create a linear model for the data in the table.\n\nwrite a linear model for the data in the table.\ny =…

create a linear model for the data in the table.\n\nwrite a linear model for the data in the table.\ny = □x+(□)\n(type integers or decimals rounded to three decimal places as needed.)

create a linear model for the data in the table.\n\nwrite a linear model for the data in the table.\ny = □x+(□)\n(type integers or decimals rounded to three decimal places as needed.)

Answer

Explanation:

Step1: Calculate the slope $m$

The formula for the slope of a line given two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let's take the first two - points $(x_1 = 4,y_1 = 7)$ and $(x_2 = 7,y_2 = 16)$. $m=\frac{16 - 7}{7 - 4}=\frac{9}{3}=3$. We can also use the least - squares formula $m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}$. $n = 6$, $\sum_{i=1}^{6}x_i=4 + 7+10 + 13+16+19=69$, $\sum_{i = 1}^{6}y_i=7 + 16+20+29+38+45 = 155$, $\sum_{i=1}^{6}x_i^{2}=4^{2}+7^{2}+10^{2}+13^{2}+16^{2}+19^{2}=16 + 49+100+169+256+361 = 951$, $\sum_{i = 1}^{6}x_iy_i=4\times7+7\times16+10\times20+13\times29+16\times38+19\times45=28+112+200+377+608+855 = 2180$. $m=\frac{6\times2180 - 69\times155}{6\times951-(69)^{2}}=\frac{13080-10695}{5706 - 4761}=\frac{2385}{945}=2.524$.

Step2: Calculate the y - intercept $b$

We use the point - slope form $y=mx + b$ and substitute one of the points, say $(x = 4,y = 7)$ and $m = 2.524$. $7=2.524\times4 + b$. $b=7-10.096=- 3.096$.

Answer:

$y = 2.524x+(-3.096)$