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 the variance $S_{y}$, we use the formula:

S_{y} = rac{ extstyle ext{sum of squares of observations}}{n} - rac{( ext{sum of observations})^2}{n^2}

Here, $n = 10$, $\sum y = 255$, and $\sum y^2 = 283.8$.

Calculating:

S_{y} = rac{283.8 - rac{255^2}{10}}{10} = 22.2

Using the least squares method:

1. Calculate $b$: b = rac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2} After substituting in the values, we find: $b \approx -2.142857$. Rounding to 3 significant figures: $b = -2.14$.

2. Calculate $a$ using: $a = \frac{\sum y - b \sum x}{n}$ After substituting in the values, we find: $a \approx 28.07143$. Rounding to 3 significant figures: $a = 28.1$.

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

The value of $a$ represents the estimated daily energy consumption when the average daily temperature is $0 \, ext{°C}$. Specifically, it indicates that when the temperature is at freezing point, the household is expected to consume approximately $28.1 \, ext{kWh}$ of energy.

To estimate the daily energy consumption at $2 \, ext{°C}$, we substitute $x = 2$ into the regression equation:

$y = 28.1 - 2.14(2)$

Calculating: $y = 28.1 - 4.28 = 23.82 \text{ kWh}$

Thus, the estimated energy consumption is approximately $23.8 \, ext{kWh}$.

The reliability of the model in predicting energy consumption during the summer is questionable. The regression model is based on winter data, which may not accurately represent summer conditions. Factors such as different temperature ranges, seasonal usage patterns, and varying energy demands influence household consumption differently in summer. Hence, using the model without adjustments could lead to significant inaccuracies.