Is this statement true?
The least squares regression line is a unique line for each data set?
I would say yes.
To determine if the statement is true, we need to understand what the least squares regression line is and how it is calculated for a given data set.
The least squares regression line is a straight line that best fits a scatter plot of data points. It is commonly used in statistics to model the relationship between two variables and make predictions. The line is determined by minimizing the sum of the squares of the vertical distances between the data points and the line.
To calculate the least squares regression line for a given data set, you can follow these steps:
1. Collect the data: Gather a set of paired observations, typically consisting of an independent variable (x) and a dependent variable (y). The number of observations should be greater than or equal to two.
2. Calculate the mean of x and y: Find the average value of the x-values (x̄) and the average value of the y-values (ȳ) from the data set.
3. Calculate the slope (β₁): Use the following formula to find the slope of the least squares regression line:
β₁ = Σ((xᵢ - x̄)(yᵢ - ȳ)) / Σ((xᵢ - x̄)²)
where Σ represents the sum of the values for all observations (i) in the data set.
4. Calculate the y-intercept (β₀): Once you have the slope, you can calculate the y-intercept of the line using the following formula:
β₀ = ȳ - β₁ * x̄
Now, coming back to the original statement, the least squares regression line is indeed unique for each data set. This is because the line is determined by the specific data points and their relationship. So, if you have two different data sets, even if they have similar ranges or patterns, the least squares regression line for each set will likely be different.
In conclusion, the statement is true. The least squares regression line is unique for each data set, and it is calculated based on the specific set of data points and their relationship.