Question

A car dealer in Big Rapids, Michigan is using Holt’s method to forecast weekly car sales. Currently the level is estimated to be 40 cars per week, and the trend is estimated to be 5 cars per week. During the current week, 20 cars are sold. After observing the current week’s sales, forecast the number of cars three weeks from now. Use a = B = 0.20

Answer 1

The forecast of the number of cars 3 weeks from now is 52 cars.

Explanation:

As per the trent the number of the cars per week is 5 cars

The current level of cars is 40 cars per week

Number of cars sold in current week=20 cars

Forecast of the cars sold 3 weeks from now is given as

`L_t=\alpha Y_t+(1-\alpha)(L_(t-1)+T_(t-1))\\`

From the data

Y_t=20 cars

L_t-1=40 cars

T_t-1=5 cars

α=β=0.2

So the equation becomes

`L_t=\alpha Y_t+(1-\alpha)(L_(t-1)+T_(t-1))\\L_t=0.2*20+(1-0.2)(40+5)\\L_t=40`

Now the trend is calculated as

`T_t=\beta(L_(t)-L_(t-1))+(1-\beta)T_(t-1)`

By putting the values the equation becomes

`T_t=\beta(L_(t)-L_(t-1))+(1-\beta)T_(t-1)\\T_t=0.2(40-40)+(1-0.2)5\\T_t=0+0.8*5\\T_t=4`

Now the forecast of the cars sale 3 weeks from now is given as

`L_(t+k)=L_t+kT_t`

where k is 3 so

`L_(t+k)=L_t+kT_t\\L_(t+3)=40+3*4\\L_(t+3)=40+12\\L_(t+3)=52\\`

So the forecast of the number of cars 3 weeks from now is 52 cars.

Answer 2

49 cars

Explanation:

Probability of cars to be sold in a week,a = 0.2

Probabiity of cars not sold in a week, b = 0.8

Number of cars estimated to be sold in a week = 20 and 60 cars in 3 weeks

Using, P(x) = nCx *(a)∧x * (b)∧n - x, where n = 3 weeks, x = 1 week

P(x=1) = 3C1 * (0.2) * (0.8)² = 3 X 0.2 X 0.64 X 60 cars = 23 cars

P(x=2) = 3C2 * (0.2)² * (0.8) = 3 X 0.04 X 0.8 X 60 cars = 6 cars

Number of cars three weeks from now: 20 + 23 + 6 = 49 cars