Question

wave height 3 6 5 1 4
number of surfers 24 61 56 15 35
using the line of best fit equation, how many approximate surfers would you expect if the wave height was 8? i am assuming this in measured in feet.
77
80
75
82

Explanation:

Step1: Define variables and data

Let $x$ = Wave Height, $y$ = Number of Surfers
Data pairs: $(3,24), (6,61), (5,56), (1,15), (4,35)$

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

$\bar{x} = \frac{3+6+5+1+4}{5} = \frac{19}{5} = 3.8$
$\bar{y} = \frac{24+61+56+15+35}{5} = \frac{191}{5} = 38.2$

Step3: Calculate slope $m$

First compute numerator and denominator:
Numerator: $\sum(x_i-\bar{x})(y_i-\bar{y})$
$=(3-3.8)(24-38.2)+(6-3.8)(61-38.2)+(5-3.8)(56-38.2)+(1-3.8)(15-38.2)+(4-3.8)(35-38.2)$
$=(-0.8)(-14.2)+(2.2)(22.8)+(1.2)(17.8)+(-2.8)(-23.2)+(0.2)(-3.2)$
$=11.36+49.96+21.36+64.96-0.64 = 146.92$
Denominator: $\sum(x_i-\bar{x})^2$
$=(3-3.8)^2+(6-3.8)^2+(5-3.8)^2+(1-3.8)^2+(4-3.8)^2$
$=0.64+4.84+1.44+7.84+0.04 = 14.8$
$m = \frac{146.92}{14.8} \approx 9.927$

Step4: Calculate y-intercept $b$

$b = \bar{y} - m\bar{x}$
$b = 38.2 - (9.927)(3.8) \approx 38.2 - 37.723 = 0.477$

Step5: Predict for $x=8$

$y = mx + b$
$y \approx 9.927(8) + 0.477 \approx 79.416 + 0.477 \approx 79.893$

Answer:

B. 80