Question

the table below shows the average heart rate, x, and calories burned, y, for seven men on an olympic rowing team during a one - hour workout class.

write the linear regression function that models the data. round all values to the nearest tenth.

enter your answer in the boxes.

$f(x)=\square x+\square$

Explanation:

Step 1: Calcular la media de \(x\) y \(y\)

Sea \(x=(135,147,150,144,146,153,143)\) y \(y=(725,812,866,761,825,863,737)\)
\(\bar{x}=\frac{135 + 147+150+144+146+153+143}{7}=\frac{1018}{7}=145.4286\)
\(\bar{y}=\frac{725 + 812+866+761+825+863+737}{7}=\frac{5599}{7}=799.8571\)

Step 2: Calcular la pendiente \(m\)

\(m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}\)
\(\sum_{i = 1}^{7}(x_i-\bar{x})(y_i-\bar{y})=(135 - 145.4286)(725 - 799.8571)+(147 - 145.4286)(812 - 799.8571)+(150 - 145.4286)(866 - 799.8571)+(144 - 145.4286)(761 - 799.8571)+(146 - 145.4286)(825 - 799.8571)+(153 - 145.4286)(863 - 799.8571)+(143 - 145.4286)(737 - 799.8571)\)
\(=(- 10.4286)(-74.8571)+(1.5714)(12.1429)+(4.5714)(66.1429)+(-1.4286)(-38.8571)+(0.5714)(25.1429)+(7.5714)(63.1429)+(-2.4286)(-62.8571)\)
\(=780.95+19.09+302.45+55.51+14.37+478.16+152.78=1793.31\)
\(\sum_{i = 1}^{7}(x_i-\bar{x})^2=(135 - 145.4286)^2+(147 - 145.4286)^2+(150 - 145.4286)^2+(144 - 145.4286)^2+(146 - 145.4286)^2+(153 - 145.4286)^2+(143 - 145.4286)^2\)
\(=(-10.4286)^2+(1.5714)^2+(4.5714)^2+(-1.4286)^2+(0.5714)^2+(7.5714)^2+(-2.4286)^2\)
\(=108.76+2.47+20.89+2.04+0.33+57.33+5.90=197.72\)
\(m=\frac{1793.31}{197.72}\approx9.1\)

Step 3: Calcular la intersección \(b\)

\(b=\bar{y}-m\bar{x}\)
\(b = 799.8571-9.1\times145.4286\)
\(b=799.8571 - 1323.4\)
\(b=-523.5\)

Answer:

\(f(x)=9.1x - 523.5\)