Question

rita is starting a running program. the table shows the total number of miles she runs in different weeks. what is the equation of the line of best fit for the data? state each number to the thousandths place. y≈□x+□ week | miles run 1 | 5 2 | 8 4 | 13 6 | 15 8 | 19 10 | 20

Explanation:

Step1: List the data points

We have the data points \((x, y)\) as: \((1, 5)\), \((2, 8)\), \((4, 13)\), \((6, 15)\), \((8, 19)\), \((10, 20)\)

Step2: Calculate the mean of \(x\) and \(y\)

First, calculate \(\bar{x}\) (mean of \(x\)):
\(x\) values: \(1, 2, 4, 6, 8, 10\)
\(\bar{x}=\frac{1 + 2+4 + 6+8 + 10}{6}=\frac{31}{6}\approx5.1667\)

\(y\) values: \(5, 8, 13, 15, 19, 20\)
\(\bar{y}=\frac{5 + 8+13 + 15+19 + 20}{6}=\frac{80}{6}\approx13.3333\)

Step3: Calculate the slope \(m\)

The formula for the slope of the line of best fit (linear regression) is:
\(m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}\)

First, calculate \((x_i-\bar{x})(y_i-\bar{y})\) for each data point:

• For \((1, 5)\): \((1 - 5.1667)(5 - 13.3333)\approx(- 4.1667)(-8.3333)\approx34.7222\)
• For \((2, 8)\): \((2 - 5.1667)(8 - 13.3333)\approx(-3.1667)(-5.3333)\approx16.8056\)
• For \((4, 13)\): \((4 - 5.1667)(13 - 13.3333)\approx(-1.1667)(-0.3333)\approx0.3889\)
• For \((6, 15)\): \((6 - 5.1667)(15 - 13.3333)\approx(0.8333)(1.6667)\approx1.3889\)
• For \((8, 19)\): \((8 - 5.1667)(19 - 13.3333)\approx(2.8333)(5.6667)\approx16.0556\)
• For \((10, 20)\): \((10 - 5.1667)(20 - 13.3333)\approx(4.8333)(6.6667)\approx32.2222\)

Sum of \((x_i-\bar{x})(y_i-\bar{y})\): \(34.7222+16.8056 + 0.3889+1.3889+16.0556+32.2222\approx101.5834\)

Now, calculate \((x_i-\bar{x})^2\) for each data point:

• For \((1, 5)\): \((1 - 5.1667)^2\approx(-4.1667)^2\approx17.3611\)
• For \((2, 8)\): \((2 - 5.1667)^2\approx(-3.1667)^2\approx10.0278\)
• For \((4, 13)\): \((4 - 5.1667)^2\approx(-1.1667)^2\approx1.3611\)
• For \((6, 15)\): \((6 - 5.1667)^2\approx(0.8333)^2\approx0.6944\)
• For \((8, 19)\): \((8 - 5.1667)^2\approx(2.8333)^2\approx8.0278\)
• For \((10, 20)\): \((10 - 5.1667)^2\approx(4.8333)^2\approx23.3611\)

Sum of \((x_i-\bar{x})^2\): \(17.3611 + 10.0278+1.3611+0.6944+8.0278+23.3611\approx60.8333\)

Now, calculate the slope \(m\):
\(m=\frac{101.5834}{60.8333}\approx1.6699\approx1.670\) (to the thousandths place)

Step4: Calculate the y - intercept \(b\)

The formula for the y - intercept is \(b=\bar{y}-m\bar{x}\)

We know that \(\bar{y}\approx13.3333\), \(m\approx1.6699\), and \(\bar{x}\approx5.1667\)

\(b = 13.3333-1.6699\times5.1667\)

\(1.6699\times5.1667\approx1.6699\times5 + 1.6699\times0.1667\approx8.3495+0.2783\approx8.6278\)

\(b=13.3333 - 8.6278\approx4.7055\approx4.706\) (to the thousandths place)

Answer:

\(y\approx\boxed{1.670}x+\boxed{4.706}\)