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 ), calculate:

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

Using provided details:

( S_y = 283.8 - \frac{255^2}{10} = 283.8 - 6460.25 = 22.2 )

1. Calculate the slope ( b ): $b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$ Substitute values: $b = \frac{10(283.8) - (0.3)(255)}{10(24.76) - (25)^2} = 28.07143$

2. Calculate the intercept ( a ): $a = \frac{\sum y - b(\sum x)}{n}$ Substitute values: $a = \frac{255 - 28.1(12)}{10} = 28.1 - 2.14 = 26.96$

3. Thus, the regression line is: ( y = 28.07143 + 2.14x )

The value of ( a ) represents the estimated daily energy consumption (in kWh) of the household when the average daily temperature is 0 °C. In this case, ( a = 28.07143 ) means that if the average daily temperature is at freezing point, the household is expected to consume approximately 28.07143 kWh of energy.

To estimate the daily energy consumption at an average temperature of 2 °C, substitute ( x = 2 ) into the regression equation:

( y = 28.07143 + 2.14(2) = 28.07143 + 4.28 = 32.35143 )

Thus, her household’s estimated daily energy consumption at this temperature is approximately 32.35 kWh.

The regression model is based on winter data and may not be reliable for summer predictions due to potential differences in energy consumption patterns influenced by higher temperatures, air conditioning use, and other seasonal factors. The model might not accurately extrapolate summer consumption because it is derived from winter conditions.