determine the regression equation for the following data set. then, use the regression equation to determine…

Explanation:

Step1: Calculate sums

Let (n = 10). Calculate (\sum_{i = 1}^{n}x_i=1 + 2+\cdots+10=\frac{10\times(10 + 1)}{2}=55), (\sum_{i = 1}^{n}y_i=73+26+\cdots+96 = 347), (\sum_{i = 1}^{n}x_i^2=1^2+2^2+\cdots+10^2=\frac{10\times(10 + 1)\times(2\times10 + 1)}{6}=385), (\sum_{i = 1}^{n}x_iy_i=1\times73+2\times26+\cdots+10\times96 = 2107).

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_iy_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}). Substitute the values: (b_1=\frac{10\times2107-55\times347}{10\times385 - 55^2}=\frac{21070-19085}{3850 - 3025}=\frac{1985}{825}\approx2.406).

Step3: Calculate intercept (b_0)

The formula for the intercept (b_0) is (b_0=\overline{y}-b_1\overline{x}), where (\overline{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{55}{10}=5.5) and (\overline{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{347}{10}=34.7). Then (b_0 = 34.7-2.406\times5.5=34.7 - 13.233=21.467). The regression equation is (y = 21.467+2.406x).

Step4: Predict (y) for (x = 15)

Substitute (x = 15) into the regression equation: (y=21.467+2.406\times15=21.467 + 36.09=776.18).

Answer:

776.18