Question

20.24 Xr20-24 The quarterly earnings (in $millions) of a large soft-drink manufacturer have been recorded for the years 2013–2016. These data are listed here. Compute the seasonal indexes given the regression line

y^=61.75+1.18t (t=1,2,...,16)

Year

2013 2014 2015 2016

Quarter

1 52 57 60 66

2   67 75 77 82

3   85 90 94 98

4  54 61 63 67

Use the seasonal indexes and trend line to forecast the quarterly earnings for the years 2014 and 2015 in Exercise 20.24 (above)

Answer 1

* In this question please check the years because to forecast the years, the years should not be same i.e if the year is 2013,2014,2015,2016 then the forecast years should be 2017 and 2018 and if the forecast is 2014 and 2015 then the years should be 2011,2012,2013,2014.

I have kept the years as it is as per the question. Please check the years and if you find the mistake in your question please change the years accordingly. *

Given :

Quarter	2013	2014	2015	2016
1	52	57	60	66
2	67	75	77	82
3	85	90	94	98
4	54	61	63	67

Solution :

Year	Quarter	Period (t)	Earnings (y)	ŷ = 61.75 + 1.18t	Ratio = y/ŷ
2013	1	1	52	62.93	0.83
2013	2	2	67	64.11	1.05
2013	3	3	85	65.29	1.30
2013	4	4	54	66.47	0.81
2014	1	5	57	67.65	0.84
2014	2	6	75	68.83	1.09
2014	3	7	90	70.01	1.29
2014	4	8	61	71.19	0.86
2015	1	9	60	72.37	0.83
2015	2	10	77	73.55	1.05
2015	3	11	94	74.73	1.26
2015	4	12	63	75.91	0.83
2016	1	13	66	77.09	0.86
2016	2	14	82	78.27	1.05
2016	3	15	98	79.45	1.23
2016	4	16	67	80.63	0.83

Average of ratio of the every quarter from the all years will the seasonal index, hence

Quarter/ Year	1	2	3	4
2013	0.83	1.05	1.30	0.81
2014	0.84	1.09	1.29	0.86
2015	0.83	1.05	1.26	0.83
2016	0.86	1.05	1.23	0.83
Seasonal Index	0.84	1.06	1.27	0.83

Forecast for the quarterly earnings for the years 2014 and 2015 will be

Year	Quarter	Period (t)	ŷ = 61.75 + 1.18t	Seasonal Index	Forecast = ŷ * Seasonal Index
2014	1	17	81.81	0.84	68.72
2014	2	18	82.99	1.06	87.97
2014	3	19	84.17	1.27	106.90
2014	4	20	85.35	0.83	70.84
2015	1	21	86.53	0.84	72.69
2015	2	22	87.71	1.06	92.97
2015	3	23	88.89	1.27	112.89
2015	4	24	90.07	0.83	74.76