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

To find $S_{y}$, we calculate the sum of the daily energy consumption values.

Using the given data:

S_{y} = rac{283.8 - 12 imes 255}{10} = 22.2

To find the regression line:

1. Calculate the means of $x$ and $y$: ar{x} = rac{ ext{sum of } x}{10} = rac{24.76}{10} = 2.476 ar{y} = rac{255}{10} = 25.5

2. The slope $b$: b = rac{(n ext{ } ext{sum of } xy) - (sum x)(sum y)/n}{(n ext{ } ext{sum } x^{2})-( ext{sum } x)^{2}/n}
   Required values give: $b = -2.14$

3. The intercept $a$:

   $$$$

ightarrow a = 28.1$$

Thus, the regression equation is: $y = 28.1 - 2.14x$

The value of $a$, which is 28.1, represents the estimated daily energy consumption, in kWh, when the average daily temperature is $0 ext{ °C}$.

Using the regression equation:

$y = 28.1 - 2.14(2) = 28.1 - 4.28 = 23.82 ext{ kWh}$

Thus, her estimated household energy consumption is approximately $23.8 ext{ kWh}$.

The regression model is based on temperature data from winter. Using it to predict energy consumption in summer is unreliable due to:

1. Seasonal variations in energy usage patterns;
2. Differences in heating and cooling needs as temperatures rise;
3. The model may not account for other summer factors influencing energy consumption (e.g., air conditioning).

Thus, extrapolating beyond the observed temperature range could lead to inaccurate predictions.