- 20 Marks
Question
A Ghanaian student studying abroad wants to develop an empirical model for energy consumption (in kilowatts per day) as a function of the daily high temperature (in degrees Celsius) in winter. For nine days the following information was obtained:
| Temperature (°C) | -0.4 | -0.2 | 0.3 | 0.8 | 1.1 | 1.4 | 1.8 | 2.1 | 2.5 |
|---|---|---|---|---|---|---|---|---|---|
| Energy used (kW) | 28 | 30 | 26 | 25 | 26 | 26 | 27 | 26 | 22 |
Required:
a) Identify the response and predictor variables, based on the purpose for developing the regression model. (2 marks)
b) Determine the coefficient of regression and the regression constant. Give your values to 2 decimal places. (5 marks)
c) Interpret your results in b) above. (3 marks)
d) Write the equation of the regression line of energy use on temperature in the form y=a+bxy = a + bx. (2 marks)
e) Estimate the student’s daily energy consumption when the daily high temperature is 2 degrees Celsius. (2 marks)
f) Determine the standard error of estimate. (6 marks)
Answer
a) Response and predictor variables:
- Energy use is the response variable.
- Temperature is the predictor variable.
(2 marks)
b) Coefficient of regression and regression constant:
| x (Temp °C) | y (Energy kW) | xy | x² | yest | (y – yest)² |
|---|---|---|---|---|---|
| -0.4 | 28 | -11.2 | 0.16 | 28.46 | 0.2116 |
| -0.2 | 30 | -6 | 0.04 | 28.15 | 3.4225 |
| 0.3 | 26 | 7.8 | 0.09 | 27.38 | 1.9044 |
| 0.8 | 25 | 20 | 0.64 | 26.60 | 2.56 |
| 1.1 | 26 | 28.6 | 1.21 | 26.14 | 0.0196 |
| 1.4 | 26 | 36.4 | 1.96 | 25.67 | 0.1089 |
| 1.8 | 27 | 48.6 | 3.24 | 25.05 | 3.8025 |
| 2.1 | 26 | 54.6 | 4.41 | 24.59 | 1.9881 |
| 2.5 | 22 | 55 | 6.25 | 23.97 | 3.8809 |
| Totals | 236 | 233.8 | 18 | 17.8985 |
b = 
a = 
c) Interpretation:
- The regression coefficient (slope) is -1.55, meaning for every additional increment of 1 degree Celsius in temperature, energy usage decreases by an average of 1.55 kW.
- The regression constant (intercept) is 27.84, meaning when the temperature is 0 degrees Celsius, energy usage is 27.84 kW.
(3 marks)
d) Equation of the regression line:
y = 27.84 − 1.55x
e) Estimate of energy consumption when temperature is 2°C:
When x = :
y = 27.84 − 1.55(2) = 24.74 kW
f) Standard error of estimate:
Thus, the standard error of estimate is 2.57.
- Tags: Energy Consumption, Regression, Standard Error of Estimate, Temperature
- Level: Level 1
- Topic: Forecasting
- Series: MAY 2018
- Uploader: Kwame Aikins