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.
Seven observations are (x_1=12, x_2=14, x_3=13, x_4=10, x_5=15, x_6=12, x_7=15) First find the mean: (\bar x = (12+14+13+10+15+12+15)/7 = 91/7 = 13) Deviations from the mean are (-1, 1, 0, -3, 2, -1, 2) For a stationary series, the sample auto-covariance at lag (k) is (C_k = \dfrac{1}{7}\sum_{t=1}^{7-k}(x_t-\bar x)(x_{t+k}-\bar x)) (C_0): ((1/7)[(-1)^2 + 1^2 + 0^2 + (-3)^2 + 2^2 + _____ _______ _________ _____ ________ __________ ___ _____ __________ _____.
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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.
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:$
$
Build a regression model by selecting appropriate regressors in the model using the Stepwise Selection method.
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.
A company has three production facilities S1, S2 and S3 with production capacity of 7, 9 and 18 units (in 100s) per week of a product, respectively. These units are to be shipped to four warehouses D1, D2, D3 and D4 with requirement of 5, 6, 7 and 14 units (in 100s) per week, respectively. The transportation costs (in Rs) per unit between factories to warehouses are given in the table below:
| $D_1$ | $D_2$ | $D_3$ | $D_4$ | Capacity | |
|---|---|---|---|---|---|
| $S_1$ | 19 | 30 | 50 | 10 | 7 |
| $S_2$ | 70 | 30 | 40 | 60 | 9 |
| $S_3$ | 40 | 8 | 70 | 20 | 18 |
| Demand | 5 | 8 | 7 | 14 | 34 |
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
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 |
Use the penalty (Big M) method to solve the following LP problem:
Minimise
Subject to the constraints.