Question

create a linear model for the data in the table.
write a linear model for the data in the table.
y = \square x + \square
(type integers or decimals rounded to three decimal places as needed.)
the table has columns x and y with rows: x=5,y=6; x=7,y=14; x=9,y=20; x=11,y=28; x=13,y=38; x=15,y=45

Explanation:

Step1: List data points

Data: $(5,6), (7,14), (9,20), (11,28), (13,38), (15,45)$

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

$\bar{x} = \frac{5+7+9+11+13+15}{6} = \frac{60}{6} = 10$
$\bar{y} = \frac{6+14+20+28+38+45}{6} = \frac{151}{6} \approx 25.1667$

Step3: Compute slope $m$

$m = \frac{\sum_{i=1}^{6}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^{6}(x_i-\bar{x})^2}$
Numerator:
$(5-10)(6-25.1667)+(7-10)(14-25.1667)+(9-10)(20-25.1667)+(11-10)(28-25.1667)+(13-10)(38-25.1667)+(15-10)(45-25.1667)$
$= (-5)(-19.1667)+(-3)(-11.1667)+(-1)(-5.1667)+(1)(2.8333)+(3)(12.8333)+(5)(19.8333)$
$= 95.8335 + 33.5001 + 5.1667 + 2.8333 + 38.4999 + 99.1665 = 275$
Denominator:
$(5-10)^2+(7-10)^2+(9-10)^2+(11-10)^2+(13-10)^2+(15-10)^2$
$=25+9+1+1+9+25=70$
$m = \frac{275}{70} \approx 3.9286$

Step4: Compute y-intercept $b$

$b = \bar{y} - m\bar{x}$
$b = 25.1667 - 3.9286\times10 = 25.1667 - 39.286 = -14.1193$

Answer:

$y=3.929x + (-14.119)$