Statistical models can provide a cheap and quick way to describe a real world situation - Edexcel - A-Level Maths Statistics - Question 4 - 2015 - Paper 1

Given the values, we can calculate:

$S_{yy} = \sum y^2 - \frac{(\sum y)^2}{n}$

Calculating the sum of $y^2$:

• $y^2$: $28^2 + 30^2 + 26^2 + 27^2 + 26^2 + 27^2 + 26^2 + 21^2 + 20^2 + 21^2 = 8160$
• $\sum y = 255$, thus:

$S_{yy} = 8160 - \frac{(255)^2}{10} = 8160 - 6502.5 = 1657.5$.

We start with the formulas for $a$ and $b$:

• $b = \frac{S_{xy}}{S_{xx}}$ = $\frac{283.8}{10.36} \approx 27.3$ (to 3 significant figures)

• To find $a$, we use: $a = \frac{\sum y - b \cdot \sum x}{n}$

• Substituting values: $a = \frac{255 - (2.142857 \cdot (\sum x = 36))}{10} = 28.07143$

• Therefore: $y = 28.1 + 2.14x$

The value of $a$ (28.1 kWh) represents the estimated energy consumption when the average daily temperature is 0 °C.

Using the regression line equation: $y = 28.1 + 2.14(2) = 28.1 + 4.28 = 32.38$ Thus, the estimated daily energy consumption at 2 °C is approximately 32.4 kWh.

The regression model is based on winter temperature data. Using it to predict summer consumption may not be reliable because different factors influence energy consumption in summer, such as cooling needs, occupancy patterns, and external temperatures rarely matching winter data.