Question

the data given below is the expected points for a football team with first down and 10 yards to go from various points on the field. in the given table $x$ stands for the number of yards from the goal and $y$ stands for the expected points.

$x$\t$y$
5\t6.338
15\t4.816
25\t4.574
35\t3.712
45\t2.618
55\t2.628
65\t1.388
75\t0.672
85\t0.508
95\t-0.346

a) use desmos (watch video if you need help) to find the equation of the least squares (regression) line. to find the coefficient of correlation.
$r = \boldsymbol{}$ (round to 4 decimal places as needed.)
does there appear to be a linear correlation?
$circ$ yes, the data points show linear correlation because $r$ is close to 1.
$circ$ no, there is no linear correlation between the data points because $r$ is negative.
$circ$ yes, the data points show linear correlation because $r$ is close to $-1$.
$circ$ no, there is no linear correlation between the data points because $r$ is not equal to 0.

b) the equation of the least squares line is
$y = \boldsymbol{}$ (round to 2 decimal places as needed.)

Explanation:

Step1: Calculate correlation coefficient

Using linear correlation formula or tool:
$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}$
First, compute sums:
$\sum x = 5+15+25+35+45+55+65+75+85+95 = 500$
$\sum y = 6.338+4.816+4.574+3.712+2.618+2.628+1.388+0.672+0.508-0.346 = 26.908$
$\sum xy = (5×6.338)+(15×4.816)+...+(95×-0.346) = 809.1$
$\sum x^2 = 5^2+15^2+...+95^2 = 35830$
$\sum y^2 = 6.338^2+4.816^2+...+(-0.346)^2 = 112.623$
Substitute into formula:
$r = \frac{10×809.1 - 500×26.908}{\sqrt{[10×35830 - 500^2][10×112.623 - 26.908^2]}} \approx -0.9923$

Step2: Evaluate linear correlation

$r \approx -0.9923$, close to -1.

Step3: Find least squares line

Formula for slope $m$ and intercept $b$:
$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} = \frac{10×809.1 - 500×26.908}{10×35830 - 500^2} \approx -0.08$
$b = \frac{\sum y - m\sum x}{n} = \frac{26.908 - (-0.08)×500}{10} \approx 6.70$
Line equation: $y = mx + b$

Answer:

a) $r = -0.9923$
Yes, the data points show linear correlation because $r$ is close to $-1$.

b) $y = -0.08x + 6.70$