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Regression Analysis: Predicting Typing Errors from Manual Dexterity, Study notes of Statistics

Arizona State University (ASU) - TempeStatistics

An in-depth explanation of regression analysis, focusing on predicting typing errors from manual dexterity. It covers the components of the regression equation, determining the slope and y-intercept, minimizing errors in prediction, and generating a prediction equation. The document also discusses evaluating the utility of a regression equation and the concept of regression to the mean.

Typology: Study notes

Pre 2010

Uploaded on 09/02/2009

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koofers-user-qyt 🇺🇸

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Okun
PSY 230
STUDY GUIDE #9: Regression
Person Manual # of Typing Errors (Y)
Dexterity (X)
______________________________________________________________
Fred 50 15
Gene 50 12
Heidi 100 12
Irene 150 6
Janet 150 0
_______________________________________________________________
Mean 100 9
Standard
Deviation 50 6
________________________________________________________________
r = -.70
1. What does a regression analysis enable us to do?
With a regression analysis, we generate a prediction equation that provides a way to
predict the scores of individuals on the quantitative dependent variable (Y), for all plausible
values of the predictor variable (X).
2. What are the components of the regression equation?
General linear equation: Yi` = [(b) (Xi)] + a
Xi = score on the predictor variable for a specific person (e.g., manual dexterity score).
Yi` = predicted score on Y for a specific person (e.g., predicted # of typing errors).
b = slope;
a = y-intercept
The slope (b) indicates the average amount of change in Y per 1 unit increase in X.
The y-intercept (a) is the value of Y` when X = 0.
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Download Regression Analysis: Predicting Typing Errors from Manual Dexterity and more Study notes Statistics in PDF only on Docsity!

Okun PSY 230 STUDY GUIDE #9: Regression Person Manual # of Typing Errors (Y) Dexterity (X)

Fred 50 15 Gene 50 12 Heidi 100 12 Irene 150 6 Janet 150 0

Mean 100 9 Standard Deviation 50 6

r = -.

  1. What does a regression analysis enable us to do? With a regression analysis, we generate a prediction equation that provides a way to predict the scores of individuals on the quantitative dependent variable (Y) , for all plausible values of the predictor variable ( X ).
  2. What are the components of the regression equation? General linear equation: Yi= [(b) (Xi)] + a Xi = score on the predictor variable for a specific person (e.g., manual dexterity score). Yi = predicted score on Y for a specific person (e.g., predicted # of typing errors). b = slope; a = y-intercept The slope (b) indicates the average amount of change in Y per 1 unit increase in X. The y-intercept (a) is the value of Y`^ when X = 0.

In this equation b and a are constants whereas X and Yare **variables**. Whereas **Yi** is the predicted score on Y (e.g., predicted # of typing errors) for an individual, Yi is the observed score on Y for the individual (e.g., # of typing errors actually made).

  1. How can the slope and the y-intercept be determined from two X,Y data points? The slope of a regression line is the ratio of the rise to the run. The rise is called the vertical movement and the run is called the horizontal movement. The rise refers to the change in Y whereas the run refers to the change in X. What are the slope and intercept for the straight line that passes through the points (0,3) and (2,6)? What are the intercept and slope for the straight line that passes through the points, (0,5) and (1,3)?
  2. What criterion did statisticians use in developing formulas for the prediction of Y from X? The criterion used in developing a regression or prediction line is to minimize errors in prediction. Errors in prediction can be defined as the distance between the actual Y scores and the predicted Y scores (Yi-Yi’). More formally, the criterion is to minimize the sum of the squared errors in prediction: S (Yi - Yi `)^2 The slope and Y-intercept are computed using formulas that minimize the squared errors in prediction, thus providing the best-fitting line.
  1. How can we use the prediction equation to solve for predicted scores (Yi) and to plot the prediction (or regression) line? If **(Manual Dexterity) X** = 0, **Y (predicted # of errors)** = [-.084 x 0] + 17.4 = 0 + 17.4 = 17. If X = 50, Y** = [-.084 x 50] + 17.4 = -4.2 + 17.4 = 13. If **X** = 100, **Y = [-.084 x 100] + 17.4 = -8.4 + 17.4 = 9 If X = 150, Y` = [-.084 x 150] + 17.4 = -12.6 + 17.4 = 4. REGRESSION LINE SUPERIMPOSED ON SCATTERPLOT

Predicting Typing Errors (Y) from the Manual Dexterity Test (X) n = 5

Name Manual Typing Predicted Error Deviation Squared Error Dexterity (X) Errors (Y) Scores (Y) Y - Y Deviation (Y - Y`)^2

Fred 50 15 13.2 +1.8 3. Gene 50 12 13.2 -1.2 1. Heidi 100 12 9.0 +3.0 9. Irene 150 6 4.8 +1.2 1. Janet 150 0 4.8 -4.8 23.

0 38.

  1. How can we evaluate the utility of a regression equation? One way to evaluate the utility of the regression line is to create a graph in which the regression line is superimposed on a scatter plot of the bivariate data. A second way to evaluate the utility of the regression line is to compare the standard error of estimate for predicting y from x to the standard deviation of the y scores.
  2. What is the conceptual formula for the population standard error of estimate? s y’ is the symbol for the population standard error of estimate. The formula for the population standard error of estimate involves taking the square root of the average squared distance between the actual Y scores and the predicted Y scores.

s y’ = Öå (Y- Y`)^2 /N

  1. How does the formula for the population standard error of estimate differ from the population standard deviation for the Y scores? ___________ sy = Öå (Y- my)^2 /N Whereas sy provides an approximate measure of the average distance that Y scores fall on either side of the mean of the Y scores, s y’ provides an approximate measure of the average distance that Y scores fall from their predicted scores (Y’).

Suppose a researcher wants to predict ratings of nervousness from # of cups of coffee consumed. N = 12, r xy = .97, R^2 xy = .95, Let’s compare how much variability exists in the Y scores (nervousness ratings) when the predictor variable (# of cups of coffee) is ignored with how much variability exists in the Y scores when the predictor variable is taken into account.

  1. How are prediction equations used in the real world? Steps in using a regression equation
  2. Collect data on several predictors in a sample of applicants to law school. X 1 = LSAT scores X 2 = College GPA X 3 = Ratings of goal statement X 4 = Ratings of letters of recommendation
  3. Determine the GPA of the applicants after the first year of law school. Let Y = GPA in Law School.
  4. Generate a multiple regression equation for predicting GPA in law school from the four predictors. Multiple regression equation: Yi` = b 1 X 1 + b 2 X 2 + b 3 X 3 + b 4 X 4 + a b 1 is the slope for predicting Law School GPA from LSAT Scores b 2 is the slope for predicting Law School GPA from College GPA b 3 is the slope for predicting Law School GPA from Ratings of goal statement b 4 is the slope for predicting Law School GPA from Ratings of letters of recommendation a is the Y-intercept.
  5. Determine the predicted values of Y for plausible values of X.
  6. Determine how well the prediction equation works (compute r^2 ).
  7. If the prediction equation works well, collect data on the predictor variables for future applicants
  8. Using the prediction equation, generate predicted GPAs (Y`) for future applicants from the values of X 1 , X 2 , X 3 , and X 4.
  9. Select applicants who are predicted to have the best GPAs (e.g., applicants predicted to have GPAs of 3.5 or higher).
  1. What is meant by the phrase, “regression to the mean”? How can predicted and observed regression to the mean be illustrated? What is the regression fallacy? How can the regression fallacy be overcome? Regression to the mean refers to a tendency for scores, particularly extreme scores, to move toward the mean.