 9) ((x_5-\bar{x})^2=(63 - 51)^2=12^2 = 144) (S_{xx}=225+1 + 1+9+144=380) The slope (b_1=\frac{S_{xy}}{S_{xx}}=\frac{27.9}{380}\approx0.073)

Step3: Calculate the y - intercept (b_0)

The y - intercept (b_0=\bar{y}-b_1\bar{x}) (b_0=1.78-0.073\times51) (b_0=1.78 - 3.723=-1.943)

Answer:

(y = 0.073x-1.943)