Question

Create a linear model for the data in the table.

x: 4,7,10,13,16,19
y: 7,16,21,29,38,43
Write a linear model for the data in the table.

Answer 1

`\sf y = \boxed{ 2.419} x + \boxed{2.152 }`

Explanation:

To create a linear model for the given data, we'll use linear regression analysis. Linear regression finds the best-fit line for the data points.

The data points are as follows:

`\begin{aligned} x &: 4, 7, 10, 13, 16, 19 \\ y &: 7, 16, 21, 29, 38, 43 \end{aligned}`

Now, let's find the linear model using linear regression:

Calculate the mean (average) of x and y:

`\sf \begin{aligned} \bar{x} &= (4 + 7 + 10 + 13 + 16 + 19)/(6) \\ &= 11.5 \end{aligned}`

`\sf \begin{aligned} \sf \bar{y} &= (7 + 16 + 21 + 29 + 38 + 43)/(6)\\ & = 25.666666666666 \end{aligned}`

Calculate the sums of the products of deviations from the means:

Calculate the sum of the squares of deviations from the means for (x):

`\begin{aligned} \sum{(x - \bar{x})^2} & = (4 - 11.5)^2 + (7 - 11.5)^2 + (10 - 11.5)^2 \\ &\quad + (13 - 11.5)^2 + (16 - 11.5)^2 + (19 - 11.5)^2 \\ & = 157.5 \end{aligned}`

Use the above values to calculate the slope (m) of the best-fit line using the formula:

`\begin{aligned} m &= \frac{\sum{(x - \bar{x})(y - \bar{y})}}{\sum{(x - \bar{x})^2}} & = (381)/(157.5) \\\\ &= 2.4190476190476\end{aligned}`

Calculate the y-intercept (b) using the mean values and the slope:

So, the linear model for the data is :

y = 2.4190476190476x + −2.1523809523814

In 3 decimal places, the linear model for tha data is:

`\sf y = \boxed{ 2.419} x + \boxed{2.152 }`

This equation represents the best-fit line for the data points. It describes the relationship between x and y based on the linear regression analysis.

Answer 2

`y=2.419x+(-2.152)`

Explanation:

Given table of values:

`\begin{array}c\cline{1-7}x&4&7&10&13&16&19\\\cline{1-7}y&7&16&21&29&38&43\\\cline{1-7}\end{array}`

In this dataset, the x-values increase by 3 units from one data point to the next, however, the y-values do not increase by a consistent amount for each corresponding x-value. Therefore, to determine a linear model for the data in the table, we need to perform linear regression.

Linear regression is a statistical method that models the relationship between a dependent variable and an independent variables as a linear equation to predict the dependent variable's values.

The simplest way to perform linear regression is to use a statistical calculator. After entering the data from the table into a statistical calculator we get:

`a = 2.41904761...`

`b = -2.15238095...`

The regression line of y on x is y = ax + b. Therefore, the linear model for the data in the table is:

`y=2.419x+(-2.152)`

where each coefficient is rounded to three decimal places.