(a) Calculate the seasonal index for the monthly sales of a product using the method of simple averages. | Months | Year 2001 | 2002 | 2003 | | --- | --- | --- | --- | | Jan | 15 | 20 | 18 | | Feb | 41 | 21 | 16 | | Mar | 25 | 27 | 20 | | Apr | 31 | 19 | 28 | | May | 29 | 17 | 24 | | June | 47 | 25 | 25 | | July | 41 | 29 | 30 | | Aug | 19 | 31 | 34 | | Sep | 35 | 35 | 30 | | Oct | 38 | 39 | 38 | | Nov | 40 | 30 | 37 | | Dec | 30 | 44 | 39 | OR (b) Consider a continuous random variable $X$ with probability density function. $f(x)=\begin{cases}2e^{-2x}, & x>0\\0, & \text{otherwise}\end{cases}$ Find $E(X)$ and $V(X)$

> > [!TLDR] > (a) Monthly averages over 3 years divided by the grand average ($29.64$) times $100$ give the twelve seasonal indices (summing to $1200$). (b) Exponential with $\lambda=2$: $E(X)=0.5$, $V(X)=0.25$. > > **Part (a) — Method of simple averages.** > > | Month | 2001 | 2002 | 2003 | Monthly avg | Seasonal index | > | --- | --- | --- | --- | --- | --- | > | Jan | 15 | 20 | 18 | 17.67 | 59.6 | > | Feb | 41 | 21 | 16 | 26.00 | 87.7 | > | Mar | 25 | 27 | 20 | 24.00 | 81.0 | > | Apr | 31 | 19 | 28 | 26.00 | 87.7 | > | May | 29 | 17 | 24 | 23.33 | 78.7 | > | June | 47 | 25 | 25 | 32.33 | 109.1 | > | July | 41 | 29 | 30 | 33.33 | 112.5 | > | Aug | 19 | 31 | 34 | 28.00 | 94.5 | > | Sep | 35 | 35 | 30 | 33.33 | 112.5 | > | Oct | 38 | 39 | 38 | 38.33 | 129.3 | > | Nov | 40 | 30 | 37 | 35.67 | 120.3 | > | Dec | 30 | 44 | 39 | 37.67 | 127.1 | > > Grand average $=\dfrac{\text{sum of the 12 monthly averages}}{12}=\dfrac{355.67}{12}=29.64.$ > Seasonal index for a month $=\dfrac{\text{monthly average}}{29.64}\times100$; e.g. Jan $=\dfrac{17.67}{29.64}\times100=59.6.$ The twelve indices sum to $1200$ (average $100$), confirming the working. > …

(a) Calculate the seasonal index for the monthly s – Class 12 Business Mathematics Statistics

Question 35 of 39

(a) Calculate the seasonal index for the monthly sales of a product using the method of simple averages.

Months	Year 2001	2002	2003
Jan	15	20	18
Feb	41	21	16
Mar	25	27	20
Apr	31	19	28
May	29	17	24
June	47	25	25
July	41	29	30
Aug	19	31	34
Sep	35	35	30
Oct	38	39	38
Nov	40	30	37
Dec	30	44	39

(b) Consider a continuous random variable $X$ with probability density function.

$f(x)=\begin{cases}2e^{-2x}, & x>0\\0, & \text{otherwise}\end{cases}$

Find $E(X)$ and $V(X)$

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(a) Monthly averages over 3 years divided by the grand average ( $29.64$ ) times $100$ give the twelve seasonal indices (summing to $1200$ ). (b) Exponential with $\lambda=2$ : $E(X)=0.5$ , $V(X)=0.25$ .

Part (a) — Method of simple averages.

Month	2001	2002	2003	Monthly avg	Seasonal index
Jan	15	20	18	17.67	59.6
Feb	41	21	16	26.00	87.7
Mar	25	27	20	24.00	81.0
Apr	31	19	28	26.00	87.7
May	29	17	24	23.33	78.7
June	47	25	25	32.33	109.1
July	41	29	30	33.33	112.5
Aug	19	31	34	28.00	94.5
Sep	35	35	30	33.33	112.5
Oct	38	39	38	38.33	129.3
Nov	40	30	37	35.67	120.3
Dec	30	44	39	37.67	127.1

Grand average $=\dfrac{\text{sum of the 12 monthly averages}}{12}=\dfrac{355.67}{12}=29.64.$

Seasonal index for a month $=\dfrac{\text{monthly average}}{29.64}\times100$ ; e.g. Jan $=\dfrac{17.67}{29.64}\times100=59.6.$ The twelve indices sum to $1200$ (average $100$ ), confirming the working.

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›Reveal solutionSolution

A time series is data arranged chronologically, i.e. recorded at successive points of time; option (b).

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