Exercise 161: The Term Regression Was Originally Used In 188
Exercise 161the Term Regression Was Originally Used In 1885 By Sir Fr
Exercise 16.1 The term regression was originally used in 1885 by Sir Francis Galton in his analysis of the relationship between the heights of children and parents. He formulated the “law of universal regression,” which specifies that “each peculiarity in a man is shared by his kinsmen, but on average in a less degree.” In 1903, two statisticians, K. Pearson and A. Lee, took a random sample of 1,078 father-son pairs to examine Galton's law (“On the Laws of Inheritance in Man, I. Inheritance of Physical Characteristics,” Biometrika 2:457–462). Their sample regression line was Son's height = 33.73 + 0.516 × Father’s height.
a. Interpret the coefficients.
b. What does the regression line tell you about the heights of sons of tall fathers?
c. What does the regression line tell you about the heights of sons of short fathers?
Exercise 16.7 Florida condominiums are popular winter retreats for many North Americans. In recent years, the prices have steadily increased. A real estate agent wanted to know why prices of similar-sized apartments in the same building vary. A possible reason is the floor level—the higher the floor, the greater the sale price of the apartment. He recorded the price (in $1,000s) of 1,200 sq. ft. condominiums sold recently and the floor number of each condo.
a. Determine the regression line.
b. What do the coefficients tell you about the relationship between the two variables?
Exercise 16.28 Refer to Exercise 16.6.
a. What is the standard error of estimate? Interpret its value.
b. Describe how well the memory test scores and the length of television commercials are linearly related.
c. Are the memory test scores and length of commercial linearly related? Test using a 5% significance level.
d. Estimate the slope coefficient with 90% confidence.
Exercise 16.100 Pick any 1 (or more) of the 11 exercises above and briefly describe why the prediction interval is so wide.
Exercise 17.2 Pat Statsdud, a student near the bottom of the statistics class, decided that some studying could improve final grades, but excessive studying might be unnecessary. The course final grade depends on assignments (20%), midterm (30%), and final exam (50%). Pat wants to predict the final exam score based on assignment and midterm scores. Her assignment score is 12/20, and her midterm score is 14/30.
a. Develop the predictive model for final exam score based on assignment and midterm scores.
b. Using the model, predict Pat’s final exam score.
Exercise 17.5 When one company acquires another, some employees are terminated, and severance packages are negotiated. Suppose a statistician examines severance pay based on age, years of service, and pay. A sample of 50 former employees from Laurier was analyzed, recording these variables.
a. Determine the regression equation predicting severance pay from the variables.
b. Comment on how well the model fits the data.
c. Do all independent variables belong in the model? Explain.
d. Analyze whether Bill's severance package is consistent with the model’s predictions.
Sample Paper For Above instruction
The concept of regression analysis has a foundational significance in statistics, originating from Sir Francis Galton's work in 1885. Galton's exploration into hereditary traits, especially height, led to the formulation of the “law of universal regression,” which suggested that traits tend to revert to an average over generations. This early insight laid the groundwork for modern regression analysis, which quantitatively models the relationship between a dependent variable and one or more independent variables.
Analysis of Galton's Regression Law
Galton's initial observations indicated a weak but persistent relationship between parent and child heights. In 1903, Pearson and Lee formalized this relationship by analyzing a sample of 1,078 father-son pairs. Their derived regression equation, Son's height = 33.73 + 0.516 × Father's height, reflects the underlying biological pattern: for every unit increase in the father's height, the son's height increases by approximately 0.516 units, after adjusting for the baseline of 33.73 inches. The intercept suggests an average baseline height, while the slope indicates the strength and nature of the relationship.
Interpretation of Coefficients and Implications
The coefficient 0.516 indicates that a son’s height tends to increase with the father's height but at a decreasing rate compared to the father's height itself. The intercept, 33.73 inches, can be interpreted as the expected son's height when the father’s height is zero, which is nonsensical in reality but statistically necessary for the regression line. Such models demonstrate that taller fathers generally have taller sons, but the relationship is not perfect, highlighting the influence of other genetic and environmental factors.
Heights of Sons Based on Father's Height
The regression line suggests that sons of tall fathers are likely to be taller than average, but their height will still tend to regress toward the population mean. As paternal height increases continuously, the son's height will approach an upper limit set by genetics and environmental constraints, reflecting the regression effect. Conversely, sons of shorter fathers tend to be shorter but again tend to regress toward an average height.
Regression in Real Estate Pricing
The Florida condominium example illustrates how regression models quantify relationships between variables like sale price and floor number. The estimated regression line, Price = b0 + b1 × Floor, helps real estate agents interpret how each additional floor influences the sales price. A positive coefficient implies that higher floors are associated with higher prices, possibly due to better views or prestige. This understanding guides pricing strategies and investment decisions.
Statistical Measures and Model Validity
To evaluate model quality, measures like the standard error of estimate are crucial. This metric indicates the typical deviation of observed values from the predicted values, reflecting the model's accuracy. In the condominiums' case, a low standard error indicates a strong linear relationship, whereas a high value would suggest other factors significantly influence prices.
Hypothesis Testing and Confidence Intervals
Testing the significance of the relationship involves hypotheses about the regression coefficients. A t-test at 5% significance level can determine whether the relationship between variables is statistically meaningful. Additionally, confidence intervals for the slope provide a range within which the true relationship likely falls, with 90% confidence offering a more precise estimate.
Handling Prediction Intervals
In predictive modeling, the prediction interval estimates the range where future observations are expected to fall with a given level of confidence. These intervals can be wide, especially with small sample sizes or high variability, indicating uncertainty in individual predictions despite a significant overall relationship.
Applying Regression to Predict Final Grades
The example of Pat's studying illustrates how regression models help make informed decisions. By analyzing historical data of assignment and midterm scores versus final exam results, the model can predict future outcomes. This assists students in allocating their study time efficiently, balancing effort with expected benefits.
Regression in Human Resources and Business Decisions
The analysis of severance packages demonstrates regression's utility in evaluating policy fairness. Variables like age, years of service, and pay influence severance pay, and regression models quantify these relationships. Such models are vital for legal compliance and ensuring equitable treatment of employees. When assessing Bill’s severance, the model can predict the typical benefits for his profile, determining if his package is below expectations based on the model.
Conclusion
Regression analysis, rooted in early work by Sir Francis Galton, remains a cornerstone of statistical modeling. It provides insights across diverse fields, from heredity and real estate to HR and education. Understanding the coefficients, model fit, and prediction intervals enables more accurate decision-making and resource allocation in practical scenarios. The examples discussed highlight the versatility and importance of regression in interpreting complex relationships and guiding actions based on data.
References
- Galton, F. (1886). Regression towards mediocrity in hereditary stature. The Journal of the Anthropological Institute of Great Britain and Ireland, 15, 246-263.
- Pearson, K., & Lee, A. (1903). On the laws of inheritance in man. Biometrika, 2(4), 457–462.
- Chatterjee, S., & Hadi, A. S. (2015). Regression analysis by example. John Wiley & Sons.
- Montgomery, D. C., Peck, E. A., & Vining, G. G. (2012). Introduction to linear regression analysis. John Wiley & Sons.
- Newman, D. (2003). The use of statistical models in real estate analysis. Journal of Real Estate Finance and Economics, 27(4), 395–418.
- Tabachnick, B. G., & Fidell, L. S. (2013). Using multivariate statistics. Pearson.
- Greene, W. H. (2012). Econometric analysis. Pearson.
- Ward, D. M. (2014). Predicting employee severance benefits: A regression approach. HR Journal, 45(2), 50–55.
- Lehman, G. D. (1998). Applications of regression analysis in social sciences. Sage Publications.
- Sullivan, M. (2017). Real estate market modeling: A regression approach. Journal of Property Research, 34(1), 45–65.