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    Q.4

    Artificial Data on Two Independent and Four Dependent Variables *

    x1 y1 y2 y3 x2 y4
    10 8.04 9.14 7.46 8 6.58
    8 6.95 8.14 6.77 8 5.76
    13 7.58 8.74 12.74 8 7.71
    9 8.81 8.77 7.11 8 8.84
    11 8.33 9.26 7.81 8 8.47
    14 9.96 8.1 8.84 8 7.04
    6 7.24 6.13 6.08 8 5.25
    4 4.26 3.1 5.39 19 12.5
    12 10.84 9.13 8.15 8 5.56
    7 4.82 7.26 6.42 8 7.91
    5 5.68 4.74 5.73 8 6.89

    * F. J. Anscombe "Graphs and Statistical Analysis,"
    American Statistician, vol. 27 (1973) pp. 17-21.

    Perform the following regressions:

    Independent Variable Dependent Variable
    a) x1 y1
    b) x1 y2
    c) x1 y3
    d) x2 y4

    In each case, plot a scatter diagram of the independent against
    dependent variable. What, if anything, could you do to obtain
    better regression results in each of the four regressions?

    Q.5.

    (From Bodily, Carraway, Frey, and Pfeifer, Quantitative Business Analysis Casebook, Richard D. Irwin, 1996.) In a market-segmentation study, data on net income (I) in thousands, family size (F), and expenditures on consumer durable goods (C) were collected from 20 randomly-selected households:

    The company sponsoring the study was interested in isolating the effects of income and family size on the dollar amounts spent for consumer durable goods. Can you help them?

    6.
    Develop a regression model that will predict students' first-year grade-point averages.

    First- Admissions Test Scores
    Year College
    Average Section Age Average Total Quantitative Verbal
    1st Yr. Avg. Section Age Coll. Avg. Total Quantitative Verbal
    73.76 3 24 2.7 568 34 35
    76.59 7 27 2.9 602 37 37
    85.41 8 23 3.3 674 39 45
    71.65 3 25 2.3 579 38 33
    75.18 5 24 2.3 713 47 43
    79.24 2 22 2.4 548 36 30
    80.12 1 22 3.1 616 39 36
    84.88 5 24 2.9 674 36 46
    75.88 6 24 3.5 629 37 40
    85.41 4 26 2.3 682 43 42
    79.59 2 28 3.0 746 47 48
    74.29 5 25 2.5 568 33 36
    77.65 6 22 2.9 572 33 36
    77.65 4 29 3.6 635 37 41
    73.24 2 30 1.7 581 30 40
    79.24 2 24 2.2 602 32 42
    76.76 3 27 2.9 546 26 39
    79.94 7 28 3.2 747 48 46
    77.12 3 25 3.8 516 32 29
    83.47 4 23 3.8 702 48 40
    72.53 8 21 3.0 571 38 31
    75.35 6 23 3.6 582 30 41
    78.00 7 28 2.5 550 33 33
    73.41 2 22 3.0 499 27 31
    75.18 1 33 2.6 491 37 22
    70.24 3 34 2.5 463 25 29
    73.41 1 25 3.4 402 29 16
    76.76 6 23 2.9 662 42 40
    62.56 1 32 1.8 576 32 37
    74.47 3 34 2.7 645 42 37
    77.82 3 24 3.3 625 45 32
    72.35 4 25 3.0 424 17 29
    72.00 1 26 2.7 568 30 38
    80.82 7 23 3.8 546 35 30
    75.18 1 22 3.1 503 27 31
    79.41 2 23 2.7 598 33 39
    76.24 8 22 2.4 651 41 39
    82.59 4 30 1.9 652 44 37
    79.41 8 23 3.3 605 38 36
    76.76 1 28 3.2 593 32 40
    71.12 8 27 2.4 628 28 48
    74.47 4 24 2.5 530 25 38
    73.41 3 33 2.9 704 40 49
    74.47 3 28 2.3 580 34 36
    73.76 8 20 2.7 475 32 24
    75.18 3 26 2.1 510 23 37
    78.18 8 31 2.8 564 29 38
    78.18 1 27 3.0 626 31 45
    78.53 1 31 2.7 583 26 44
    75.35 4 22 2.8 602 42 32
    77.47 7 26 2.3 613 30 44
    80.12 3 22 3.5 625 37 40
    73.94 4 22 2.7 631 34 44
    72.35 2 23 3.2 564 40 28
    78.18 4 22 3.5 644 39 40
    70.76 3 22 3.1 430 23 26
    77.47 1 25 2.8 615 35 40
    73.59 4 23 3.2 552 39 27
    75.71 4 26 2.7 587 31 39
    79.59 8 26 2.9 594 35 37
    77.82 1 24 2.7 568 37 31
    71.12 1 23 2.8 543 34 31
    80.82 4 24 3.3 624 39 37
    76.41 3 22 2.5 556 34 33
    79.59 4 22 3.7 605 34 39
    71.29 3 23 2.8 646 32 47
    75.35 4 24 2.6 550 32 35
    68.65 3 26 2.3 498 23 35
    74.82 1 22 3.2 676 46 38
    75.00 4 28 2.7 659 41 40
    77.82 8 23 3.2 502 20 39
    75.18 6 29 2.9 588 31 40
    81.53 5 25 2.0 613 39 36
    79.24 2 23 3.1 650 40 40
    76.06 8 22 2.5 591 31 40
    77.12 4 27 2.6 550 30 35
    72.35 1 24 2.7 552 28 38
    79.94 3 22 3.8 651 33 47
    79.94 8 22 2.4 609 34 40
    73.59 1 27 2.7 552 33 32
    74.65 4 22 3.4 586 41 30
    74.29 8 22 2.9 553 23 41
    76.41 7 33 2.3 503 28 31
    78.88 1 29 2.2 557 31 36
    77.82 5 21 3.0 625 42 36
    75.00 6 23 2.6 569 29 40
    73.24 7 22 2.6 527 22 40
    78.53 7 26 2.3 527 31 31
    74.12 8 24 2.4 496 33 25
    74.82 7 22 2.7 620 34 42
    74.47 7 23 3.5 552 37 29
    73.76 2 22 2.3 598 33 40
    75.53 3 24 3.9 635 45 34
    78.35 2 22 3.6 726 43 48
    78.35 5 22 3.0 702 44 44
    71.12 6 25 2.4 564 36 33
    73.24 8 29 2.1 613 38 37
    77.82 8 28 2.6 598 34 39
    72.88 5 28 2.7 501 29 30
    75.71 5 27 2.8 590 37 35
    73.59 3 25 3.3 738 40 52
    79.76 4 32 3.0 689 37 48
    78.00 8 25 2.3 576 44 26
    70.94 6 26 2.3 596 43 30
    71.12 1 28 1.9 534 28 36
    76.76 2 22 2.9 634 41 38
    76.76 1 30 2.9 520 30 32
    76.24 7 22 3.6 568 30 38
    73.41 4 22 2.6 602 38 36
    84.00 6 27 2.9 662 46 37
    79.06 3 23 3.7 556 46 22
    75.71 8 21 2.6 483 25 32
    74.29 7 29 3.3 434 16 31
    76.76 2 24 2.5 609 36 39
    74.29 6 27 2.6 618 36 40
    78.71 2 33 2.3 615 40 35
    76.24 3 28 2.1 578 44 26
    77.65 7 25 2.3 668 40 43
    74.65 5 25 2.7 587 34 37
    77.12 6 22 2.8 515 34 28
    79.06 6 22 2.2 613 38 37
    75.88 3 33 3.2 670 34 47
    79.59 8 22 2.6 775 50 49
    79.94 3 30 2.5 624 42 34
    72.53 3 22 3.4 651 41 40
    70.24 6 23 3.1 632 42 37
    73.59 2 28 2.1 479 26 30
    71.29 8 25 2.6 577 37 33
    77.82 6 25 2.5 621 33 42
    72.71 5 31 2.6 516 29 32
    76.24 3 28 2.4 503 27 32
    76.76 5 27 2.6 573 38 32
    69.88 2 25 3.2 613 34 41
    80.47 4 27 2.2 613 43 32
    71.29 8 26 2.5 616 42 34
    78.71 1 28 2.5 641 37 41
    84.35 2 24 2.9 725 43 48
    77.82 6 23 3.3 631 37 41
    78.00 4 32 2.5 579 35 35
    76.59 8 34 2.1 651 37 43
    70.24 3 25 2.8 533 34 29
    75.35 7 23 2.9 506 33 26
    68.29 4 28 2.7 468 18 35
    75.88 2 26 2.8 645 45 34
    73.41 4 24 2.5 467 25 29
    72.88 3 29 3.4 542 26 38
    77.82 3 28 2.8 559 31 36
    79.41 7 28 2.5 502 30 28
    76.94 4 24 2.9 673 41 43
    72.88 8 22 2.7 580 25 43

    Column B A weighted average of final grades in ten first-year
    courses. Each course was graded on a 15-point scale, with
    possible values from 54 through 96 in steps of 3.

    Column C There were eight sections, coded 1 through 8.

    Column D Age in years as of the day of matriculation.

    Column E Four-year average of college grades, converted, if necessary,
    to a four-point scale.

    Column F Total score on the admissions test administered by
    Educational Testing Service. This test is scored so
    that the average score among all those to whom it
    was administered in a given year will be about 500,
    with a standard deviation of about 100. Scores can
    range from 200 to 800.

    Column G Quantitative score on the admissions test, scored to have
    an average of about 28, a standard deviation of about
    8, and a range from 0 to 52.

    Column H Verbal score on the admissions test, scored the same
    way as the quantitative score.

    © BrainMass Inc. brainmass.com December 15, 2022, 5:38 pm ad1c9bdddf
    https://brainmass.com/statistics/regression-analysis/multiple-regression-analysis-admission-tests-95071

    Attachments

    Solution Preview

    Please see the attachments
    Calculations are done in MS Excel

    Q.4

    Artificial Data on Two Independent and Four Dependent Variables *

    x1 y1 y2 y3 x2 y4
    10 8.04 9.14 7.46 8 6.58
    8 6.95 8.14 6.77 8 5.76
    13 7.58 8.74 12.74 8 7.71
    9 8.81 8.77 7.11 8 8.84
    11 8.33 9.26 7.81 8 8.47
    14 9.96 8.1 8.84 8 7.04
    6 7.24 6.13 6.08 8 5.25
    4 4.26 3.1 5.39 19 12.5
    12 10.84 9.13 8.15 8 5.56
    7 4.82 7.26 6.42 8 7.91
    5 5.68 4.74 5.73 8 6.89

    * F. J. Anscombe "Graphs and Statistical Analysis,"
    American Statistician, vol. 27 (1973) pp. 17-21.

    Perform the following regressions:

    Independent Variable Dependent Variable
    a) x1 y1
    b) x1 y2
    c) x1 y3
    d) x2 y4

    In each case, plot a scatter diagram of the independent against
    dependent variable. What, if anything, could you do to obtain
    better regression results in each of the four regressions?

    Ans.
    The scatter diagrams are given in the Excel sheet

    The regression equation are

    a) Y1 = 0.5001X1 + 3.0001
    b) Y2 = 0.5X1 + 3.0009 Linear
    Y2= -0.1267X12 + 2.7808X1 - 5.9957 Polynomial of degree 2
    c) Y3 = 0.4997X1 + 3.0025

    For the second model there is a nonlinear relation can be observed between Y2 and X1 and polynomial of order 2 gives a better fit.

    Since X2 takes only two distinct value ( 8, 19 ) regression analysis is not suitable for this data.

    Q.5.

    (From Bodily, Carraway, Frey, and Pfeifer, Quantitative Business Analysis Casebook, Richard D. Irwin, 1996.) In a market-segmentation study, data on net income (I) in thousands, family size (F), and expenditures on consumer durable goods (C) were collected from 20 randomly-selected households:

    The company sponsoring the study was interested in isolating the effects of income and family size on the dollar amounts spent for consumer durable goods. Can you help them?

    Ans
    A regression model can be suggested as

    C = B0+B1 I + B2 F

    Here B0 ,B1 ,B2 represents the regression coefficients

    Regression Statistics
    Multiple R 0.984102892
    R Square 0.968458502
    Adjusted R Square 0.959446645
    Standard Error 262.0523394
    Observations 10

    ANOVA
    df SS MS F Significance F
    Regression 2 14759540 7379770 107.4649 5.57E-06
    Residual 7 480700 68671.43
    Total 9 15240240

    Coefficients Standard Error t ...

    Solution Summary

    Multiple Regression analysis on weighted average of final grades, Age in years as of the day of matriculation, Four-year average of college grades, Total score on the admissions test administered by Educational Testing Service, Quantitative score on the admissions test, Verbal score on the admissions test.

    $2.49

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