A Sales Manager For An Advertising Agency Believes There Is
A Sales Manager For An Advertising Agency Believes There Is a Relation
A sales manager for an advertising agency hypothesizes that there is a relationship between the number of contacts made by salespeople and the total sales generated. To test this hypothesis, data was collected, and various statistical analyses were performed to examine the relationship and determine the nature of this link. The data focuses on key variables such as the number of contacts (independent variable), the sales amount (dependent variable), and other statistical measures to verify this relationship.
Assignment instructions:
- Identify the dependent variable
- Identify the independent variable
- Determine the Y-intercept of the linear regression equation
- Calculate the slope of the linear regression equation
- Find the standard error of estimate
- Determine the coefficient of correlation
- Find the coefficient of determination
- Establish the 95% confidence interval for 30 calls
- Find the 95% prediction interval for a person making 30 calls
- Specify the regression equation
- Understand the assumptions required for a valid multiple regression analysis
- Measure the degree of association between independent variables and the dependent variable
Paper For Above instruction
The investigation into the relationship between the number of contacts and sales in an advertising agency relies heavily on regression analysis to quantify and validate the hypothesized connection. The analysis begins with identifying the dependent and independent variables, which are fundamental for setting the stage for further statistical testing. In this context, the dependent variable (Y) is the amount of sales generated, as it depends on the number of contacts made, which serves as the independent variable (X).
Dependent and Independent Variables
In regression analysis, the dependent variable is the outcome we aim to predict or explain. Here, the sales amount (in thousands) is the dependent variable because it is influenced by the number of contacts. Conversely, the independent variable is the predictor; in this case, the number of contacts made by salespeople. Therefore, the correct answers are:
- Dependent variable: C) Amount of sales
- Independent variable: B) Number of contacts
Regression Equation Parameters
The regression equation models the relationship between contacts and sales as \(Y' = b_0 + b_1X\), where \(b_0\) is the Y-intercept and \(b_1\) is the slope. Based on the given data, the Y-intercept is approximately -12.201, and the slope of the line is roughly 2.1946, indicating that each additional contact is associated with an increase of about 2.1946 thousand dollars in sales.
Statistical Measures
The standard error of estimate, which reflects the typical deviation of observed sales from predicted sales, is about 8.778. The coefficient of correlation (\(r\)) measures the strength and direction of the linear relationship; an \(r\) of approximately 0.9754 suggests a very strong positive correlation. The coefficient of determination (\(R^2\)) of roughly 0.9513 indicates that nearly 95% of the variation in sales is explained by the number of contacts.
Interval Estimates
A 95% confidence interval provides a range within which we expect the true mean sales for calls totaling 30 with 95% confidence; the interval is approximately 51.4 to 55.9 thousand dollars. The prediction interval, which forecasts the sales for a single individual making 30 calls, is broader, roughly from 46.7 to 60.6 thousand dollars, reflecting the uncertainty around individual predictions.
Regression Equation
Considering the parameter estimates, the regression equation is:
\[Y' = -12.201 + 2.1946 \times X\]
This equation allows predicting sales based on the number of contacts.
Assumptions in Regression Analysis
A valid multiple regression analysis requires several assumptions:
- The dependent variable should be measured on at least an interval scale.
- The residuals (errors) are normally distributed with constant variance (homoscedasticity).
- There must be a linear relationship between predictors and the dependent variable.
- The observations should be independent, meaning no autocorrelation.
Measuring the Degree of Association
The strength of the relationship between the independent variables and the dependent variable is quantified using the coefficient of multiple determination (\(R^2\)). In simple linear regression, the square of the correlation coefficient (\(r^2\)) measures the proportion of variance in sales explained by contacts, indicating a very strong association.
Conclusion
The statistical analysis reveals a significant, positive linear relationship between the number of contacts and sales in the advertising agency. The data supports the sales manager’s belief that increasing contacts can effectively boost sales, with a high degree of confidence. Nonetheless, the assumptions underlying regression analysis must be satisfied for these results to be valid, emphasizing the importance of proper model diagnostics in practical applications.
References
- Agresti, A., & Franklin, C. (2017). Statistics: The Art and Science of Learning from Data. Pearson.
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
- Hair, J. F., Black, W. C., Babin, B. J., & Anderson, R. E. (2019). Multivariate Data Analysis. Cengage Learning.
- Montgomery, D. C., Peck, J. P., & Vining, G. G. (2012). Introduction to Linear Regression Analysis. Wiley.
- Neter, J., Kutner, M. H., Nachtsheim, C. J., & Wasserman, W. (1996). Applied Linear Regression Models. McGraw-Hill.
- Sheskin, D. J. (2011). Survey Research Methods. CRC Press.
- Tabachnick, B. G., & Fidell, L. S. (2013). Using Multivariate Statistics. Pearson.
- Wooldridge, J. M. (2013). Introductory Econometrics: A Modern Approach. Cengage Learning.
- Ott, R. L., & Longnecker, M. (2015). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
- Myers, R. H. (2011). Classical and Modern Regression with Applications. PWS-Kent Publishing.