The following data represents the monthly natural gas consumption (in hundreds of cubic feet) for a family for a random sample of nine months selected from five years of data. The other variable is Degree-days which is a measure of temperature. Degree-days are the number of degrees the average temperature falls below 65 F.
Degree-
Days 6, 4, 12, 43, 0, 52, 26, 30, 33
Gas 2.1, 1.7, 3.1, 8.9, 1.2, 11.0, 5.3, 6.9, 7.5
The regression equation is
Predictor Coef SE Coef T P
Constant 0.9595 0.1678 5.72 0.001
D-Days 0.189634 0.005845 32.44 0.000
S = 0.303674 R-Sq = 99.3%
Write the regression model sothat you can predict gass consumption krowing Degree-day. Write the equation in the terms of the actual variables.
Discuss the form, direction and strength of the relationship and verify. Give correct coefficient.
Calculate the residual for the first data point in the table. What is the purposew of residual analysis? State two assumptions necessary for regression analysis.
To write the regression model, we use the coefficients of the regression equation provided:
Regression equation: Gas = 0.9595 + 0.189634 * Degree-days
This equation allows us to predict gas consumption based on the degree-days.
The relationship between gas consumption and degree-days is positive, as indicated by the positive coefficient (0.189634). As the degree-days increase, gas consumption is expected to increase.
The strength of the relationship can be determined by looking at the coefficient of determination (R-Sq). In this case, R-Sq is 99.3%, which means that 99.3% of the variation in gas consumption can be explained by the variation in degree-days. This indicates a strong relationship between the two variables.
To calculate the residual for the first data point in the table, we need to substitute the observed values into the regression model:
Degree-days for first data point: 6
Gas consumption for first data point: 2.1
Using the regression equation: Gas = 0.9595 + 0.189634 * Degree-days
Gas = 0.9595 + 0.189634 * 6
Gas = 0.9595 + 1.137804
Gas = 2.097304
The predicted gas consumption for the first data point is 2.097304.
The residual can be calculated as the difference between the predicted value and the observed value:
Residual = Observed value - Predicted value
Residual = 2.1 - 2.097304
Residual ≈ 0.003696
The purpose of residual analysis is to assess the accuracy of the regression model by examining the differences between the observed values and the predicted values (residuals). Residuals should ideally be random and have a mean of zero. If there is a pattern or bias in the residuals, it indicates that the model may not be capturing all the relationships or factors affecting the dependent variable.
Two assumptions necessary for regression analysis are:
1. Linearity: The relationship between the dependent variable and the independent variable(s) is assumed to be linear. This means that a straight line can reasonably represent the relationship.
2. Independence: The observations should be independent of each other. Each data point should be unrelated to any other data point.