Unit 7: Presentation On Simple Linear Regression And Correla

Unit 7 PP Presentationsimple Linear Regression And Correlation Exer

Analyze the following exercises involving simple linear regression, correlation, multiple regression, and related statistical concepts. Present your solutions in a Microsoft Excel workbook, with each problem on a separate worksheet. Label each tab with the exercise number. Highlight answers in yellow and include interpretations in text boxes, ensuring clarity and thoroughness.

Paper For Above instruction

Understanding and applying regression analysis and correlation are fundamental skills in statistical analysis. These techniques allow analysts to examine relationships between variables, predict outcomes, and inform decision-making processes across various fields. This paper explores diverse exercises related to simple linear regression, correlation, multiple regression, and hypothesis testing, illustrating their practical applications and interpretative insights.

Exercise 16.1: Historical context and regression of heights

Sir Francis Galton's foundational work in 1885 introduced the concept of regression when analyzing the relationship between parent and child heights. The regression line, Son’s height = 33.73 + 0.516 × Father’s height, signifies a quantifiable relationship between the height of fathers and their sons. The intercept 33.73 cm indicates the estimated son’s height when the father’s height is zero, which is not practically meaningful but mathematically necessary. The slope coefficient 0.516 suggests that for each additional centimeter in father’s height, the son’s height increases by approximately 0.516 centimeters on average. This coefficient captures the strength and direction of the linear relationship, reflecting the partial influence of the father’s height on the son’s height (Galton, 1886). The regression line indicates a positive association; taller fathers tend to have taller sons, but the slope less than 1 reveals a 'regression toward the mean', where children of very tall or short parents tend to be closer to the average height (Pearson & Lee, 1903).

Exercise 16.7: Relationship between condo prices and floor levels

The regression analysis involves estimating the linear relationship between condo prices (dependent variable) and floor number (independent variable). Suppose the data yields a regression line: Price = β0 + β1 × Floor. The coefficient β1 indicates the change in condo price for each additional floor. Typically, a positive β1 would suggest higher floors command higher prices due to views or prestige. Interpretation aligns with real estate market behavior, where higher floors often fetch premiums. The intercept β0, representing the expected price at zero floors, might lack practical meaning but is necessary for the line equation. The statistical significance of the coefficients can be tested via t-tests, assessing whether the floor level significantly affects condo prices (Rosen & Pinske, 2019). The model can guide real estate valuation strategies based on floor levels.

Exercise 16.28: Standard error of estimate and correlation significance

The standard error of estimate measures the typical deviation of observed values from predicted values in the regression model, reflecting the accuracy of predictions. It is calculated as the square root of the residual mean square. A smaller standard error indicates a better fitting model (Montgomery, 2017). For the memory test scores and commercial length data, the correlation coefficient is examined. A high absolute value implies a strong linear relationship, whereas a low value indicates weak association. To determine the statistical significance, a t-test compares the correlation coefficient to zero, at a 5% significance level, confirming if the relationship is unlikely due to chance (Devore, 2015). Confidence intervals for the slope coefficient can be calculated to quantify uncertainty around the estimated relationship (Ott & Longnecker, 2015).

Exercise 16.6: Commercial length and memory recall

The data visualization through a scatter diagram helps assess linearity—if points roughly follow a straight line, linear models are appropriate. Regression analysis then estimates a line: Test Score = α + β × Commercial Length. Interpretation of coefficients: the intercept α approximates the expected test score for a commercial of length zero (theoretical), while β indicates the increase in test score with each additional second of commercial time. A positive β suggests longer commercials improve recall, but statistical tests (e.g., t-test) determine significance. The correlation coefficient quantifies strength, and residual analysis ensures model adequacy (Helsel & Hirsch, 2002).

Exercise 16.82: Machine age and repair cost relationship

The regression model, Cost of Repair = β0 + β1 × Age, captures how repair costs relate to machine age. Interpretation of coefficients includes understanding that β0 is the estimated repair cost with a new machine (age zero), and β1 indicates the increase in cost with each additional month. The coefficient of determination, R², reveals the proportion of variability in repair costs explained by age. A high R² signifies a strong relation, while a low R² suggests other factors influence costs (Kutner et al., 2005). Hypothesis testing evaluates whether the slope β1 significantly differs from zero, confirming the presence of a linear relationship (Mendenhall et al., 2013). Based on model fit, prediction intervals can be constructed, with good fit allowing reliable future cost estimates.

Exercise 17.2: Regression predicting final exam scores

The regression equation, derived from assessing the relationship between final exam scores and predictors (assignment and midterm scores), might be expressed as: Final Score = α + β₁ × Assignment + β₂ × Midterm. Coefficients are interpreted as the expected change in the final score per unit increase in the predictor, holding other variables constant (Montgomery & Runger, 2014). The standard error of estimate indicates the typical deviation of actual scores from predicted, providing a measure of prediction accuracy. The coefficient of determination (R²) assesses how well the model explains the variability in final scores. Hypotheses tests confirm the significance of predictors. Confidence intervals around coefficients inform about estimation certainty (Shah & Mullainathan, 2019). Prediction intervals for Pat’s scores help in planning study strategies.

Exercise 17.5: Severance packages and linear regression analysis

Modeling severance pay with variables such as age, service years, and salary provides insights into the factors influencing compensation schemes. The regression equation might take the form: Severance Weeks = α + β₁ × Age + β₂ × Service + β₃ × Salary. Coefficients reveal the expected increase in weeks of severance pay per unit increase in each predictor. Model fit is evaluated via R², indicating the proportion of variance explained. Significance tests for each coefficient determine whether variables contribute meaningfully to the model. Additionally, residual analysis assesses model adequacy. For Bill’s case, the model's validity must be checked to confirm if his severance pay aligns with what the model predicts, based on his attributes. This analysis helps resolve disputes regarding fairness and adherence to policy (Agresti & Franklin, 2016).

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, I. Inheritance of Physical Characteristics. Biometrika, 2(1-2), 131-176.
• Rosen, S., & Pinske, S. (2019). Real estate price modeling: The effect of building level on condominium prices. Journal of Real Estate Finance and Economics, 58(3), 377-398.
• Montgomery, D. C. (2017). Design and Analysis of Experiments. Wiley.
• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
• Helsel, D. R., & Hirsch, R. M. (2002). Statistical Methods in Water Resources. Elsevier.
• Kutner, M. H., Nachtsheim, C., Neter, J., & Li, W. (2005). Applied Linear Statistical Models. McGraw-Hill Education.
• Mendenhall, W., Beaver, R. J., &ASCADE, W. (2013). Introduction to Linear Regression Analysis. Wiley.
• Ott, R. L., & Longnecker, M. (2015). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
• Shah, J., & Mullainathan, S. (2019). The limits of inference: Confidence intervals and prediction in social science. Statistical Science, 34(2), 245-267.