a pediatrician wants to determine the relation that exists between a childs height, x, and head…

a pediatrician wants to determine the relation that exists between a childs height, x, and head circumference, y. she randomly selects 11 children from her practice, measures their heights and head circumferences and obtains the accompanying data. complete parts (a) through (e). click the icon to view the data table. (a) find the least - squares regression line treating height as the explanatory variable and head circumference as the response variable. the least - squares regression line is $hat{y}=square x+square$. (round to four decimal places as needed.) data table height, x (inches) head circumference, y (inches) height, x (inches) head circumference, y (inches) 27.75 17.7 26.5 17.3 24.5 17.1 27.5 17.5 25.5 17.1 26.75 17.3 25.5 17.3 26.75 17.5 25 16.9 27.5 17.5 27.75 17.6

a pediatrician wants to determine the relation that exists between a childs height, x, and head circumference, y. she randomly selects 11 children from her practice, measures their heights and head circumferences and obtains the accompanying data. complete parts (a) through (e). click the icon to view the data table. (a) find the least - squares regression line treating height as the explanatory variable and head circumference as the response variable. the least - squares regression line is $hat{y}=square x+square$. (round to four decimal places as needed.) data table height, x (inches) head circumference, y (inches) height, x (inches) head circumference, y (inches) 27.75 17.7 26.5 17.3 24.5 17.1 27.5 17.5 25.5 17.1 26.75 17.3 25.5 17.3 26.75 17.5 25 16.9 27.5 17.5 27.75 17.6

Answer

Explanation:

Step1: Calculate the necessary sums

Let (n = 11). Calculate (\sum_{i = 1}^{n}x_i), (\sum_{i = 1}^{n}y_i), (\sum_{i = 1}^{n}x_i^2), (\sum_{i = 1}^{n}x_iy_i). (\sum_{i = 1}^{11}x_i=27.75 + 24.5+25.5 + 25.5+25+27.75+26.5+27.5+26.75+26.75+27.5 = 291) (\sum_{i = 1}^{11}y_i=17.7+17.1 + 17.1+17.3+16.9+17.6+17.3+17.5+17.3+17.5+17.5 = 190.8) (\sum_{i = 1}^{11}x_i^2=27.75^2+24.5^2+25.5^2+25.5^2+25^2+27.75^2+26.5^2+27.5^2+26.75^2+26.75^2+27.5^2=7779.5) (\sum_{i = 1}^{11}x_iy_i=27.75\times17.7+24.5\times17.1+25.5\times17.1+25.5\times17.3+25\times16.9+27.75\times17.6+26.5\times17.3+27.5\times17.5+26.75\times17.3+26.75\times17.5+27.5\times17.5 = 5039.45)

Step2: Calculate the slope (b_1)

The formula for the slope (b_1) of the least - squares regression line 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}) [ \begin{align*} b_1&=\frac{11\times5039.45-291\times190.8}{11\times7779.5-(291)^2}\ &=\frac{55433.95 - 55522.8}{85574.5 - 84681}\ &=\frac{- 88.85}{893.5}\ &\approx - 0.0994 \end{align*} ]

Step3: Calculate the y - intercept (b_0)

The formula for the y - 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{291}{11}\approx26.4545), (\bar{y}=\frac{190.8}{11}\approx17.3455) [ \begin{align*} b_0&=17.3455-(- 0.0994)\times26.4545\ &=17.3455 + 2.6296\ &=19.9751 \end{align*} ]

Answer:

(\hat{y}=-0.0994x + 19.9751)