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.
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.
