École De Gesteon Telfers School Of M A ✓ Solved

É C O L E D E G E S T I O N T E L F E R S C H O O L O F M A N A

Consider the daily percent change in the stock price of two companies, A and B, in an investment portfolio. The dataset is called Investment Portfolio. Answer the following questions manually. Use statistical software or MS Excel for help with the computation of any summary statistics needed for manual computations.

a) Draw a scatterplot of the company A daily percent changes against the company B daily percent changes. Describe the relationship between daily percent changes that you see in this scatterplot.

b) Determine the regression equation to predict the daily percent change in the stock price of company A from the daily percent change in the stock price of company B. Interpret the value of the slope coefficient.

c) Find the correlation between the percent changes. Does the correlation value support your description of the scatterplot in part a)?

d) Compute the corresponding coefficient of determination and interpret its value. In financial terms, it represents the proportion of non-diversifiable risk in company A.

e) Compute the 95% confidence interval for the slope coefficient.

f) Test at the 5% significance level whether the slope coefficient is significantly different from 1, representing the beta of a highly diversified portfolio.

Using Minitab or any other statistical software, run a simple linear regression model to predict Operating Margin based on Distance and answer the following questions:

a) Using an appropriate graph, plot Operating Margin versus Distance and comment on the relationship between these two variables.

b) Write down your estimation of the regression equation for predicting Operating Margin from Distance. Draw the regression line on the plot in part a).

c) Assuming α = 0.01, test whether Distance has statistically significant predictive power in estimating Operating Margin. State the hypotheses, provide a test statistic and p-value, and state your conclusion. Show your calculations.

d) Interpret the values of the regression coefficients (slope and intercept).

Using Minitab or any other statistical software, now perform a multiple linear regression analysis of Operating Margin (response variable) against all the remaining variables as predictors, excluding Location ID Number.

a) Write down the regression equation and provide at least two summary measures of the fit of the model. Based on the summary measures, does the model provide a good fit for the data? Explain.

b) Plot the residuals against the fitted values and comment on whether the usual model conditions are met.

c) The variable Operating Margin New in the dataset corresponds to the Operating Margin variable from which some values have been recorded as missing values. Identify those missing values and explain what they are and why they were recorded as missing.

Using statistical software, run the same multiple linear regression model as in Part 2 above but this time using Operating Margin New as the response variable. Then, answer the following questions:

a) Briefly compare the resulting regression equation and fit with those obtained in Part 2.

b) Plot the residuals against the fitted values and comment on whether the model complies with the usual conditions for multiple linear regression.

c) Provide an interpretation for the model intercept and for the regression coefficients associated with variables Income and Distance. Is an interpretation of the model intercept appropriate in this case? Compare the value of the regression coefficient for Distance with the one obtained in Part 1 above and clearly explain any difference.

d) Do you see any justification for dropping any variable(s) from the model? Explain (hint: multicollinearity; the significance of predictors).

e) Run a final model using Operating Margin New as the response variable and including only the significant predictors (hint: those with a p-value ≤ 5%).

f) Test the overall significance of the final model in part e). Use a 1% significance level and follow all the steps for hypothesis testing indicated in the Instructions section.

Based on your final model in Part 3 above, answer the following questions:

a) Test the marginal contribution of Quality, assuming that the other variables in the model remain constant. Use a 1% significance level, and make sure you follow all the steps for hypothesis testing indicated in the Instructions section. Show the computation of the t-statistics (i.e., the ratio used to compute it).

b) Calculate the 99% prediction interval for the actual operating margin of a new location with the same characteristics as those for Location ID Number 3098 in the data file. Check if the prediction interval includes the actual operating margin associated with Location ID Number 3098 and explain why it does or does not.

c) Calculate the 99% confidence interval for the mean operating margin of a new location with the same characteristics as those for Location ID Number 3098 in the data file. Explain any difference between the size of this interval and the one in part b) above.

Paper For Above Instructions

The investment decisions made by portfolio managers heavily rely on analytical techniques to evaluate the performance of stocks accurately. This paper addresses the assignment prompt by analyzing the stock prices of two companies, referred to as Company A and Company B, paired with their respective daily percent changes. Consequently, the first question requires visual representation and interpretation of the daily percent changes.

To begin with, a scatterplot will be created to examine the relationship between the daily percent changes of Company A against Company B. The scatterplot will portray Company A's changes on the Y-axis while Company B's changes will be on the X-axis. By analyzing the scatterplot, we can infer how the changes in one company's stock price may be influencing or related to changes in the other company's stock price. A linear relationship observed could imply a correlation where an increase in Company B's stock changes similarly affects Company A's price changes.

Next, the regression equation is derived to predict Company A’s percent change based on Company B's. Using statistical software (e.g., Minitab or Excel), the Ordinary Least Squares method will be utilized to compute the slope and intercept of the regression line. The slope coefficient indicates the expected change in the stock price percentage of Company A for every one percent change in Company B. An interpretation of the coefficient helps to conduct a thorough interpretation of the relationship between the two variables.

Following the regression, the correlation coefficient will be calculated to establish the strength of the relationship. A high positive correlation close to 1 signifies a strong positive relationship, which likely aligns with the initial scatterplot interpretation. A determination of correlation supports our earlier observation by quantifying the relationship strength.

Furthermore, the coefficient of determination (R²) will be computed, which reflects how well the independent variable (Company B's changes) explains the variability in Company A's percent changes. In financial terms, this coefficient illustrates the proportion of non-diversifiable risk faced by investors in Company A, based on movements observed in Company B.

Additionally, a 95% confidence interval for the slope coefficient will be established to mark the range in which we anticipate the true slope lies, showing statistical reliability in our discovered relation. Finally, we will perform hypothesis testing at a 5% significance level to see if the slope coefficient significantly differs from 1. The decision to reject or not reject the null hypothesis will depend on the calculated p-value obtained from the regression analysis.

Moving to the location analysis portion, we will apply statistical techniques to predict the operating margin of a motel chain based on the identified predictive variables relevant to effective site selection. A linear regression model will be fit to assess operating margin's relationship with distance to the nearest competitor.

Initially, a plot of Operating Margin versus Distance will showcase the potential correlation between operating margins and proximity to competitors. The estimation of the regression equation will then be performed, which allows predictions of operating margins based on the distances measured. Furthermore, the statistical significance of distance as a predictor will be investigated through hypothesis testing, calculating a p-value to measure significance.

In the subsequent parts, a multiple linear regression model will analyze various predictors towards operating margin beyond distance alone. It is essential to scrutinize the model's fit quality using summary measures like R² and adjusted R². The residuals will also be analyzed to ensure regression conditions are met, meaning that the assumptions of linearity, homoscedasticity, and normality are not violated.

In concluding the analysis, marginal contributions of specific variables (such as Quality of service) will be assessed, showcasing how well they predict operating margins in conjunction to other variables. `This includes calculating prediction intervals and confidence intervals around predicted operating margins, offering richer insights on associated variability and expected performance of new sites compared to existing ones.

Throughout this analysis, it will be crucial to document findings, including statistical outputs from Minitab or Excel, to substantiate the computations and interpretations derived from raw data. References will be included to back statistical methods and theories as required, allowing for a comprehensive understanding of the statistical relevance in managerial decisions.

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