sample quantitative questions chapter 3 ted mitchell

Sample Quantitative QuestionsChapter 3

Ted Mitchell

The Eventual Goal is

• To take two observations of marketing performance by replaying the same turn (changing only the Price, P, and thus the Quantity sold, Q) and

• To calculate the optimal selling price (Input)that will maximize the amount of gross profit generated (output).

Single Point of Observation with Calculated Conversion Rate and Forecasted Point

H = Hours Open

Quantity, Q

x x x

x x x

x x

x x

x x x

x

x +x

Q = r x HQ = (Q/H) x H

X

Two Points of Observation with Calculated Meta-Conversion Rate in Slope-Origin Form

∆H = change in Hours Open

∆Q = change in Quantity

x x x

x x x

x x

x x

x x x

x

x

x

∆Q = m x ∆H∆Q = (∆Q/∆H) x ∆H

X

Meta-machine in Slope-Intercept Form with Conversion Rate, m, and Intercept, a

H = Hours Open

Quantity, Q

x x x

x x x

x x

x xx x

x x x

x

x

x x

x

Q = a + (m)(H)

Xa

X

Multi-Point Estimation

H = Hours Open

Quantity, Q

x x x

x x x

x x

x xx x

x x x

x

x

x x

x

Q = kHh

Multi-point Meta-Conversion Rate

H = Hours Open

Quantity, Q

x x x

x x x

x x

x xx x

x x x

x

x

x x

x

Q = 6,000 + 40cps(H)

Q = kHh


Profits From Store Hours

H = Hours Open

Profit, Z

Reaching the Goal of Profit Analysis takes Four steps

• To take two observations of marketing performance by replaying the same turn (Price and Quantity sold) and

• 1) calculate the meta-conversion rate in the Slope-Origin Format

• Can you make a forecast of the change in output (demand, quantity sold) given a proposed change in input (Price) and a Meta-Marketing machine in Slope-Origin Format?

Reaching the Goal of Profit Analysis takes Four steps

• 2) Conceptualize the Slope-intercept Form of a Meta-Marketing Machine

• Can you convert the Slope-Origin Version into a Slope-Intercept Version by calculating the y-intercept?

• Can you use slope-intercept equation of the meta-demand machine to forecast the amount of output (demand, quantity sold) to be expected from a proposed amount of input (Price)?

Reaching the Goal takes Four steps

• 3) Conceptualize the demand model and convert it to a Revenue machine

• Can you Forecast the amount of revenue for any proposed level of price?

• Calculate the optimal amount of input (price) to produce the maximum amount of revenue (output)

Reaching the Goal takes Four steps

• 4) Conceptualize the Revenue model and convert it to a Profit machine

• Can you Forecast the amount of gross profit for any proposed level of price?

• Calculate the optimal amount of input (price) to produce the maximum amount of gross profit (output)

Today We are at Step #2

• To take two observations of marketing performance by replaying the same turn (Price and Quantity sold) and

• 2) calculate the y-intercept and use slope-intercept equation of the meta-demand machine

• forecast the amount of output (demand, quantity sold) to be expected from a proposed amount of input (Price)

1) Calculate the meta-conversion ratefrom two observations

• You have observed two coffee shop performances

• Observation #1: open 100 hours and sold 10,000 cups of coffee at a rate of 100 cups per hour, cph

• Observation #2 open 120 hours and sold 10,800 cups of coffee at a rate of 80 cups per hour, cph

• What is the meta-conversion rate, m?


• Meta-coffee from hours marketing machine is• Difference in output =

(meta-conversion rate, m)x difference in input• ∆Q = (meta-conversion rate, m) x ∆hours• Measure Difference in output, ∆Q = Q2-Q1

∆Q = 10,800 – 10,000 = 800 cups• Measure Difference in Input, ∆H = H2-H1

∆H = 120 hours -100 hours = 20 hours• What is the Meta-conversion rate?

Single Point Meta-Marketing Machine

Input Factor, ∆π

Output, ∆ø

0, 0

∆π

∆ø X

Calibration point

(∆π, ∆ø)Meta-Conversion rate

m = rise/run

m = ∆ø/∆π

Slope-Point Equation for Two-Factor Meta-machine

Rise

Run


Observation 1 Observation 2 Meta-demand marketing machine

Hours open, H

100 120

Cups per hour, cps = Q/H

100 80 m = ?

Cups sold, Q 10,000 10,800

What is the meta-conversion rate, m, for meta-demand marketing machine?


Observation 1 Observation 2 Meta-Demand Marketing Machine

Hours open, H

100 120 ∆H = 20

Cups per hour, cph = Q/H

100Not Used

80Not Used

m = ∆Q/∆Hm = 800/20 =m = 40 cph

Cups sold, Q 10,000 10,800 ∆Q = 800

Remember the single point r = cph is not used

1) Calculate the meta-conversion ratefrom two observed performances

• You don’t have to use the table format• ∆Q = (meta-conversion rate, m) x ∆hours• Measure Difference in output, ∆Q = Q2-Q1

∆Q = 10,800 – 10,000 = 800 cups• Measure Difference in Input, ∆H = H2-H1

∆H = 120 hours -100 hours = 20 hours• Calculate Meta-conversion rate, m = ∆O/∆H• Meta-conversion rate, m = 800 cups/20 hrs =

40cph

#2 Forecast a change in output, ∆O, from a Meta-Marketing Machine

• You know the calibrate meta-machine that explains the differences in observed performances for coffee sales, ∆Q, from store hours, ∆H is represented as

• Output, ∆Q = 40cph x Input, ∆H• The boss is proposing a decrease in the number

of store of ∆H = -5 hours• What is the forecasted change in the number

of number of cups sold, ∆Q?

Single Point Meta-Marketing Machine

Input Factor, ∆π

Output, ∆ø

0, 0

∆π = 20

∆ø = 800 X

Calibration point

(800, 20)Meta-Conversion rate

m = rise/run

m = 40 cph

Slope-Point Equation for Two-Factor Meta-machine

Rise

Run

#2 Forecast a change in output, ∆O, from a Meta-Marketing Machine

• You know the calibrate meta-machine that explains the differences in observed performances for coffee sales, ∆Q, from store hours, ∆H is represented asOutput, ∆Q = 40cph x Input, ∆HThe boss is proposing a decrease in the number of store of ∆H = -5 hoursWhat is the forecasted change in the number of number of cups sold, ∆Q?Answer

• ∆Q = 40cph x -5 hours = -200 cups

2) Forecast a Change in output, ∆O, from a Meta-Marketing Machine

Observation 1 Observation 2

Meta-demand marketing machine

Forecasted change

Hours open, H

100 120 ∆H = 20 ∆H = -5


100 80 m = ∆Q/∆Hm = 800/20 =m = 40 cph

m = 40 cph

Cups sold, Q

10,000 10,800 ∆Q = 800 cups

∆Q = -200 cups

Forecasting the change in quantity sold due to

a change in one of the marketing inputs• Works in concert with the analysis of a

breakeven quantity, BEQ, to calculate the minimum quantity that must be sold to cover the increase in the expenditure of marketing input

• In a future chapter we shall calculated breakeven quantity, BEQ

#3 Forecast an Actual Output• In week 2 you are open for 120 hours a week and are selling

10,800 cups of coffee at at rate of 80 cups per hour.• The Boss is proposing a reduction in store hours of 5 hours a

week and you have forecasted a 200 cup reduction in sales in week 3.

• Three related questions:• 1) What is the proposed number of hours in week 3?• H3 = 120 hours – 5 hours = 115 hours• 2) What is the forecasted sales volume in week 3?• Q3 = 10,800 cups - 200 cups = 10,600 cups• 3) What is the forecasted rate of sales per hour• Conversion rate, r = Q/H = 10,600/115 = 92.17 cph

Modification: to forecast an outcome from a current point of performance.

• Market research has provided you with a calibrated equation of the single point slope equation of a meta-marketing machine

• Change in Coffee sales, ∆Q = (40 cph) x Change in store hours, ∆H∆Q = 40 cph x ∆H∆Q= (40 cph) x -5 hours

• Convert the ∆Q and the ∆H to the proposed point and forecast minus the current point (input, output)(Q3 – Q2) = (40cph) x (H3 - H2)(Q3 – 10,800) = (40cph) x (H3 – 120)

• ∆H = H3 – 120 hours-5 = H3 – 120 hoursH3 = 115 hours

• (Q3 – 10,800) = (40cph) x (115 – 120)

• Q3 = 10,800 + (40cph) x -5 hours

• Q3 = 10,800 – 200 = 10,600 cups

#3 Forecast a Specific Output and Conversion Rate

Week 2 Forecasted change

Forecasted week 3

Meta-marketing machine

Hours open, H

120 hours

∆H = -5 hours Proposed Input 115 hours

∆H


80 cph m = 40 cph Forecasted r ≠ m

m = ∆Q/∆Hm = 40 cph

Cups sold, Q 10,800 cups

∆Q = -200 cups

Forecasted Output = Q3 = 10,600 cups

∆Q

The forecasted conversion rate, r, is not the meta-conversion rate, m

#3 Forecast a Specific Output and Conversion Rate

Week 2 Forecasted change

Forecasted week 3

Meta-marketing machine

Hours open, H

120 hours

∆H = -5 hours Proposed Input 115 hours

∆H


80 cph m = 40 cph Forecasted conversion rate,r=10,600/115r = 92.17 cph

m = ∆Q/∆Hm = 40 cph

Cups sold, Q 10,800 cups

∆Q = -200 cups

Forecasted Output = 10,600 cups

∆Q

The forecasted conversion rate, r, is not the meta-conversion rate, m

Meta-Conversion Rate, m

H = Hours Open

Quantity, Q

x x x

x x x

x x

x xx x

x x x

x

x

x x

x

Q = a + (m cph)(H)

+


OX

When you have to do this calculation many times

• 1) Forecasting a specific outcome is awkward when using the single point Slope-Origin equation of the meta-marketing ‘coffee sales from hours’ machine

• Does not help us easily identify optimal levels of store hours for maximum profit

We want a slope-intercept equation

H = Hours Open

Quantity, Q

x x x

x x x

x x

x xx x

x x x

x

x

x x

xLinear Meta-Machine is a secant that approximates the Quantity sold as a function of hours open

Q = a + m(H)Q = 6,000 + (40cph)(H)

a = 6,000 cups

5) Construct a slope-intercept equation of a meta-marketing machine

• Market research has provided you with a calibrated equation of the single point slope equation of a meta-marketing machine

• Change in Coffee sales, ∆Q = (meta-conversion rate, 40 cph) x Change in store hours, ∆H

∆Q = 40 cph x ∆H• (Q3-10,800 cups) = 40cph x (H3 – 120 hours)• What is the y-intercept of the slope-intercept equation of the meta-marketing

machine?• Set H3 = zero and Q3 = value of y-intercept, a• a -10,800 cups = 40 cph x (0-120 hours)• a = 10,800 cups – 4,800 cups• a = 6,000 cups sold when the store is closed and only the drive through is open• What is the slope-intercept equation of the meta-marketing machine?• Forecasted Cups sold, Q = 6,000 cups + (40 cps)(Hours open, H)

Linear versus Q = kHh

H = Hours Open

Quantity, Q

x x x

x x x

x x

x xx x

x x x

x

x

x x

xLinear Meta-Machine is a secant that approximates the Quantity sold as a function of hours open

Q = 6,000 + (40cph)(H)

Q = kHh

6) Forecast an output from the slope-intercept equation given a proposed level of input

• Market research department has provided an estimate of the relationship between hours the store is open and the number of cups that is sold as a slope-intercept equation

• Q = a + m(H)• Forecasted Cups sold, Q = 6,000 cups + (40 cph)(Hours open,

H)• It is proposed to stay open for 105 hours a week. What is the

forecasted sales volume for staying open that many hours?Answer:

• Cups sold, Q = 6,000 cups + (40 cph)(105 hours)• Cups sold, Q = 6,000 cups + 4,200 cups = 10,200 cups

We need to forecast demand, quantity sold, Q

• From changes in any one of the 4 P’s as input• Promotion (advertising, radio spots, servers)• Product quality ( coffee, servers)• Place ( ambience, location, hours open• and • Price Tag

• slope-intercept equations for representing meta-marketing machines from two observations are always constructed in the same way to provide:

• Demand, Q = a + m(positive, π)• Demand, Q = a – m(negative, P)

7) Forecast a Quantity sold from a proposed Price tag

• Market Research has provided you with the slope-intercept equation which they are calling a demand curve given the price tag on each cup, P, is an input

• Q = a – m(P)• Quantity Demanded, Q

= 6,000 cups -900 cpP x Price tag, P• When the price is $4.00 for a cup what is the

forecasted quantity of cups sold, Q?• Quantity, Q = 6000 cups – 900 cp$ x ($4)• Quantity, Q = 6000 cups – 3,600 cups = 2,400 cups

Lower Price Sells More Units

Price per Cup$3.90 $4.00

Quantity

Sold

2,400

Demand Equation

Q = 6,000 – 900(P)

Revenue = 2,400 x $4.00Revenue = $9,600

TJM

The Demand Equation

• Q = a – m(P)Often called demand equation or price response function

• The Slope-Intercept equation of the meta-marketing machine that produces a demand or quantity sold, Q, using the price tag, P, as an input

• Any Questions?

sample quantitative questions chapter 3 ted mitchell

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