Question

using the regression calculator ria 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 = \boxed{} x + \boxed{} week miles run 1 5 2 8 4 13 6 15 8 19 10 20

Explanation:

Step1: Identify data points

Let \( x \) be the week and \( y \) be miles run. The data points are \((1, 5)\), \((2, 8)\), \((4, 13)\), \((6, 15)\), \((8, 19)\), \((10, 20)\).

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

Mean of \( x \) (\(\bar{x}\)):
\(\bar{x} = \frac{1 + 2 + 4 + 6 + 8 + 10}{6} = \frac{31}{6} \approx 5.1667\)

Mean of \( y \) (\(\bar{y}\)):
\(\bar{y} = \frac{5 + 8 + 13 + 15 + 19 + 20}{6} = \frac{80}{6} \approx 13.3333\)

Step3: Calculate slope (\(m\))

Slope formula: \( m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \)

Compute \((x_i - \bar{x})(y_i - \bar{y})\) for each point:

• \((1 - 5.1667)(5 - 13.3333) \approx (-4.1667)(-8.3333) \approx 34.7222\)
• \((2 - 5.1667)(8 - 13.3333) \approx (-3.1667)(-5.3333) \approx 16.8889\)
• \((4 - 5.1667)(13 - 13.3333) \approx (-1.1667)(-0.3333) \approx 0.3889\)
• \((6 - 5.1667)(15 - 13.3333) \approx (0.8333)(1.6667) \approx 1.3889\)
• \((8 - 5.1667)(19 - 13.3333) \approx (2.8333)(5.6667) \approx 16.0556\)
• \((10 - 5.1667)(20 - 13.3333) \approx (4.8333)(6.6667) \approx 32.2222\)

Sum of these: \( 34.7222 + 16.8889 + 0.3889 + 1.3889 + 16.0556 + 32.2222 \approx 101.6667 \)

Compute \((x_i - \bar{x})^2\) for each point:

• \((1 - 5.1667)^2 \approx 17.3611\)
• \((2 - 5.1667)^2 \approx 10.0278\)
• \((4 - 5.1667)^2 \approx 1.3611\)
• \((6 - 5.1667)^2 \approx 0.6944\)
• \((8 - 5.1667)^2 \approx 8.0278\)
• \((10 - 5.1667)^2 \approx 23.3611\)

Sum of these: \( 17.3611 + 10.0278 + 1.3611 + 0.6944 + 8.0278 + 23.3611 \approx 60.8333 \)

Slope \( m = \frac{101.6667}{60.8333} \approx 1.671\) (to thousandths)

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

Using \( \bar{y} = m\bar{x} + b \):
\( 13.3333 = 1.671(5.1667) + b \)
\( 13.3333 \approx 8.633 + b \)
\( b \approx 13.3333 - 8.633 \approx 4.700\) (to thousandths)

Answer:

\( y = \boldsymbol{1.671}x + \boldsymbol{4.700} \)