Create a linear model for the data in the table.

Question

x
5
7
9
11
13
15

y
7
16
20
28
38
45
Question content area bottom
Part 1
Write a linear model for the data in the table.

Answer 1

To create a linear model for the given data, we can first identify the independent variable $x$ and the dependent variable $y$, and then use the method of least squares to find the slope ($m$) and y-intercept ($b$) of the linear equation in the form $y=mx+b$.

Given Data:

$$\text{Misplaced hline}$$

Step 1: Calculate the Means

Calculate the means of $x$ and $y$:

$$\overline{x}=\frac{5+7+9+11+13+15}{6}=\frac{60}{6}=10$$

$$\overline{y}=\frac{7+16+20+28+38+45}{6}=\frac{154}{6}\approx 25.67$$

Step 2: Calculate the Slope ($m$)

To calculate the slope $m$:

$$m=\frac{\sum (x_i-\overline{x})(y_i-\overline{y})}{\sum (x_i-\overline{x}{)}^2}$$

Calculating the sums:

• $\sum (x_i-\overline{x})(y_i-\overline{y})$

$$=(5-10)(7-25.67)+(7-10)(16-25.67)+(9-10)(20-25.67)+(11-10)(28-25.67)+(13-10)(38-25.67)+(15-10)(45-25.67)$$

$$=(-5)(-18.67)+(-3)(-9.67)+(-1)(-5.67)+(1)(2.33)+(3)(12.33)+(5)(19.33)$$

$$=93.35+29.01+5.67+2.33+37.00+96.65=263.01$$

• $\sum (x_i-\overline{x}{)}^2$

$$=(5-10{)}^2+(7-10{)}^2+(9-10{)}^2+(11-10{)}^2+(13-10{)}^2+(15-10{)}^2$$

$$=25+9+1+1+9+25=70$$

Now substitute these sums into the formula for $m$:

$$m=\frac{263.01}{70}\approx 3.77$$

Step 3: Calculate the Intercept ($b$)

We can calculate $b$ using the formula:

$$b=\overline{y}-m\overline{x}$$

Substituting the values:

$$b=25.67-(3.77\times 10)=25.67-37.7\approx -12.03$$

Final Model

Combining the slope and intercept, the linear model for the data is:

$$y=3.77x-12.03$$

This linear equation models the relationship between $x$ and $y$ based on the provided dataset.