Answer to Question #168011 in Statistics and Probability for Nida

Question

Trevor Harty , an avid mountain biker always wanted to start a business selling top of the line mountain bikes and other outdoor supplies , A little over 6 yearv ago , he and a silent partner opened  a store called Hale and Harty Trail bikes and supplies. Growth was rapid in the first 2 yeard, but since that time , growth in sales has slowed a bit , as expressed , The quarterly sales in ( $1000) for past 4 years.

Year1 year2 year3 year4

274  282     282      296

172   178      182       210

130    136     134      158

16    168.      170       182

(a) Develop atrend using data ,Use this to forecast sales for each quarter of year 5.What doess slope of the line indicates?

(b) use the multiplicative decompostion model to incorporate both trend and seasonal components into the forecast, What does the slope of this line indicate ?

Compare the slope of trend line in part(a) to the slope in the trend line for decomposition for trend that based on deseasonalized sales figure , Discuss why thes are so different and explain which one is best route?

Answer 1

Using online calculator (https://www.socscistatistics.com/tests/regression/default.aspx) for regression line we get:

quarter 1: $y=6.6x+267$ , $x=5,y=300$

quarter 2: $y=11.8x+156$ , $x=5,y=215$

quarter 3: $y=8.2x+119$ , $x=5,y=160$

quarter 4: $y=50x+9$ , $x=5,y=259$

Slope of the line indicates speed of sales growth.

b) The multiplicative decomposition has the form:

$Y_t=T_t\cdot S_t\cdot E_t$

$Y_t$ - actual data at period t;

$T_t$ - deterministic trend-cycle or general movement component;

$S_t$ - deterministic seasonal component;

$E_t$ - irregular (remainder or residual) (stationary) component

Using online calculator:

Slope of line is rate of change of sales growth.

c) Multiplicative decomposition is more prevalent with economic series

because most seasonal economic series do have seasonal variations which

increase with the level of the series.