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.
For the given data, the number of observations is (n = 10). The regression of (Y) on (X) is of the form [ Y - \bar{Y} = b_{YX}(X - \bar{X}) ] where [ b_{YX} = \frac{SS_{XY}}{SS_X}. ] First compute the means: [ \bar{X} = \frac{\sum X}{n} = \frac{265}{10} = 26.5, \quad \bar{Y} = \frac{\sum Y}{n} = \frac{27.73}{10} ______ __________ ___ _________ _____ ___ _____.
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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.
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 |
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 |
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.
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.
| Sequence | A | B | C | D | E | |
|---|---|---|---|---|---|---|
| Job 1 | Time (in hours) | 1 | 2 | 3 | 5 | 4 |
| Job 2 | Sequence | C | A | D | E | B |
| Time (in hours) | 3 | 4 | 2 | 1 | 5 |
The following table represents the sales (in thousands) of mobile sets of a shop for 16 quarters over four years:
| Year | Quarter | |||
|---|---|---|---|---|
| Q_1 | Q_2 | Q_3 | Q_4 | |
| 2011 | 554 | 590 | 616 | 653 |
| 2012 | 472 | 501 | 521 | 552 |
| 2013 | 501 | 531 | 553 | 595 |
| 2014 | 403 | 448 | 460 | 480 |