Question

The following data represent a company's yearly sales volume and its advertising expenditure over a period of 8 years.

Sales in Advertising Millions of Dollars in ($10,000) 32 15 16 18 17 16 19 19 24 35 34 36 37 39 42

1. Develop a scatter diagram of sales versus advertising and explain what it shows regarding the relationship between sales and advertising. Make sure your axes are labeled and that you have the axes correct (notice that x and y are above the labels). The graph should have a title that mentions the two variables.
2. Use the method of least squares to compute an estimated regression line between sales and advertising OR just show it on the graph along with your r-squared value..
3. If the company's advertising expenditure is $200,000, what are the predicted sales? Give the answer in dollars. (This is found using algebra and the equation of the trendline / regression line - make sure to watch your units)
4. What does the slope of the estimated regression line indicate?

Answer 1

To develop a scatter diagram of Sales (Y) versus advertising we measure Y (which is called response variable) on the vertical axis and advertising (X) along the horizontal axis. The scatter diagram is shown in the following diagram.

The corresponding observations of Y and X reveals approximately a linear path. therefore, the scatter diagram reveals that a linear relationship between sales and advertising and hence indicating that a linear regression model might be appropriate.

Let Y -a+bX be the equation of the best fitting straight line by the method of least squares. The constraints a and b are obtained by solving the normal equations Where n is the number of pair of observations 15 16 18 17 16 19 19 24 144 1024 1089 1225 1156 1296 1369 1521 1764 288 10444 xy 480 528 630 578 576 703 741 1008 5244 32 35 34 36 37 39 42 Total Putting ΣΥ-144, n-8, ΣΧ-288, ΣΧΥ-5244 and ΣΧ2-10444 in the normal equations, we have 144 = 8a + 288b (1) 5244 288a 10444b Solving (1) and (2) for a and b we get A -22.26 and b-1.12 Therefore the estimated regression equation is Y-22.26 1.12X To find the value of r2 we use the following formula Cov(X,Y) #xOY Now Cor(XY) ←-(?)(?) _ s244-(288)(14)-7.5 7.5 Therefore, r- 3.08 x 2.650.92