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Assuming you have noted the prices for paperback books along with the number of pages each contains, you are tasked with developing a least-squares estimated regression line. The assignment involves calculating the coefficient of determination, interpreting its meaning, computing the correlation coefficient between the book prices and pages, and conducting a significance test to determine if these variables are related at a specified significance level.

Additionally, you are to analyze data on a company's yearly sales volume and advertising expenditure over eight years. This includes creating a scatter diagram to visualize the relationship, calculating the regression line via the least squares method, predicting sales from a specified advertising expenditure, and interpreting the slope of the regression.

Furthermore, using ROI data for two majors, you are to plot scatter diagrams of ROI against cost, derive regression coefficients, draw fitted lines, predict ROI for a given cost, and perform hypothesis testing on the regression slope. You are also expected to reflect on the findings regarding the estimates and statistical tests, considering the coefficient of determination, regression plots, and hypotheses results to make business insights.

Paper For Above instruction

The analysis of the relationship between variables such as book prices and pages, advertising expenditures and sales, and ROI against costs involves applying fundamental statistical regression techniques. These methods help quantify the strength and nature of the relationships, enabling a deeper understanding of the underlying business or economic phenomena.

Part 1: Regression Analysis of Book Prices and Pages

To develop a least-squares regression line predicting book price based on the number of pages, we first compile the data points for each book. Using these data, we calculate the regression coefficients b0 (intercept) and b1 (slope). The formulas for these are:

• b1 = Σ(x - x̄)(y - ȳ) / Σ(x - x̄)^2
• b0 = ȳ - b1 * x̄

where x̄ and ȳ are the means of the number of pages and prices, respectively. Once the regression line is established, the coefficient of determination (r²) is computed to estimate how well the regression line fits the data. R² is interpreted as the proportion of variance in the dependent variable (price) explained by the independent variable (pages).

The correlation coefficient (r) is derived from r², taking the square root (considering the sign of b1) to understand the strength and direction of the relationship. A significant correlation suggests that the number of pages and price are related, subject to hypothesis testing at α = 0.10. A t-test on the correlation coefficient determines whether the observed correlation is statistically significant, indicating whether these variables are genuinely related or if the association could be due to chance.

Part 2: Regression of Sales and Advertising Expenditure

For the company's sales and advertising data, plotting a scatter diagram visually assesses the linear relationship, where a positive trend would suggest that increased advertising correlates with higher sales. Using the least squares method, we derive the regression line ŷ = b0 + b1X, where ŷ is sales predicted based on advertising expenditure X.

Calculations for b0 and b1 follow similar formulas as above, based on the sums of products and sums of squares. Once obtained, the regression line allows for prediction; specifically, with an advertising expenditure of $400,000, the predicted sales are calculated by plugging this value into the regression equation, resulting in a dollar amount of predicted sales.

The slope b1 indicates the expected change in sales for each additional dollar spent on advertising. A positive slope reflects that increased advertising expenditure tends to increase sales, which is a common business insight. The magnitude of this slope reveals the efficiency or return on investment in advertising efforts.

Part 3: ROI Data and Regression Analysis

Using ROI data categorized by business major and school type, scatter diagrams are created plotting ROI against the cost of education. From these, regression coefficients b0 (intercept) and b1 (slope) are obtained, typically via regression analysis software or Excel output. These coefficients quantify how ROIs change as costs vary.

The fitted regression lines are superimposed onto the scatter diagrams, visually illustrating the relationship. For a given cost of $160,000, the estimated ROI is calculated using the regression equation, providing insight into expected financial return upon investment.

To test the significance of the cost coefficient β1, hypotheses are set:

• H0: β1 = 0 (cost does not affect ROI)
• Ha: β1 ≠ 0 (cost significantly affects ROI)

A t-test on the regression coefficient determines the p-value, quantifying the likelihood that the observed relationship occurs due to chance. If the p-value is below the chosen significance level (α = 0.10), H0 is rejected, indicating a statistically significant relationship between cost and ROI.

Reflecting on the regression estimates, coefficient of determination, plots, and hypothesis test outcomes provides valuable insights: for instance, whether the relationship is strong enough to inform decision-making, whether ROI increases or decreases with costs, and the overall model significance.

Conclusion

Application of regression analysis across these various datasets demonstrates its power in revealing relationships between business and economic variables. The coefficient of determination illuminates the explanatory power of the models, while hypothesis testing assesses the significance of predictors. Visual plots such as scatter diagrams complement statistical estimates, facilitating interpretability and strategic decision-making.

References

• Montgomery, D. C., & Peck, E. A. (2012). Introduction to Linear Regression Analysis. Wiley.
• Weisberg, S. (2005). Applied Linear Regression. Wiley.
• Chatterjee, S., & Hadi, A. S. (2015). Regression Analysis by Example. Wiley.
• Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage.
• Yamane, T. (1967). Statistics: An Introductory Analysis. Harper and Row.
• Payscale.com. (2013). Best College ROI by Major. Retrieved from https://www.payscale.com
• Excel Data Analysis Toolpak. (Available in Excel for performing regression analysis).
• Statistics.com. Regression tutorial. Retrieved from https://www.statistics.com/tutorials/regression/
• Freedman, D., Pisani, R., & Purves, R. (2007). Statistics. Norton & Company.
• Gelman, A., & Hill, J. (2006). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.