Least squares regression line (I'm very lost)

In summary: Once you have the values for a and b, you can use the equation y= ax+ b to estimate the value of x when y=15.In summary, the least squares line of best fit is a line that "best fits" in a very specific way. It is found by minimizing the total square error of the given data. Using this method, the value of x can be estimated when y=15.
  • #1
Melody55
1
0
Hi! Basically this is the exercise:

Given the covariance of x and y is -12 and the variance of x is 6,5, using the least squares line of best fit connecting x and y yo estimate the value of x when y=15

x25979107
y251711108713
any help would mean everything, I'm desperate :(
 
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  • #2
Do you know what "least squares best fit" means?
It is the line y= ax+ b that "best fits" in very specific way. When x= 2, that equation gives y= 2a+ b while the correct value is 25. The "error", if any, is 2a+ b- 25. If we want to find a "total error" by adding those, some might be negative and cancel positive errors giving too small a total error. We could fix that by taking the absolute value but the absolute value function is not differentiable at 0. So instead we fix the sign problem by squaring. The "square error" at x= 2 is $(2a+ b- 25)^2$.

Using all of the given data,

$(2a+ b- 25)^2$

$(5a+ b- 17)^2$

$(9a+ b- 11)^2$

$(7a+ b- 10)^2$

$(9a+ b- 8)^2$

$(10a+ b- 7)^2$

$(7a+ b- 13)^2$
The total square error is

$(2a+ b- 25)^2+(5a+ b- 17)^2+ (9a+ b- 11)^2+ (7a+ b- 10)^2+ (9a+ b- 8)^2+ (10a+ b- 7)^2+ (7a+ b- 13)^2$.
That's a function of the two variables, a and b. Find the minimum by taking the partial derivatives with respect to a and b and setting them equal to 0,
 

FAQ: Least squares regression line (I'm very lost)

1. What is a least squares regression line?

The least squares regression line is a statistical method used to find the best fitting line through a set of data points. It minimizes the sum of the squared distances between the line and the data points, making it the most accurate representation of the relationship between the variables.

2. How is a least squares regression line calculated?

To calculate a least squares regression line, the following steps are typically followed:

  • Plot the data points on a scatter plot.
  • Draw a line that best represents the trend of the data.
  • Calculate the distance between each data point and the line.
  • Square each distance and add them together to get the sum of the squared distances.
  • Adjust the line's slope and y-intercept until the sum of the squared distances is minimized.

3. What is the purpose of a least squares regression line?

The purpose of a least squares regression line is to find the best fit for a set of data points, allowing for the prediction of future values and the analysis of the relationship between variables. It is commonly used in fields such as economics, finance, and social sciences.

4. What are the assumptions of a least squares regression line?

There are several assumptions that must be met for a least squares regression line to be valid:

  • The relationship between the variables is linear.
  • The residuals (the difference between the actual data points and the predicted values) are normally distributed.
  • The variability of the residuals is constant across all values of the independent variable.
  • The residuals are independent of each other.

5. How do you interpret the slope and y-intercept of a least squares regression line?

The slope of a least squares regression line represents the change in the dependent variable for every one unit change in the independent variable. The y-intercept represents the value of the dependent variable when the independent variable is equal to zero. It is important to note that the interpretation of the slope and y-intercept may vary depending on the context and the variables being analyzed.

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