A researcher is interested in developing a linear model for the electricity consumption of a household having an AC (1.5 ton) so that she can predict the electricity consumption. For this purpose, she selects 25 houses and records the electricity consumption (in kWh), size of house (in square feet) and AC hours for one month during summers. The results obtained are:
SS(B0) = 12526.08, SS(B0,B1) = 17908.47, SS(B0,B2) = 17125.23, SS(B0, B1, B2) = 18079.0,
Build a regression model by selecting appropriate regressors in the model using the Stepwise Selection method.
The Stepwise Selection method involves iteratively adding or removing predictors based on statistical criteria. Let's apply it to the given data. --- ### Step 1: Define the Regression Model The general multiple linear regression model is: \[ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon \] where: - \( Y \) = Electricity consumption (kWh) - \( X_1 \) = Size of the house (sq. ft) - \( X_2 \) = AC hours - \( \beta_0, \beta_1, \beta_2 ___ ___ __________ _____ _______ ________ ________ _____ _______.
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Use the penalty (Big M) method to solve the following LP problem:
Minimise Z 5x1+ 3x2
Subject to the constraints:
Using the graphical method to minimise the time required to process Job 1 and Job 2 on five machines A, B, C, D and E, find the minimum elapsed times an idle times to complete both jobs.
| Job 1 | Sequence | A | B | C | D | E | ||||
| Time (in hours) | 1 | 2 | 3 | 5 | 4 | |||||
| Job 2 | Sequence | C | A | D | E | B | ||||
| Time (in hours) | 3 | 4 | 2 | 1 | 5 | |||||
A firm wants to know whether there is any linear relationship between the sales (X) and its yearly revenue (Y). The records for 10 years were examined and the following results were obtained:
(a) Fit a regression line taking Y as the dependent variable and X as the independent variable.
(b) Test whether the sales have any effect on revenue at 5% level of significance. (c) Comment on the goodness of fit of the regression line.
Four professors are capable of teaching any one of four different courses. Class preparation time in hours for different topics varies from professor to professor and is given in the table below:
| Professor | A | B | C | D | ||||
| Linear Programming | 2 | 15 | 13 | 4 | ||||
| Queuing Theory | 10 | 4 | 14 | 15 | ||||
| Transportation Problem | 9 | 14 | 16 | 13 | ||||
| Regression Analysis | 7 | 8 | 11 | 9 | ||||
In a railway marshalling yard, goods trains arrive at a rate of 36 trains per day. Assuming that the inter-arrival and service time distributions both follow exponential distribution with an average of 30 minutes, calculate the following:
(i) Traffic intensity
(ii) The mean queue length
(iii) Probability that the queue size exceeds
Seven successive observations on a stationary time-series are as follows:
12, 14, 13, 10, 15, 12, 15
(a) Calculate auto-covariances C0, C1, C2, C3 and C4.
(b) Calculate auto-correlation coefficients r1, r2, r3 and r4.
(c) Plot the correlogram.
State whether the following statements are True or False. Give reasons in support of your answers.
(a) The solution of a transportation problem with 5 rows (supplies) and 4 columns (destinations) is feasible if number of possible allocations are 8.
(b) The moving averages of suitable period in a time-series are free from the influences of seasonal and cyclic variations.
(c) If the basic solutions for a system of equations are (-2, 0, 1), (0, 1, 3), (-2, 3, 0), then only (0, 1, 3) is feasible.
(d) In the stepwise selection method of multiple regression model, once a variable enters in the model then it always remains in the model.
(e) An enterprise requires 1000 units per month. The ordering cost is estimated to be 50 per order. The purchase price is 20 per unit and the carrying cost per unit is 10% of it. Then the economic lot size to be ordered is 775.