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 +

Explanation:

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

Let \(x\) be the week number and \(y\) be the miles run.
\(\bar{x}=\frac{1 + 2+4+6+8+10}{6}=\frac{31}{6}\approx5.167\)
\(\bar{y}=\frac{5 + 8+13+15+19+20}{6}=\frac{80}{6}\approx13.333\)

Step2: Calculate the numerator and denominator for the slope \(m\)

The formula for the slope \(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
\((x_1-\bar{x})(y_1 - \bar{y})=(1 - 5.167)(5 - 13.333)=(- 4.167)(-8.333)\approx34.722\)
\((x_2-\bar{x})(y_2 - \bar{y})=(2 - 5.167)(8 - 13.333)=(-3.167)(-5.333)\approx16.944\)
\((x_3-\bar{x})(y_3 - \bar{y})=(4 - 5.167)(13 - 13.333)=(-1.167)(-0.333)\approx0.389\)
\((x_4-\bar{x})(y_4 - \bar{y})=(6 - 5.167)(15 - 13.333)=(0.833)(1.667)\approx1.390\)
\((x_5-\bar{x})(y_5 - \bar{y})=(8 - 5.167)(19 - 13.333)=(2.833)(5.667)\approx16.074\)
\((x_6-\bar{x})(y_6 - \bar{y})=(10 - 5.167)(20 - 13.333)=(4.833)(6.667)\approx32.222\)
\(\sum_{i = 1}^{6}(x_{i}-\bar{x})(y_{i}-\bar{y})=34.722 + 16.944+0.389+1.390+16.074+32.222 = 101.741\)

\((x_1-\bar{x})^{2}=(1 - 5.167)^{2}=(-4.167)^{2}\approx17.364\)
\((x_2-\bar{x})^{2}=(2 - 5.167)^{2}=(-3.167)^{2}\approx10.029\)
\((x_3-\bar{x})^{2}=(4 - 5.167)^{2}=(-1.167)^{2}\approx1.362\)
\((x_4-\bar{x})^{2}=(6 - 5.167)^{2}=(0.833)^{2}\approx0.694\)
\((x_5-\bar{x})^{2}=(8 - 5.167)^{2}=(2.833)^{2}\approx8.026\)
\((x_6-\bar{x})^{2}=(10 - 5.167)^{2}=(4.833)^{2}\approx23.358\)
\(\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=17.364+10.029 + 1.362+0.694+8.026+23.358=60.833\)

\(m=\frac{101.741}{60.833}\approx1.672\)

Step3: Calculate the \(y\) - intercept \(b\)

Using the formula \(y=mx + b\), substitute \(m\), \(\bar{x}\), and \(\bar{y}\)
\(13.333=1.672\times5.167+b\)
\(13.333 = 8.630+b\)
\(b=13.333 - 8.630=4.703\)

Answer:

\(y = 1.672x+4.703\)