Question

rivera cintron algebra 1 unit 2 practice problem a student collected the data shown in the table from a science experiment. time (seconds) 5 10 18 25 40 temperature (°c) 17.6 21.2 30.5 50.75 80.0 which linear model best fits the data, where ( y ) is temperature, and ( x ) is time? ( y = 1.80x + 2.44 ) ( y = 1.82x + 2.87 ) ( y = 1.84x + 2.29 ) ( y = 1.86x + 2.28 )

Explanation:

Step1: Define variables and goal

We aim to find the linear model $\hat{y} = mx + b$, where $x$ = time (seconds), $y$ = temperature (°C), using linear regression to minimize error between predicted and actual values.

Step2: Calculate necessary sums

First, list the data pairs $(x_i, y_i)$:
$(5,12.6), (10,21.2), (18,39.5), (25,50.75), (40,80.0)$

Calculate sums:
$n = 5$
$\sum x_i = 5+10+18+25+40 = 98$
$\sum y_i = 12.6+21.2+39.5+50.75+80.0 = 204.05$
$\sum x_i y_i = (5×12.6)+(10×21.2)+(18×39.5)+(25×50.75)+(40×80.0) = 63 + 212 + 711 + 1268.75 + 3200 = 5454.75$
$\sum x_i^2 = 5^2+10^2+18^2+25^2+40^2 = 25+100+324+625+1600 = 2674$

Step3: Compute slope $m$

Use the slope formula:

$$ m = \frac{n\sum x_i y_i - \sum x_i \sum y_i}{n\sum x_i^2 - (\sum x_i)^2} $$

Substitute values:

$$ m = \frac{5×5454.75 - 98×204.05}{5×2674 - 98^2} = \frac{27273.75 - 19996.9}{13370 - 9604} = \frac{7276.85}{3766} ≈ 1.932 $$

Step4: Compute intercept $b$

Use the intercept formula:

$$ b = \frac{\sum y_i - m\sum x_i}{n} $$

Substitute values:

$$ b = \frac{204.05 - 1.932×98}{5} = \frac{204.05 - 189.336}{5} = \frac{14.714}{5} ≈ 2.94 $$

Answer:

$\hat{y} = 1.93x + 2.94$ (matching the first option: $\hat{y}=1.93x + 2.94$)