Section 38-39 Linear Regression

Question 1

Linear Regression

Answer 1

LINEAR REGRESSION looks at sets of data points defined by two criteria (For example that which could be charted in a SCATTERPLOT) to determine if there is a relationship between the two sets of data (EX: SAT scores and College GPA). LINEAR REGRESSION determines the equation for a single straight line that best describes the dots.

• Tells the DIRECTION (Positive or Negative) and STRENGTH (Slope Steepness) of the relationship.
• Then, when we make predictions, we use the equation to get a single predicted score.
• First, take the equation for a straight line: Y = a + bX (similar to y = mx + b)
  
  • Where Y is the REGRESSION score on Variable Y (the score to be predicted).
  • a is the y-intercept (the point where the line meets the y-axis)
  • b is the slope (which determines the angle of the line)
  • and X is the score on Variable X (which will be given).
    
    • In general, if given the value of X, we can determine the value of Y.

NOTE: LINEAR REGRESSION is useful only if the dots form a pattern that follows a straight line (linear, NOT curvilinear).

Also, In general, the lower the correlation between two variables, the greater the error that will be made when using linear regression.

Question 2

Linear Regression Computation

Answer 2

To COMPUTE the LINEAR REGRESSION equation, set up a table like Table 1. below. X and Y are given, the rest will need to be calculated.

• This is the same example used in the section on PEARSON *r*, which we found to be -.77, a strong inverse relationship. So we would expect a LINEAR REGRESSION LINE to be downward sloping from left to right.

1. Calculate the SLOPE (b) (Refer to the example below).
2. Calculate the MEAN (M) of both data sets X and Y.
3. Calculate the REGRESSION SLOPE (a)
4. Insert the values of a and b into the REGRESSION equation Y = a + bX to get the COMPLETE regression equation.
   
   • ​​Insert any value for X to get a Predicted Y value.