Question

A study was carried out to relate sales revenue \(y\) (in thousands of dollars) to advertising expenditure \(x\) (also in thousands of dollars) for fast-food outlets during a 3-month period. A sample of 15 outlets yielded the accompanying summary quantities. $$ \begin{aligned} &\sum x=14.10 \quad \sum y=1438.50 \quad \sum x^{2}=13.92 \\ &\sum y^{2}=140,354 \quad \sum x y=1387.20 \\ &\sum(y-\bar{y})^{2}=2401.85 \quad \sum(y-\hat{y})^{2}=561,46 \end{aligned} $$ a. What proportion of observed variation in sales revenue can be attributed to the linear relationship between revenue and advertising expenditure? b. Calculate \(s\), and \(s_{b}\). c. Obtain a \(90 \%\) confidence interval for \(\beta\), the average change in revenue associated with a \(\$ 1000\) (that is, 1 -unit) increase in advertising expenditure.

Short answer

a) R-square value represents the proportion of variation. b) \(s\) and \(s_{b}\) are the estimated standard deviation of residuals and standard error of the regression slope respectively. c) The 90% confidence interval for beta gives us the range of plausible values for the average change in revenue for a $1000 increase in expenditure.

Step-by-step solution

Calculate Beta

First, we need to calculate beta, the slope of the regression line, using the formula \(\beta = \frac{\sum xy - n \bar{x}\bar{y}}{\sum x^2 - n \bar{x}^2} = \frac{1387.2 - (1/15)*(14.1)*(1438.5)}{13.92 - (1/15)*(14.1)^2}\)

Calculate R-Square

The proportion of observed variation explained by the model is represented by R-square. It can be computed using the formula \(R^2 = 1 - \frac{\sum (y - \hat{y})^2}{\sum (y - \bar{y})^2}\) Plug in the given values from the problem into this formula.

Calculate S

To calculate the standard deviation of residuals, or S, use the following formula: \(s = \sqrt{\frac{\sum(y-\hat{y})^2}{n-2}} = \sqrt{\frac{561.46}{15-2}}\).

Calculate Sb

Next, use the calculated s from step 3 and the given \(\bar{x}\) to compute sb, the standard error of the slope coefficient, using the formula \(s_{b} = \frac{s}{\sqrt{\sum (x - \bar{x})^2}}\).

Confidence Interval for Beta

The formula to find a 90% confidence interval for beta is \(\beta \pm t_{n-2,1-\alpha/2} * s_{b}\) where \(t_{n-2,1-\alpha/2}\) is the \(1-\alpha/2\) quantile from a t-distribution with \(n-2\) degrees of freedom. Take \(\alpha = 0.1\) for 90% confidence interval and use a t-table to find the quantile.