The table shows the advertising expenditures x and sales...

The table shows the advertising expenditures $x$ and sales volumes $y$ for a company for nine randomly selected months. Both are measured in thousands of dollars. $$\begin{array}{|c|c|c|} \hline \text { Month } & \begin{array}{c} \text { Advertising } \\ \text { expenditures, } x \end{array} & \begin{array}{c} \text { Sales } \\ \text { volume, } y \end{array} \\ \hline 1 & 2.4 & 215 \\ 2 & 1.6 & 182 \\ 3 & 1.4 & 172 \\ 4 & 2.3 & 214 \\ 5 & 2.0 & 201 \\ 6 & 2.6 & 230 \\ 7 & 1.4 & 176 \\ 8 & 1.6 & 177 \\ 9 & 2.0 & 196 \\ \hline \end{array}$$ a. Use the regression feature of a graphing utility to find a linear model for the data. b. Use the graphing utility to plot the data and graph the model in the same viewing window. c. Interpret the slope of the model in the context of the problem. d. Use the model to estimate sales for advertising expenditures of $\$ 1500 .$

The table shows the advertising expenditures $x$ and sales volumes $y$ for a company for nine randomly selected months. Both are measured in thousands of dollars.
$$\begin{array}{|c|c|c|}
\hline \text { Month } & \begin{array}{c}
\text { Advertising } \\
\text { expenditures, } x
\end{array} & \begin{array}{c}
\text { Sales } \\
\text { volume, } y
\end{array} \\
\hline 1 & 2.4 & 215 \\
2 & 1.6 & 182 \\
3 & 1.4 & 172 \\
4 & 2.3 & 214 \\
5 & 2.0 & 201 \\
6 & 2.6 & 230 \\
7 & 1.4 & 176 \\
8 & 1.6 & 177 \\
9 & 2.0 & 196 \\
\hline
\end{array}$$
a. Use the regression feature of a graphing utility to find a linear model for the data.
b. Use the graphing utility to plot the data and graph the model in the same viewing window.
c. Interpret the slope of the model in the context of the problem.
d. Use the model to estimate sales for advertising expenditures of $\$ 1500 .$

Key Concepts

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It involves finding the best-fitting straight line through the data points such that the sum of the squared differences between the observed values and the values predicted by the line is minimized.

Least Squares Method

The least squares method is a mathematical procedure used in linear regression to determine the line of best fit. By minimizing the sum of the squares of the vertical distances (residuals) between the actual data points and the predicted values, this method produces estimates of the slope and y-intercept that best represent the data.

Scatter Plot

A scatter plot is a type of graph that displays data points on a coordinate system, with one variable on the x-axis and the other on the y-axis. This visual representation helps in identifying the potential relationship or correlation between the variables and is a crucial step in determining if linear regression may be appropriate.

Slope Interpretation

The slope in a linear regression model represents the rate of change in the dependent variable for a one-unit change in the independent variable. It is a key component in understanding how variations in the predictor impact the outcome, indicating the magnitude and direction of the relationship.

Regression Prediction

Regression prediction involves using the established linear model to estimate the value of the dependent variable for a given value of the independent variable. This predictive capability allows for forecasting and decision-making based on trends identified in the historical data.

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