Question

The weekly statistics about number of passengers (Y in 1000s) per week in terms of the number electric train cars (X) were collected through 30 weeks. The data were collected to assess the relationship between the number of passengers and the number of electric train cars. The process of data indicates the following results:

ΣX= 2589, ΣY= 762.748, ΣX^²=373055

ΣY^²=34905.9243 ΣYX= 112979.248

Σ(Y-Ya)*(X-Xa)= 14241.5194,

Σ(X-Xa)^² =37908.95

SST= 5816.7, SSE= 466.488, SSX= 37908.95

Where: Xa stands for Average of X

Ya Stands for Average of Y

For any hypothesis testing undertaken in this model, use the level of significance α=5%

1. What is the expected sign of the independent variable?
2. Develop the linear regression model Using the least square method of estimation and interpret the meaning of the intercept and the regression coefficient.
3. Compute R² and interpret its meaning
4. Is the number of passengers sensitive to the number of electric train cars?
5. Compute the confidence interval of regression coefficient and interpret its meaning
6. Forecast the dependent variable if the independent variable is 165.
7. Find out and interpret the confidence interval of the predicted value computed in question (6).
8. Based on the previous questions, how well the model fits to this phenomenon

Answer 1

i) The expected sign of the independent variable is positive as the relationship between these two variables seems to be positive.

ii) The least-square line is given by:

here

and

Thu s the equation is

iii) The R² is given by:

Interpretation: It means that 91.93% of the variation in the dependent variable is explained by the linear regression model obtained here.

iv) To find if the number of passengers sensitive to the number of electric train cars, we should test if the regression coefficient of the independent variable is significant i.e. we should test that:

To test the above hypothesis we should t-test for the regression coefficient whose test statistic is given by:

which is highly significant. Thus we can conclude that the number of passengers sensitive to the number of electric train cars.

v) 95% confidence interval is given by: .

Interpretation: With an increase in the electric train there is a 95% chance of the true increase in the number of passengers (in 1000s) increase in the above interval.

vi) Using the regression line the predicted value is 123.338.