Question

a researcher hopes to determine whether the number of hours a person jogs per week is related to the persons age.
age, x | 49 | 46 | 18 | 52 | 62
hours, y | 1.5 | 2 | 5.5 | 1.5 | 1
r ≈ - 0.98
part: 0 / 3
part 1 of 3
find the equation of the regression line and draw the line on the scatter plot, but only if r is significant. round the slope and y - intercept to four decimal places, if necessary.
r is significant at the 5% level.
the equation of the regression line is y = □ - □x.

Explanation:

Step1: Recall regression - line formula

The regression - line equation is $y = a+bx$, where $b=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}$ and $a=\bar{y}-b\bar{x}$, $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$, $\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}$, and $n$ is the number of data points.
Let $x$ be age and $y$ be hours of jogging. Here $n = 5$, $\sum_{i=1}^{5}x_i=49 + 46+18+52+62=227$, $\sum_{i = 1}^{5}y_i=1.5 + 2+5.5+1.5+1=11.5$, $\sum_{i=1}^{5}x_i^{2}=49^{2}+46^{2}+18^{2}+52^{2}+62^{2}=2401+2116+324+2704+3844 = 11389$, $\sum_{i = 1}^{5}x_iy_i=49\times1.5+46\times2+18\times5.5+52\times1.5+62\times1=73.5 + 92+99+78+62=404.5$.

Step2: Calculate $\bar{x}$ and $\bar{y}$

$\bar{x}=\frac{\sum_{i = 1}^{5}x_i}{5}=\frac{227}{5}=45.4$, $\bar{y}=\frac{\sum_{i = 1}^{5}y_i}{5}=\frac{11.5}{5}=2.3$.

Step3: Calculate the slope $b$

$b=\frac{5\times404.5-227\times11.5}{5\times11389-(227)^{2}}=\frac{2022.5-2610.5}{56945 - 51529}=\frac{- 588}{5416}\approx - 0.1086$.

Step4: Calculate the $y$ - intercept $a$

$a=\bar{y}-b\bar{x}=2.3-(-0.1086)\times45.4=2.3 + 4.9204=7.2204$.

Answer:

$y = 7.2204-0.1086x$