Question

during the summer, rafaels toy shop collects data that implies a relationship between the weekly average temperature and the number of kites sold.
weekly temp °f (x) kites sold (y)
73.8 6
75.7 6
88.7 8
92.1 8
72.5 6
74.6 5

choose the correct equation for the least squares line.
a. $y=4.08x - 0.133$
b. $y=0.133x - 4.08$
c. $y=0.133x + 4.08$

Explanation:

Step1: Calculate $\bar{x}$ and $\bar{y}$

First, find the mean of $x$ (temperatures) and $y$ (kites sold):
$\bar{x} = \frac{73.8 + 75.7 + 88.7 + 92.1 + 72.5 + 74.6}{6} = \frac{477.4}{6} \approx 79.57$
$\bar{y} = \frac{6 + 6 + 8 + 8 + 6 + 5}{6} = \frac{39}{6} = 6.5$

Step2: Calculate slope $m$

Use the formula $m = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sum(x_i - \bar{x})^2}$:
First compute numerator:
$(73.8-79.57)(6-6.5)+(75.7-79.57)(6-6.5)+(88.7-79.57)(8-6.5)+(92.1-79.57)(8-6.5)+(72.5-79.57)(6-6.5)+(74.6-79.57)(5-6.5)$
$= (-5.77)(-0.5)+(-3.87)(-0.5)+(9.13)(1.5)+(12.53)(1.5)+(-7.07)(-0.5)+(-4.97)(-1.5)$
$= 2.885 + 1.935 + 13.695 + 18.795 + 3.535 + 7.455 = 48.3$

Denominator:
$(73.8-79.57)^2+(75.7-79.57)^2+(88.7-79.57)^2+(92.1-79.57)^2+(72.5-79.57)^2+(74.6-79.57)^2$
$= (-5.77)^2+(-3.87)^2+(9.13)^2+(12.53)^2+(-7.07)^2+(-4.97)^2$
$= 33.29 + 14.98 + 83.36 + 156.99 + 49.98 + 24.70 = 363.3$
$m = \frac{48.3}{363.3} \approx 0.133$

Step3: Calculate intercept $b$

Use $\bar{y} = m\bar{x} + b$:
$6.5 = 0.133 \times 79.57 + b$
$6.5 \approx 10.58 + b$
$b = 6.5 - 10.58 \approx -4.08$

Step4: Form the regression line

The line is $y = mx + b$

Answer:

B. $y=0.133x - 4.08$