Statistsics Mathematics Problem: Linear Regression

[iamblessed20062]

Find the equation of the regression line for the given data. then construct A SCATTER PLOT of the data and draw the regression line. (each pair of variables has a significant correlation.) then use the regression equation to predict the value of y for each of the given x- values, if meaningful. the caloric content and the sodium content(in milligrams) for 6 beef ho dogs are shown in the table below. find the regression equation. y=_________________________ x +______________________________. (round to three decimal places as needed.) (a) x = 160 calories (b) x = 100 calories (c) x = 140 calories (d) x = 60 calories calories, x, 150, 170, 130, 120, 90, 180 sodium, y, 415, 465, 340, 370, 270, 550

 

[MarkFL]

Hello and welcome to MHB, iamblessed20062! :D

It appears that given the calories of a hot dog, the sodium content can be predicted from this. I am going to present the data in tabular format, so that it is more easily read:

Calories	Sodium Content (mg)
150	415
170	465
130	340
120	370
90	270
180	550

Now, according to my old Stats textbook, we have:

[box=green]Regression equation: \(\displaystyle \hat{y}=b_0+b_1x\), where:

\(\displaystyle b_1=\frac{S_{xy}}{S_{xx}}\)

\(\displaystyle b_0=\frac{1}{n}\left(\sum y-b_1\sum x\right)\)

\(\displaystyle S_{xx}=\sum\left(x-\overline{x}\right)^2\)

\(\displaystyle S_{xy}=\sum\left(\left(x-\overline{x}\right)\left(y-\overline{y}\right)\right)\)[/box]

Can you proceed?

 

[iamblessed20062]

No, I cannot proceed from here.:-(

Hello and welcome to MHB, iamblessed20062! :D

It appears that given the calories of a hot dog, the sodium content can be predicted from this. I am going to present the data in tabular format, so that it is more easily read:

Calories	Sodium Content (mg)
150	415
170	465
130	340
120	370
90	270
180	550

Now, according to my old Stats textbook, we have:

[box=green]Regression equation: \(\displaystyle \hat{y}=b_0+b_1x\), where:

\(\displaystyle b_1=\frac{S_{xy}}{S_{xx}}\)

\(\displaystyle b_0=\frac{1}{n}\left(\sum y-b_1\sum x\right)\)

\(\displaystyle S_{xx}=\sum\left(x-\overline{x}\right)^2\)

\(\displaystyle S_{xy}=\sum\left(\left(x-\overline{x}\right)\left(y-\overline{y}\right)\right)\)[/box]

Can you proceed?

 

[MarkFL]

It seems like we should first compute $S_{xx}$. So, we need to identify $n$ and $\overline{x}$...can you find these?

 

[CellCoree]

Linear regression is a statistical method used to analyze the relationship between two variables and to predict the value of one variable based on the value of the other variable. In this problem, we are given a set of data for the caloric content and sodium content of 6 beef hot dogs. The first step in finding the regression equation is to plot the data on a scatter plot and draw the regression line.

After plotting the data, we can see that there is a positive correlation between the caloric content and sodium content of the hot dogs. This means that as the caloric content increases, the sodium content also tends to increase. To find the regression equation, we can use the least squares method to determine the line of best fit for the data points. The equation for the regression line can be written as y = mx + b, where m is the slope of the line and b is the y-intercept.

To find the values of m and b, we can use the following formulas:

m = (nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2)

b = (Σy - mΣx) / n

Where n is the number of data points, Σxy is the sum of the products of x and y, Σx is the sum of x values, and Σy is the sum of y values.

Using the given data, we can calculate the values of m and b as follows:

n = 6

Σx = 150 + 170 + 130 + 120 + 90 + 180 = 840

Σy = 415 + 465 + 340 + 370 + 270 + 550 = 2410

Σxy = (150*415) + (170*465) + (130*340) + (120*370) + (90*270) + (180*550) = 107450

Σx^2 = (150^2) + (170^2) + (130^2) + (120^2) + (90^2) + (180^2) = 86400

Using the formulas, we can calculate:

m = (6*107450 - 840*2410) / (6*86400 - (840)^2) = 0.387

b = (2410 - 0