Question

Use the following question to answer #12-15 A marketing manager conducted a study to determine whether there is a linear relationship between money spent on advertising and company sales. The data are shown in the table below.give a brief explanation of the type of relationship. assume all assumptions have been met

Answer 1

The question deals with identifying a linear relationship between two variables using statistical methods such as correlation and regression, relevant in college-level mathematics and statistics.

Step-by-step explanation:

The student's question pertains to the study of the relationship between two measured quantities, which suggests a topic within mathematics, particularly in the field of statistics and regression analysis. This is a common subject at the college level where students learn how to analyze data, describe relationships between variables, and use linear regression techniques to model such relationships.

When addressing the student's questions, it is important to identify the independent variable and the dependent variable, which will guide the creation of a scatter plot. This visual representation allows us to initially assess whether there could be a linear relationship between the two variables. With a scatter plot, one can visually inspect the pattern that the data points make when graphed, which may suggest some form of association. If the points tend to cluster around a straight line, this indicates a linear relationship.

To formally measure the strength and direction of the linear relationship, we calculate the least-squares regression line and the correlation coefficient. The regression line takes the form ŷ = a + bx, with 'a' representing the y-intercept and 'b' the slope of the line. The correlation coefficient, denoted by 'r', quantifies the strength and direction of the linear relationship between the independent and dependent variables. A value of 'r' close to 1 or -1 suggests a strong relationship, whereas a value close to 0 indicates a weak relationship.

In addition, any outliers or points with large residuals should be investigated, as they can significantly affect the regression analysis. If one finds that there is a significant linear relationship, we can use the regression line to make predictions about the dependent variable based on new observations of the independent variable.