91Ó°ÊÓ

The accompanying data on \(x=\) advertising share and \(y=\) market share for a particular brand of cigarettes during 10 randomly selected years are from the article "Testing Alternative Econometric Models on the Existence of Advertising Threshold Effect" (Journal of Marketing Research \([1984]: 298-308)\). \(\begin{array}{lllllllllll}x & .103 & .072 & .071 & .077 & .086 & .047 & .060 & .050 & .070 & .052\end{array}\) \(\begin{array}{rlllllllll}y & .135 & .125 & .120 & .086 & .079 & .076 & .065 & .059 & .051 & .039\end{array}\) a. Construct a scatterplot for these data. Do you think the simple linear regression model would be appropriate for describing the relationship between \(x\) and \(y ?\) b. Calculate the equation of the estimated regression line and use it to obtain the predicted market share when the advertising share is . 09 . c. Compute \(r^{2}\). How would you interpret this value? d. Calculate a point estimate of \(\sigma .\) On how many degrees of freedom is your estimate based?

Step by step solution

01

Construct a Scatterplot

Use statistical or plotting software (or a graphing calculator) to plot \(y\) (market share) as a function of \(x\) (advertising share). The dots on the plot should appear to form a roughly straight line indicating a positive linear relationship.

02

Calculate the Equation of the Estimated Regression Line

Apply the method of least squares to calculate the equation of the regression line. The equation is of the form \(y = mx + b\), where \(m\) is the slope, obtained from the formula \(\frac{∑(x_i - \bar{x})(y_i - \bar{y})}{∑(x_i - \bar{x})^2}\) and \(b\) is the y-intercept, calculated from the formula \(\bar{y} - m\bar{x}\). Here, \(\bar{x}\) and \(\bar{y}\) are the means of \(x\) and \(y\) respectively, and \(x_i\) and \(y_i\) are individual observations.

03

Use the Regression Line to Predict Values

Substitute \(x = 0.09\) into the above equation to predict the market share.

04

Compute \(r^{2}\)

The coefficient of determination, or \(r^{2}\), is calculated as \(r^{2} = (1 - \frac{SS_{res}}{SS_{tot}})\), where \(SS_{res} = ∑(y_i - \hat{y_i})^2\) and \(SS_{tot} = ∑(y_i - \bar{y})^2\). Here, \(\hat{y_i}\) are the predicted values of \(y\) obtained from the regression model.

05

Calculate a Point Estimate of \(\sigma\)

The estimator of \(\sigma\) (standard deviation) is given by \(\sqrt{\frac{SS_{res}}{n-2}}\), where \(n\) is the number of observations. The degrees of freedom for this estimate is \(n-2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatterplot Construction

When examining the relationship between two variables, a scatterplot is a fundamental starting point. By plotting each pair of observations on a graph, with one variable on the x-axis and the other on the y-axis, we're able to visualize any apparent relationship between them.

For the cigarette brand data, plotting advertising share (x) against market share (y) will allow us to observe patterns; whether there's a linear relationship, clusters or outliers. A positive linear relationship would result in points that trend upward from left to right. The scatterplot acts as a crucial diagnostic tool to determine if the simple linear regression could be a suitable model—indeed necessary before proceeding with mathematical analysis. If data points form a roughly straight line, it suggests that simple linear regression could effectively describe the relationship.

Least Squares Method

In determining the best-fitting line through a scatterplot, we use the least squares method. This involves finding the line that minimizes the sum of the squares of the vertical distances of the points from the line.

This method provides us with the slope (\(m\)) and y-intercept (\(b\)) of our regression line. The slope is indicative of how much we expect the dependent variable (y) to change with a one-unit change in the independent variable (x). For the provided data, computing these using the given formulas will yield our regression equation, denoted as \(y = mx + b\). Each observed value of the independent variable can then be plugged into this equation to predict the corresponding value of the dependent variable, providing insight into the nature of their relationship.

Coefficient of Determination

The coefficient of determination, denoted as \(r^2\), measures the proportion of variability in the dependent variable that is explained by the regression model. It's a powerful indicator of the model's strength; a value closer to 1 implies a strong linear relationship, while a value near 0 suggests a weak relationship.

In simple linear regression, calculating \(r^2\) is straightforward using the residual sum of squares and the total sum of squares. Interpreting this statistic provides us valuable insight—specifically, for the given advertising and market share data, we can understand the extent to which changes in advertising share explain the variability in market share.

Point Estimate of Standard Deviation

The point estimate of standard deviation, also known as the standard error of the estimate, denotes the typical distance that observed values fall from the regression line. It gives us a measure of the precision of our predictions and is calculated using the residual sum of squares.

With the given formula \(\sqrt{\frac{SS_{res}}{n-2}}\), where \(n\) is the number of observations, we obtain an estimate of \(\sigma\), the standard deviation of the residuals. The degrees of freedom, \(n-2\), are used because two parameters (the slope and the intercept) are estimated from the data. This estimate helps in assessing the variability of the observation from the predicted regression line, informing us about the spread and accuracy of our model's predictions.