Question

1. this table shows the temperature, in °f, x minutes after the sun sets.

based on the best - fit linear model, what is the temperature one hour after the sun sets?
47.3°f
60°f
55.5°f
67.7°f

Explanation:

Step1: Calculate mean of x and y

$x = [5,19,24,49], \bar{x} = \frac{5+19+24+49}{4} = 24.25$
$y = [67,64,62,58], \bar{y} = \frac{67+64+62+58}{4} = 62.75$

Step2: Compute slope (m)

$m = \frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum(x_i-\bar{x})^2}$
Numerator: $(5-24.25)(67-62.75)+(19-24.25)(64-62.75)+(24-24.25)(62-62.75)+(49-24.25)(58-62.75) = (-19.25)(4.25)+(-5.25)(1.25)+(-0.25)(-0.75)+(24.75)(-4.75) ≈ -81.8125-6.5625+0.1875-117.5625 ≈ -205.75$
Denominator: $(5-24.25)^2+(19-24.25)^2+(24-24.25)^2+(49-24.25)^2 ≈ 370.5625+27.5625+0.0625+612.5625 ≈ 1010.75$
$m ≈ \frac{-205.75}{1010.75} ≈ -0.2035$

Step3: Find intercept (b)

$b = \bar{y} - m\bar{x} ≈ 62.75 - (-0.2035)(24.25) ≈ 62.75 + 4.93 ≈ 67.68$

Step4: Predict at x=60

$y = -0.2035(60) + 67.68 ≈ -12.21 + 67.68 ≈ 55.47 ≈ 55.5$

Answer:

C. 55.5°F