Question

1. the table displays the walking heart rate and running heart rate of eight girls in beats per minute (bpm). determine the linear best - fit model for the data. (round your answer to the hundredths place.) y = \square x + \square

Explanation:

Step1: Identify variables

Let \( x \) be Walking Heart Rate (independent variable) and \( y \) be Running Heart Rate (dependent variable). The data points are: \((66, 128)\), \((72, 136)\), \((74, 134)\), \((78, 138)\), \((80, 142)\), \((84, 146)\), \((86, 148)\), \((88, 152)\).

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

Mean of \( x \) (\(\bar{x}\)):
\(\bar{x} = \frac{66 + 72 + 74 + 78 + 80 + 84 + 86 + 88}{8} = \frac{628}{8} = 78.5\)

Mean of \( y \) (\(\bar{y}\)):
\(\bar{y} = \frac{128 + 136 + 134 + 138 + 142 + 146 + 148 + 152}{8} = \frac{1124}{8} = 140.5\)

Step3: Calculate slope (\(m\))

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

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

• \((66 - 78.5)(128 - 140.5) = (-12.5)(-12.5) = 156.25\)
• \((72 - 78.5)(136 - 140.5) = (-6.5)(-4.5) = 29.25\)
• \((74 - 78.5)(134 - 140.5) = (-4.5)(-6.5) = 29.25\)
• \((78 - 78.5)(138 - 140.5) = (-0.5)(-2.5) = 1.25\)
• \((80 - 78.5)(142 - 140.5) = (1.5)(1.5) = 2.25\)
• \((84 - 78.5)(146 - 140.5) = (5.5)(5.5) = 30.25\)
• \((86 - 78.5)(148 - 140.5) = (7.5)(7.5) = 56.25\)
• \((88 - 78.5)(152 - 140.5) = (9.5)(11.5) = 109.25\)

Sum of \((x_i - \bar{x})(y_i - \bar{y})\):
\(156.25 + 29.25 + 29.25 + 1.25 + 2.25 + 30.25 + 56.25 + 109.25 = 414\)

Sum of \((x_i - \bar{x})^2\):
\((-12.5)^2 + (-6.5)^2 + (-4.5)^2 + (-0.5)^2 + (1.5)^2 + (5.5)^2 + (7.5)^2 + (9.5)^2\)
\(= 156.25 + 42.25 + 20.25 + 0.25 + 2.25 + 30.25 + 56.25 + 90.25 = 398\)

Slope \(m = \frac{414}{398} \approx 1.04\) (rounded to hundredths).

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

Using \(y = mx + b\) and \(\bar{y} = m\bar{x} + b\):
\(140.5 = 1.04(78.5) + b\)
\(140.5 = 81.64 + b\)
\(b = 140.5 - 81.64 = 58.86\)

Answer:

\(y = 1.04x + 58.86\) (slope \(1.04\), y-intercept \(58.86\))