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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.

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

Answer :

Answer:

[tex]\sf y = \boxed{ 2.419} x + \boxed{2.152 }[/tex]

Step-by-step 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:

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

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

Calculate the mean (average) of x and y:

[tex]\sf \begin{aligned} \bar{x} &= \dfrac{4 + 7 + 10 + 13 + 16 + 19}{6} \\ &= 11.5 \end{aligned}[/tex]

[tex]\sf \begin{aligned} \sf \bar{y} &= \dfrac{7 + 16 + 21 + 29 + 38 + 43}{6}\\ & = 25.666666666666 \end{aligned} [/tex]

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

[tex] \begin{aligned} \sum{(x - \bar{x})(y - \bar{y})} &= (4 - 11.5)(7 - 25.666666666666) + (7 - 11.5)(16 -25.666666666666) \\ &\quad +(10 - 11.5)(21 - 25.666666666666) + (13 - 11.5)(29 - 25.666666666666) +\\ &\quad (16 - 11.5)(38 -25.666666666666) + (19 - 11.5)(43 - 25.666666666666) \\ &= 381 \end{aligned} [/tex]

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

[tex] \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} [/tex]

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

[tex] \begin{aligned} m &= \dfrac{\sum{(x - \bar{x})(y - \bar{y})}}{\sum{(x - \bar{x})^2}} & = \dfrac{381}{157.5} \\\\ &= 2.4190476190476\end{aligned} [/tex]

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

[tex] \begin{aligned} b & = \bar{y} - m\bar{x} & = 25.666666666666 -11.5 \cdot 2.4190476190476 \\ & = 25.666666666666 - 27.8190476190474\\& = −2.1523809523814 \end{aligned} [/tex]

So, the linear model for the data is :

y = 2.4190476190476x + −2.1523809523814

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

[tex]\sf y = \boxed{ 2.419} x + \boxed{2.152 }[/tex]

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.

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Rewritten by : Barada

Answer:

[tex]y=2.419x+(-2.152)[/tex]

Step-by-step explanation:

Given table of values:

[tex]\begin{array}{|c|c|c|c|c|c|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}[/tex]

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:

[tex]a = 2.41904761...[/tex]

[tex]b = -2.15238095...[/tex]

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

[tex]y=2.419x+(-2.152)[/tex]

where each coefficient is rounded to three decimal places.