Question

(b) find the equation of the least - squares regression line.
for the least squares regression line $hat{y}=a + bx$, the slope $b$ is calculated using the equation below.
$b=\frac{sum(x - \bar{x})(y - \bar{y})}{sum(x - \bar{x})^2}$
refer to the data below.

age group	representative age (midpoint of age group)	median six - minute walk distance (meters)
6 - 8	7.0	582.0
9 - 11	10.0	665.3
12 - 15	13.5	699.1
16 - 18	17.0	725.6

calculate $\bar{x}$ and $\bar{y}$.
$\bar{x}=square$
$\bar{y}=square$

Explanation:

Step1: Calculate the mean of \(x\) values

The \(x\) - values are \(4.0,7.0,10.0,13.5,17.0\). The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n = 5\), \(\sum_{i=1}^{5}x_{i}=4.0 + 7.0+10.0 + 13.5+17.0=51.5\). So \(\bar{x}=\frac{51.5}{5}=10.3\).

Step2: Calculate the mean of \(y\) values

The \(y\) - values are \(541.3,582.0,665.3,699.1,725.6\). The formula for the mean \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\), where \(n = 5\), \(\sum_{i=1}^{5}y_{i}=541.3+582.0 + 665.3+699.1+725.6=3213.3\). So \(\bar{y}=\frac{3213.3}{5}=642.66\).

Answer:

\(\bar{x}=10.3\), \(\bar{y}=642.66\)