Which of the following is the equation for the slope of the regression line?

Hint: To calculate the slope and intercept of a regression line, we are going to take the set of data having $x$ and $y$ values. We are taking $x$ and $y$ values because we have to calculate the slope of a straight line and the straight line is of the form $y = mx + c$, where "$m$ and $c$" corresponds to slope and intercept of the straight line. We are taking the set of $x$ and $y$ values $\left( {x,y} \right)$ as: $\left( { - 3,7} \right),\left( {0,4} \right),\left( {3, - 2} \right),\left( {5,2} \right),\left( {3, - 3} \right)$. Then, find the slope of the regression line for the given set of data by putting the values of $x,y,n$ in the formula for slope of the regression line.
Formula used:
The formula for slope of the regression line is as follows:
$m = \dfrac{{n\sum {xy} - \left( {\sum x } \right)\left( {\sum y } \right)}}{{n\sum {{x^2}} - {{\left( {\sum x } \right)}^2}}}$

Complete step by step answer:
Let us assume that we have drawn the regression line using the following set of $x$ and $y$ values:
$\left( { - 3,7} \right),\left( {0,4} \right),\left( {3, - 2} \right),\left( {5,2} \right),\left( {3, - 3} \right)$
The above coordinates are plotted on the graph where in each bracket, first coordinate is the $x$ coordinate and the second coordinate is the $y$ coordinate.
We know that the equation of a straight line contains a slope and intercept and in the below, we are writing the formula for slope and intercept of a regression line.
The formula for slope of the regression line is as follows:
$m = \dfrac{{n\sum {xy} - \left( {\sum x } \right)\left( {\sum y } \right)}}{{n\sum {{x^2}} - {{\left( {\sum x } \right)}^2}}}$
Now, using the above set of $x$ and $y$ in the above equation we get,
$m = \dfrac{{5\left( { - 21 + 0 - 6 + 10 - 9} \right) - \left( { - 3 + 0 + 3 + 5 + 3} \right)\left( {7 + 4 - 2 + 2 - 3} \right)}}{{5\left( {9 + 0 + 9 + 25 + 9} \right) - {{\left( { - 3 + 0 + 3 + 5 + 3} \right)}^2}}}$
$ \Rightarrow m = \dfrac{{5 \times \left( { - 26} \right) - 8 \times 8}}{{5 \times 52 - {8^2}}}$
$ \Rightarrow m = \dfrac{{ - 130 - 64}}{{260 - 64}}$
$ \Rightarrow m = \dfrac{{ - 194}}{{ - 196}}$
$\therefore m = \dfrac{{97}}{{98}} \approx 0.989$
Final solution: Hence, the slope of the regression line for the given set of data is $\dfrac{{97}}{{98}}$ or $0.989$.

Nội dung chính Show

Key Takeaways
Regression Line Explained
Frequently Asked Questions (FAQs)
Recommended Articles
Which of the following is the equation for the slope of the regression line quizlet?
What is the equation of the regression line quizlet?
What is the function of the slope in a regression line?

Note:
Regression line we have to draw when we have a dependent and independent variable. The independent variable we have plotted on the $x$-axis and the dependent variable we have plotted on the $y$-axis and then to get the best fit line which is passing through these points we need the slope and intercept formula. This is the example where we require calculating the slope and intercept of a regression line.

A regression line indicates a linear relationship between the dependent variables on the y-axis and the independent variables on the x-axis. The correlation is established by analyzing the data pattern formed by the variables.

The regression line is plotted closest to the data points in a regression graph. This statistical tool helps analyze the behavior of a dependent variable y when there is a change in the independent variable x—by substituting different values of x in the regression equation.

Regression Line Definition
- Regression Line Explained
- Formula
- Example
- Frequently Asked Questions (FAQs)
- Recommended Articles

Key Takeaways

The regression line establishes a linear relationship between two sets of variables. The change in one variable is dependent on the changes to the other (independent variable).
The Least Squares Regression Line (LSRL) is plotted nearest to the data points (x, y) on a regression graph.
Regression is widely used in financial models like CAPM and investing measures like Beta to determine the feasibility of a project. It is also used for creating projections of investments and financial returns.
If Y is the dependent variable and X is the independent variable, the Y on X regression line equation is represented as follows:
‘Y = a + bX + ɛ.’

Regression Line Explained

A regression line is a statistical tool that depicts the correlation between two variables. Specifically, it is used when variation in one (dependent variable) depends on the change in the value of the other (independent variable).

There can be two cases of simple linear regression:

The equation is Y on X, where the value of Y changes with a variation in the value of X.
The equation is X on Y, where the change in X variable depends upon the Y variable’s deviation.

Which of the following is the equation for the slope of the regression line?

You are free to use this image on your website, templates, etc., Please provide us with an attribution linkHow to Provide Attribution?Article Link to be Hyperlinked
For eg:
Source: Regression Line (wallstreetmojo.com)

Regression is extensively applied to various real-world scenarios—business, investment, finance, and marketing. For example, in finance, regression is majorly employed in the BetaBetaBeta is a financial metric that determines how sensitive a stock's price is to changes in the market price (index). It's used to analyze the systematic risks associated with a specific investment. In statistics, beta is the slope of a line that can be calculated by regressing stock returns against market returns.read more and Capital Asset Pricing Model (CAPMCAPMThe Capital Asset Pricing Model (CAPM) defines the expected return from a portfolio of various securities with varying degrees of risk. It also considers the volatility of a particular security in relation to the market.read more)—for estimating returns and budgeting.

Using regressionRegressionRegression Analysis is a statistical approach for evaluating the relationship between 1 dependent variable & 1 or more independent variables. It is widely used in investing & financing sectors to improve the products & services further. read more, the company can determine the appropriate asset price with respect to the cost of capital. In the stock market, it is used for determining the impact of stock price changes on the price of underlying commodities.

In marketing, regression analysis can be used to determine how price fluctuation results in the increase or decrease in goods sales. It is very effective in creating sales projections for a future period—by correlating market conditions, weather predictions, economic conditions, and past sales.

Formula

The formula to determine the Least Squares Regression Line (LSRL) of Y on X is as follows:

Y=a + bX + ɛ

Here,

Y is the dependent variable.
a is the Y-intercept.
b is the slope of the regression line.
X is the independent variable.
ɛ is the residual (error).

Also,

b = (N∑XY-(∑X)(∑Y) / (N∑X2– (∑X)2) ;

And,

a = (∑Y – b ∑X) / N

Where N is the total number of observations.

Example

Let us look at a hypothetical example to understand real-world applications of the theory.

The finance manager of ABC Motors wants to correlate variation in sales and variation in the price of electric bikes. For this purpose, he analyzes data pertaining to the last five years.

We assume there is no error. The price and sales volume for the previous five years are as follows:

YearPrice (in $)Sales Volume20172100150002018205016500201920002100020202200190002021205020000

Based on the given data, determine the regression line of Y on X,

Solution:

Let us determine the regression line of Y on X:

Given:

Y = Sales Volume
X = Profit
N = 5
ɛ = 0

YearPrice (in $) (X)Sales Volume (Y)X2XY20172100150004410000315000002018205016500420250033825000201920002100040000004200000020202200190004840000418000002021205020000420250041000000–104009150021655000190125000

Y = a + bX + ɛ

Let us first find out the value of b and a:

b = (N∑XY-(∑X)(∑Y) / (N∑X2– (∑X)2)

b = ((5×190125000) – (10400×91500)) / ( (5×21655000) – 104002 )
b = (950625000-951600000) / (08275000 -108160000)
b = – 8.478

a = (∑Y – b ∑X) / N

a = 91500 – ( – 8.478 × 10400) / 5
a = 35935
Y = 35935 + ( – 8.478 X) + 0
Y = 35935 – 8.478X

The data is represented as a regression line graph:

(Source)

Visualization of collected data makes data interpretation easier. The regression line is sometimes called the line of best fit.

It is important to note that real-world data cannot always be expressed with a regression equation. If the majority of observations follow a pattern, then the outliers can be eliminated. But sometimes, there is no obvious pattern. If there are random irregularities in collected data—the regression method is not suitable.

Frequently Asked Questions (FAQs)

What is a regression line?

A regression line depicts the relationship between two variables. It is applied in scenarios where the change in the value of the independent variable causes changes in the value of the dependent variable.

How to find a regression line?

The formula of the regression line for Y on X is as follows:

Y = a + bX + ɛ
Here Y is the dependent variable, a is the Y-intercept, b is the slope of the regression line, X is the independent variable, and ɛ is the residual (error).

What is the slope of a regression line?

The slope of a regression line is denoted by ‘b,’ which shows the variation in the dependent variable y brought out by changes in the independent variable x. The formula to determine the slope of the regression line for Y on X is as follows:

b = (N∑XY-(∑X)(∑Y) / (N∑X2– (∑X)2)