Question

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.

The data given in the above question seem to indicate that the difference between the two sample variances seems to indicate that the population variances differ. Test to see if this is correct or not at α = 0.05.​

(COULD YOU PLEASE DESCRIBE ALL THE STEPS ONE BY ONE IN YOUR CALCULATION?) Thank you in advance for your help.

Answer 1

Following table shows the calculations for sample mean and sd for sample 1:

	X1	(X1-mean)^2
	568	1297.4404
	646	1762.3204
	607	8.8804
	555	2402.9604
	530	5478.9604
	714	12095.6004
	593	121.4404
	647	1847.2804
	650	2114.1604
	498	11240.2404
	636	1022.7204
	529	5628.0004
	565	1522.5604
	566	1445.5204
	639	1223.6004
	551	2811.1204
	580	576.9604
	629	624.0004
	589	225.6004
	739	18219.6004
	637	1087.6804
	568	1297.4404
	687	6885.6804
	693	7917.4404
	683	6237.8404
	532	5186.8804
	651	2207.1204
	681	5925.9204
	539	4227.6004
	617	168.4804
	584	400.8004
	694	8096.4004
	556	2305.9204
	667	3966.4804
	467	18774.4804
	540	4098.5604
	596	64.3204
	633	839.8404
	607	8.8804
	566	1445.5204
	473	17166.2404
	649	2023.2004
	622	323.2804
Total	25973	172324.9772

Sample size: n=43

Following table shows the calculations for sample mean and sd for sample 2:

	X2	(X2-mean)^2
	705	5150.9329
	819	34510.4929
	706	5295.4729
	509	15433.0929
	613	409.2529
	582	2624.5129
	601	1038.7729
	608	636.5529
	787	23645.2129
	573	3627.6529
	428	42119.3529
	754	14585.3929
	741	11614.3729
	628	27.3529
	537	9260.2129
	748	13172.1529
	663	886.2529
	526	11498.2729
	541	8506.3729
	462	29319.7129
	719	7356.4929
	754	14585.3929
	740	11399.8329
	688	2999.7529
	539	8879.2929
	725	8421.7329
	711	6048.1729
	607	688.0129
	816	33404.8729
	426	42944.2729
	773	19535.6529
	480	23479.4329
	632	1.5129
	569	4125.4929
	547	7435.6129
	710	5893.6329
	679	2094.8929
	674	1662.1929
	505	16442.9329
	527	11284.8129
	679	2094.8929
	830	38718.4329
	602	975.3129
	765	17363.3329
	593	1618.4529
	723	8058.6529
	730	9364.4329
	701	4592.7729
	672	1503.1129
	685	2680.1329
	800	27812.2329
	663	886.2529
	739	11187.2929
	369	69817.4929
	596	1386.0729
	758	15567.5529
	637	14.2129
	553	6436.8529
	620	175.0329
	679	2094.8929
	599	1171.6929
	566	4519.8729
	484	22269.5929
	603	913.8529
	717	7017.4129
	642	76.9129
	663	886.2529
	563	4932.2529
	733	9954.0529
	664	946.7929
	625	67.7329
	655	473.9329
	466	27965.8729
	462	29319.7129
	726	8606.2729
	646	163.0729
	588	2045.7529
	476	24721.2729
	421	45041.5729
	812	31958.7129
	547	7435.6129
	643	95.4529
	693	3572.4529
	624	85.1929
	549	7094.6929
	623	104.6529
	645	138.5329
	794	25846.9929
	490	20514.8329
	514	14215.7929
	580	2833.4329
	624	85.1929
	566	4519.8729
	709	5741.0929
	518	13277.9529
	554	6277.3929
	788	23953.7529
	747	12943.6129
	583	2523.0529
	573	3627.6529
	536	9453.6729
	833	39908.0529
	644	115.9929
	594	1538.9929
	596	1386.0729
	750	13635.2329
	582	2624.5129
Total	67756	1130995.16

Sample size: n=107

Null hypothesis:

Alternative hypothesis is claim.

Since it is not given that variances are equal so degree of freedom of the test is

So df is 122.

And test statistics will be

The p-value is: 0.0386

Since p-value is less than 0.05 so we reject the null hypothesis. That is we can conclude that the result of the study is statistically significant at 5% significance level.