In: Math
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.
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
x̅1= first sample mean
s12 = first sample standard deviation
N2 = second sample size
x̅2= 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!