Find the average change in the sales.

Q: How is the average change related to the trend line?

In the linear trend $Y = a + bX$, the average change per year is exactly the slope b. From the previous part $b = 2$ (in Rs thousands), so the average change in sales is Rs 2,000 per year.

Time Series Analysis

Interpretation of Trend Slope

Official Answer

The average change in sales per year is the slope of the trend line $Y = 90 + 2X$ .

Slope $b = 2$ (in Rs thousands per year).

Average change in sales = Rs 2,000 per year (an increase of 2 thousand rupees each year).

Because the slope is positive, the trend shows steadily rising sales.

average changeslope of trend lineb=2Rs 2000 per yearrate of changetime seriesincreasing trend

Marking Scheme

11 mark: stating the average change equals the slope $b = 2$ (thousand), i.e. Rs 2,000 per year.
2Accept '2 thousand rupees per year' or 'increase of Rs 2,000 annually'.

Hint

The average change per year in a linear trend $Y = a + bX$ is exactly the slope b. Read it off the trend line found in the previous part.

Quick Oral Answer

The average change in sales is simply the slope of the trend line, $b = 2$ , meaning sales grow on average by Rs 2,000 every year.

Analysis & Explanation

This short part tests interpretation rather than fresh calculation.

Concept

In a linear trend $Y = a + bX$ , the coefficient b is the average rate of change of Y per unit change in X. Since X is measured in years, $b = 2$ directly gives the average annual change.
The units follow the data: sales are in Rs thousands, so the average change is Rs 2,000 per year.

Exam trap

Students sometimes recompute averages of the raw sales instead of simply reading the slope; the 'average change' is the slope, not the mean of the values.
Dropping the units (thousands) turns a correct 2 into a wrongly stated Rs 2.

Real-world

The slope is the single most-quoted figure in business trend reports — 'sales are growing by Rs 2,000 a year' is exactly this coefficient.

Common Mistakes

1Averaging the raw sales figures instead of reading the trend slope.
2Omitting the unit 'thousands', so the answer becomes Rs 2 instead of Rs 2,000.
3Confusing the intercept $a = 90$ with the rate of change $b = 2$ .

Interesting Facts

In a least-squares trend, the slope is the only parameter that describes growth; the intercept just fixes the level at the origin year.

A constant positive slope implies compound-free linear growth — a useful baseline against which businesses judge whether sales are accelerating or slowing.

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Frequently Asked Questions

How is the average change related to the trend line?

In the linear trend $Y = a + bX$ , the average change per year is exactly the slope b. From the previous part $b = 2$ (in Rs thousands), so the average change in sales is Rs 2,000 per year.

Why is the average change not just the difference between first and last year's sales?

The raw endpoints fluctuate year to year. The least-squares slope averages the change over ALL seven years, giving a more reliable trend figure — here a steady Rs 2,000 per year — rather than being distorted by a single high or low year.