Forecasting The Size of A Change A Sample Question Ted Mitchell

Market research has established that there is a linear relationship between the number of server hours, S, and the quantity of coffee, Q, that is sold in your café. You normally pay for S 1 = 200 server hours each week and sell Q 1 = 2,000 cups of coffee. However, in the past few weeks you have been experimenting with hiring more servers to improve the service level in the shop. You have found that when employ S 2 = 220 server hours in a week, your sales volume increases to Q 2 = 2,160 cups a week. What is the forecasted change in the number of cups that will sold, ∆Q 3, when you increase the number of server hours by ∆S 3 = 10 hours?

Characteristics of a Linear Relationship 1) You need two observations of the machine’s performance (X 1, Y 1 ) and (X 2, Y 2 ) 2) The ratio of the differences between the two performances has a constant value, m ∆Y/∆X = m

Linear Relationship Has Two Observations of Machine’s Performances Performance 1 Performance 2 Linear Performance Forecasted Size of Change in Output InputS 1 =200 server hours S 2 = 220 server hours ∆S = 20∆S 3 = 10 hours You can ignore the direct conversion rates, r = Q/S ∆Q/∆S = 160/20 m= 8 cups per change in server hour Conversion rate m = 8 cups per change in server hour OutputQ 1 =2,000 cups Q 2 = 2,160 cups ∆Q = 160 cups∆Q 3 = 8 x 10 ∆Q 3 = 80 cup change

Linear Relationship You need 1) the difference between the two observed Inputs ∆S = (S 2 -S 1 ) 2) the difference between the two observed Outputs ∆Q = (Q 2 -Q 1 ) To calculate the definitional characteristic of a linear relationship: ratio of the differences is a constant over the entire relationship ∆Q/∆S = meta-conversion rate, m

Calculating the Meta-conversion rate, ∆Q/∆S = m Performance 1 Performance 2 Linear Performance Forecasted Size of Change in Output InputS 1 =200 server hours S 2 = 220 server hours ∆S = 20∆S 3 = 10 hours Meta-conversion rate needed for a linear relationship, m = ∆Q/∆S ∆Q/∆S = 160/20 m= 8 cups per hour change in server hours Conversion rate m = 8 cups per change in server hour OutputQ 1 =2,000 cups Q 2 = 2,160 cups ∆Q = 160 cups∆Q 3 = 8 x 10 ∆Q 3 = 80 cup change

Forecasting The Size of Change Performance 1 Performance 2 Linear Performance Forecasted Size of Change in Output InputS 1 =200 server hours S 2 = 220 server hours ∆S = 20Proposed size of ∆S 3 = 10 hours Meta-conversion rate, m = ∆Q/∆S∆Q/∆S = 160/20 m= 8 cups per hour change in server hours Conversion rate m = 8 cups per hour change in server hours OutputQ 1 =2,000 cups Q 2 = 2,160 cups ∆Q = 160 cups ∆Q 3 = 8 x 10 ∆Q 3 = 80 cup change

Forecasting The Size of Change Performance 1 Performance 2 Linear Performance Forecasted Size of Change in Output InputS 1 =200 server hours S 2 = 220 server hours ∆S = 20Proposed size of ∆S 3 = 10 hours Meta-conversion rate, m = ∆Q/∆S∆Q/∆S = 160/20 m= 8 cups per hour change in server hours Conversion rate m = 8 cups per hour change in server hours OutputQ 1 =2,000 cups Q 2 = 2,160 cups ∆Q = 160 cups ∆Q 3 = m x ∆S 3 ∆Q 3 = 8 x 10 hours ∆Q 3 = 80 cup change

Forecasting the Change with a Linear Relationship Will be on the exam!