When observed over a long period of time, a time series data can predict trends that can forecast increase or decrease or stagnation of a variable under consideration. Such analytical studies can benefit a business for forecasting or prediction of future estimated sales or production. Mathematically, for finding a line of best-fit to represent a trend, many methods are available. Methods like moving averages and least squares are some of the techniques to predict such trends. Mr. Nitin runs a soap-making factory and the record of his sales of soaps for the period 2018 -2024 is as follows : | Year | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | 2024 | |---|---|---|---|---|---|---|---| | Sales (in ₹ thousands) | 80 | 90 | 92 | 83 | 94 | 99 | 92 | Based on the above information, answer the following questions : (i) Obtain the trend line to the given data. (ii) Find the average change in the sales. (iii) (a) Find the sum of the differences between the actual sales and the trend values (for 2018 - 2024). OR (iii) (b) What are the expected sales for the year 2025 ?

Answer: (i) The trend line is $Y = 90 + 2X$ . (ii) The average change in sales is ₹2 thousand per year. (iii) (b) The expected sales for the year 2025 are ₹98 thousand.

Step-by-step solution

Step 1: Calculate coded years and sums for trend line

To find the trend line using the method of least squares, we first need to code the years. Since there are 7 years (an odd number), the middle year 2021 is assigned a coded value of $x=0$ . The years before 2021 are assigned negative integers, and the years after are assigned positive integers. We then calculate $xy$ and $x^2$ for each year and find the sum of $y$ , $x$ , $xy$ , and $x^2$ . Here, $\sum y = 630$ , $\sum x = 0$ , $\sum xy = 56$ , and $\sum x^2 = 28$ . The number of data points is $n=7$ .

Step 2: Calculate constants 'a' and 'b' for the trend line

The equation of the trend line is given by $Y = a + bX$ , where $Y$ represents the sales and $X$ represents the coded year. The constants $a$ and $b$ are calculated using the formulas $a = \frac{\sum y}{n}$ and $b = \frac{\sum xy}{\sum x^2}$ . Substituting the calculated sums, we find $a = 90$ and $b = 2$ .

Step 3: Obtain the trend line equation

Substituting the values of $a$ and $b$ into the trend line equation $Y = a + bX$ , we get the trend line equation as $Y = 90 + 2X$ . This equation can be used to estimate sales for any given coded year.

Step 4: Calculate average change in sales

In the trend line equation $Y = a + bX$ , the constant $b$ represents the average change in sales per unit change in the coded year. Since $b=2$ , the average change in sales is ₹2 thousand per year.

Step 5: Calculate expected sales for 2025

To find the expected sales for the year 2025, we first determine its coded year value. Since 2024 has a coded value of $X=3$ , 2025 will have a coded value of $X=4$ . Substituting $X=4$ into the trend line equation $Y = 90 + 2X$ , we get $Y_{2025} = 90 + 2(4) = 98$ . Therefore, the expected sales for 2025 are ₹98 thousand.

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