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This question is modified from an actual experiment published in a medical journal. A study claimed...

This question is modified from an actual experiment published in a medical journal. A study claimed that people who eat high-fibre cereal for breakfast will on average consume fewer calories for lunch than people who do not eat high-fibre cereal for breakfast. A group of 150 people were randomly selected. Each person was identified as either a consumer or a non-consumer of high-fibre cereal at breakfast, and the number of calories consumed at lunch was measured and recorded. Here are the data. (Numbers are fictitious.)
(a) Calories consumed at lunch by high-fibre breakfast consumers:

568 646 607 555 530 714 593 647 650 498 636 529 565 566 639 551 580 629
589 739 637 568 687 693 683 532 651 681 539 617 584 694 556 667 467 540

596 633 607 566 473 649 622

(b) Calories consumed at lunch by low-fibre breakfast consumers:
705 754 740 569 593 637 563 421 514 536
819 741 688 547 723 553 733 812 580 833
706 628 539 710 730 620 664 547 624 644
509 537 725 679 701 679 625 643 566 594
613 748 711 674 672 599 655 693 709 596

582 663 607 505 685 566 466 624 518 750
601 526 816 527 800 484 462 549 554 582
608 541 426 679 663739 603 726 623 788
787 462 773 830 369 717 646 645 747
573 719 480 602 596 642 588 794 583
428 754 632 765 758 663 476 490 573

Test if the result of the study is statistically significant at 5% significance level. (COULD YOU PLEASE DESCRIBE ALL THE STEPS ONE BY ONE IN YOUR CALCULATION?) Thank you in advance for your help.

Solutions

Expert Solution

Using - T-test for difference between two population means

Hypothesis

Now let

μ1 = Mean Calories consumed at lunch by low-fibre breakfast consumers

μ2 = Mean Calories consumed at lunch by high-fibre breakfast consumers

The null hypothesis is given by:

H0: There is no difference between the two population means i.e. Mean Calories consumed at lunch by high-fibre breakfast consumers is less than Mean Calories consumed at lunch by low-fibre breakfast consumers i.e.

μ1 = μ2

while the alternative hypothesis is given by:

H1: There is no difference between the two population means i.e. Mean Calories consumed at lunch by high-fibre breakfast consumers is less than Mean Calories consumed at lunch by low-fibre breakfast consumers i.e

μ1 > μ2

Test Statistic

Under H0the test statistic is given by:

Where N1 = first sample size

1= first sample mean

s12 = first sample standard deviation

N2 = second sample size

2= second sample mean

s22 = second sample standard deviation

Now Low Fibre constitute sample 1, while high fibre constitute sample 2

Sample 1

X X - Mean (X - Mean)2
705 71.77 5150.41
819 185.77 34509.14
706 72.77 5294.94
509 -124.23 15434
613 -20.23 409.4
582 -51.23 2624.89
601 -32.23 1039.01
608 -25.23 636.74
787 153.77 23644.09
573 -60.23 3628.09
428 -205.23 42120.85
754 120.77 14584.51
741 107.77 11613.59
628 -5.23 27.39
537 -96.23 9260.91
748 114.77 13171.32
663 29.77 886.04
526 -107.23 11499.05
541 -92.23 8507.05
462 -171.23 29320.96
719 85.77 7355.87
754 120.77 14584.51
740 106.77 11399.05
688 54.77 2999.35
539 -94.23 8879.98
725 91.77 8421.06
711 77.77 6047.61
607 -26.23 688.2
816 182.77 33403.54
426 -207.23 42945.78
773 139.77 19534.63
480 -153.23 23480.55
632 -1.23 1.52
569 -64.23 4125.96
547 -86.23 7436.24
710 76.77 5893.07
679 45.77 2094.56
674 40.77 1661.9
505 -128.23 16443.87
527 -106.23 11285.59
679 45.77 2094.56
830 196.77 38717
602 -31.23 975.54
765 131.77 17362.37
593 -40.23 1618.75
723 89.77 8058
730 96.77 9363.73
701 67.77 4592.28
672 38.77 1502.83
685 51.77 2679.76
800 166.77 27811.02
663 29.77 886.04
369 -264.23 69819.42
596 -37.23 1386.34
758 124.77 15566.64
637 3.77 14.19
553 -80.23 6437.44
620 -13.23 175.13
679 45.77 2094.56
599 -34.23 1171.94
566 -67.23 4520.36
484 -149.23 22270.68
739 105.77 11186.52
717 83.77 7016.8
642 8.77 76.85
663 29.77 886.04
563 -70.23 4932.76
733 99.77 9953.33
664 30.77 946.57
625 -8.23 67.79
655 21.77 473.77
466 -167.23 27967.09
462 -171.23 29320.96
603 -30.23 914.07
646 12.77 162.98
588 -45.23 2046.08
476 -157.23 24722.42
421 -212.23 45043.12
812 178.77 31957.41
547 -86.23 7436.24
643 9.77 95.38
693 59.77 3572.02
624 -9.23 85.26
549 -84.23 7095.31
726 92.77 8605.6
645 11.77 138.45
794 160.77 25845.82
490 -143.23 20515.88
514 -119.23 14216.66
580 -53.23 2833.82
624 -9.23 85.26
566 -67.23 4520.36
709 75.77 5740.54
518 -115.23 13278.79
554 -79.23 6277.97
623 -10.23 104.73
747 113.77 12942.78
583 -50.23 2523.42
573 -60.23 3628.09
536 -97.23 9454.38
833 199.77 39906.6
644 10.77 115.91
594 -39.23 1539.28
596 -37.23 1386.34
750 116.77 13634.38
582 -51.23 2624.89
788 154.77 23952.62

Sample 2

X X - Mean (X1 - Mean)^2
568 -36.02 1297.67
646 41.98 1762.05
607 2.98 8.86
555 -49.02 2403.28
530 -74.02 5479.44
714 109.98 12094.88
593 -11.02 121.51
647 42.98 1847
650 45.98 2113.86
498 -106.02 11240.93
636 31.98 1022.51
529 -75.02 5628.49
565 -39.02 1522.81
566 -38.02 1445.77
639 34.98 1223.37
551 -53.02 2811.47
580 -24.02 577.12
629 24.98 623.84
589 -15.02 225.7
739 134.98 18218.72
637 32.98 1087.47
568 -36.02 1297.67
687 82.98 6885.14
693 88.98 7916.86
683 78.98 6237.33
532 -72.02 5187.35
651 46.98 2206.81
681 76.98 5925.42
539 -65.02 4228.02
617 12.98 168.4
584 -20.02 400.93
694 89.98 8095.81
556 -48.02 2306.23
667 62.98 3966.07
467 -137.02 18775.37
540 -64.02 4098.98
596 -8.02 64.37
633 28.98 839.65
607 2.98 8.86
566 -38.02 1445.77
473 -131.02 17167.09
649 44.98 2022.91
622 17.98 323.16

The calculations are :

Now tn1+n2-2, 0.05 = t148,0.05 = 1.655

Since t >t148,0.05

therefore we reject Ho at 5% level of significance and conclude that Mean Calories consumed at lunch by high-fibre breakfast consumers is less than Mean Calories consumed at lunch by low-fibre breakfast consumers.

Please upvote! Thanks!


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