A company sets different prices for a particular DVD system...

A company sets different prices for a particular DVD system in eight different regions of the country. The accompanying table shows the numbers of units sold and the corresponding prices (in dollars). $$ \begin{array}{lrrrrrrrr} \hline \text { Sales } & 420 & 380 & 350 & 400 & 440 & 380 & 450 & 420 \\ \hline \text { Price } & 104 & 195 & 148 & 204 & 96 & 256 & 141 & 109 \\ \hline \end{array} $$ a. Graph these data, and estimate the linear regression of sales on price. b. What effect would you expect a $$\$ 50$$ increase in price to have on sales?

A company sets different prices for a particular DVD system in eight different regions of the country. The accompanying table shows the numbers of units sold and the corresponding prices (in dollars).
$$
\begin{array}{lrrrrrrrr}
\hline \text { Sales } & 420 & 380 & 350 & 400 & 440 & 380 & 450 & 420 \\
\hline \text { Price } & 104 & 195 & 148 & 204 & 96 & 256 & 141 & 109 \\
\hline
\end{array}
$$
a. Graph these data, and estimate the linear regression of sales on price.
b. What effect would you expect a $$\$ 50$$ increase in price to have on sales?

Key Concepts

Scatter Plot

A scatter plot is a graph that displays paired numerical data to visually assess the relationship between two quantitative variables. It is an essential tool for initial data analysis, as it helps to determine whether a linear or some other form of relationship exists between the variables.

Simple Linear Regression

Simple linear regression is a statistical method used to model the relationship between a dependent variable and a single independent variable using a straight line. The method estimates the linear equation that best describes the observed data, providing a basis for prediction and analysis.

Least Squares Estimation

Least squares estimation is the procedure used in regression analysis to determine the best-fitting line by minimizing the sum of the squares of the vertical distances (errors) between the observed data points and the line. This method ensures that the overall prediction error is minimized.

Regression Coefficient Interpretation

In the context of a linear regression model, the slope represents the rate of change in the dependent variable for a one-unit increase in the independent variable, while the intercept represents the expected value of the dependent variable when the independent variable is zero. Understanding the slope is crucial for interpreting the effect of changes, such as assessing the impact of a $50 increase in price on sales.

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