T.MATH REFLECTIO

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FILENAME(S): T.MATH REFLECTIO

FILE TYPE(S): SEQ

FILE SIZE: 1.7K

FIRST SEEN: 2025-11-30 18:12:04

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5557c08033b7b801426f8edab9de849ceeaa648f825856008bc3b9aa70368b0b

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DISK TITLE	FILENAME	FILE TYPE	COLLECTION	TRACK	SECTOR	ACTIONS
LOADSTAR 22 S 2	T.MATH REFLECTIO	SEQ	Sailor, Ganheden	25	0	DOWNLOAD FILE
LOADSTAR 22 S 2	T.MATH REFLECTIO	SEQ	Curtcool, Frank	25	0	DOWNLOAD FILE

FILE CONTENT & ANALYSIS

00000000: 0D 20 20 20 20 20 D4 48  45 20 C3 45 4E 54 52 41  |.     .HE .ENTRA|
00000010: 4C 20 CC 49 4D 49 54 20  D4 48 45 4F 52 45 4D 0D  |L .IMIT .HEOREM.|
00000020: 0D 0D 20 20 C9 46 20 53  41 4D 50 4C 45 53 20 4F  |..  .F SAMPLES O|
00000030: 46 20 4E 20 4E 55 4D 42  45 52 53 20 41 52 45 0D  |F N NUMBERS ARE.|
00000040: 0D 52 45 50 45 41 54 45  44 4C 59 20 43 48 4F 53  |.REPEATEDLY CHOS|
00000050: 45 4E 20 49 4E 20 41 20  52 41 4E 44 4F 4D 20 46  |EN IN A RANDOM F|
00000060: 41 53 48 49 4F 4E 0D 0D  46 52 4F 4D 20 41 20 47  |ASHION..FROM A G|
00000070: 49 56 45 4E 20 50 4F 50  55 4C 41 54 49 4F 4E 20  |IVEN POPULATION |
00000080: 57 49 54 48 20 4D 45 41  4E 20 CD 0D 0D 41 4E 44  |WITH MEAN ...AND|
00000090: 20 56 41 52 49 41 4E 43  45 0D 20 20 20 20 20 20  | VARIANCE.      |
000000A0: 20 20 20 20 20 20 20 20  32 0D 20 20 20 20 20 20  |        2.      |
000000B0: 20 20 20 20 20 20 20 53  0D 0D 41 4E 44 20 49 46  |       S..AND IF|
000000C0: 20 46 52 4F 4D 20 45 41  43 48 20 53 41 4D 50 4C  | FROM EACH SAMPL|
000000D0: 45 20 54 48 45 20 4D 45  41 4E 20 49 53 0D 0D 46  |E THE MEAN IS..F|
000000E0: 4F 52 4D 45 44 20 28 54  48 41 54 20 49 53 2C 20  |ORMED (THAT IS, |
000000F0: 54 48 45 20 4E 55 4D 42  45 52 53 20 49 4E 20 54  |THE NUMBERS IN T|
00000100: 48 45 0D 0D 53 41 4D 50  4C 45 20 41 52 45 20 41  |HE..SAMPLE ARE A|
00000110: 44 44 45 44 20 41 4E 44  20 44 49 56 49 44 45 44  |DDED AND DIVIDED|
00000120: 20 42 59 20 4E 29 2C 0D  0D 54 48 45 4E 20 54 48  | BY N),..THEN TH|
00000130: 45 53 45 20 4D 45 41 4E  53 20 46 4F 52 4D 20 41  |ESE MEANS FORM A|
00000140: 20 4E 45 57 0D 0D 44 49  53 54 52 49 42 55 54 49  | NEW..DISTRIBUTI|
00000150: 4F 4E 2E 0D 0D 20 20 D4  48 45 52 45 20 49 53 20  |ON...  .HERE IS |
00000160: 41 20 54 48 45 4F 52 45  4D 20 49 4E 20 4D 41 54  |A THEOREM IN MAT|
00000170: 48 45 4D 41 54 49 43 41  4C 0D 0D 53 54 41 54 49  |HEMATICAL..STATI|
00000180: 53 54 49 43 53 20 57 48  49 43 48 20 53 54 41 54  |STICS WHICH STAT|
00000190: 45 53 20 54 48 41 54 20  54 48 45 20 4E 45 57 0D  |ES THAT THE NEW.|
000001A0: 0D 44 49 53 54 52 49 42  55 54 49 4F 4E 20 4F 46  |.DISTRIBUTION OF|
000001B0: 20 53 41 4D 50 4C 45 20  4D 45 41 4E 53 20 57 49  | SAMPLE MEANS WI|
000001C0: 4C 4C 20 42 45 0D 0D 41  50 50 52 4F 58 49 4D 41  |LL BE..APPROXIMA|
000001D0: 54 45 4C 59 20 4E 4F 52  4D 41 4C 20 28 42 45 4C  |TELY NORMAL (BEL|
000001E0: 4C 2D 53 48 41 50 45 44  29 0D 0D 52 45 47 41 52  |L-SHAPED)..REGAR|
000001F0: 44 4C 45 53 53 20 4F 46  20 54 48 45 20 53 48 41  |DLESS OF THE SHA|
00000200: 50 45 20 4F 46 20 54 48  45 0D 0D 4F 52 49 47 49  |PE OF THE..ORIGI|
00000210: 4E 41 4C 20 44 49 53 54  52 49 42 55 54 49 4F 4E  |NAL DISTRIBUTION|
00000220: 20 46 52 4F 4D 20 57 48  49 43 48 20 54 48 45 0D  | FROM WHICH THE.|
00000230: 0D 53 41 4D 50 4C 45 53  20 57 45 52 45 20 44 52  |.SAMPLES WERE DR|
00000240: 41 57 4E 2E 20 20 C6 55  52 54 48 45 52 4D 4F 52  |AWN.  .URTHERMOR|
00000250: 45 2C 20 54 48 45 0D 0D  44 49 53 54 52 49 42 55  |E, THE..DISTRIBU|
00000260: 54 49 4F 4E 20 4F 46 20  53 41 4D 50 4C 45 20 4D  |TION OF SAMPLE M|
00000270: 45 41 4E 53 20 57 49 4C  4C 20 48 41 56 45 0D 0D  |EANS WILL HAVE..|
00000280: 54 48 45 20 53 41 4D 45  20 4D 45 41 4E 2C 20 CD  |THE SAME MEAN, .|
00000290: 2C 20 41 53 20 54 48 45  20 4F 52 49 47 49 4E 41  |, AS THE ORIGINA|
000002A0: 4C 0D 0D 44 49 53 54 52  49 42 55 54 49 4F 4E 2C  |L..DISTRIBUTION,|
000002B0: 20 41 4E 44 20 48 41 56  45 20 56 41 52 49 41 4E  | AND HAVE VARIAN|
000002C0: 43 45 20 45 51 55 41 4C  0D 0D 54 4F 3A 0D 20 20  |CE EQUAL..TO:.  |
000002D0: 20 20 20 20 20 20 20 20  32 0D E0 20 20 20 20 20  |        2..     |
000002E0: 20 20 20 53 20 2F 4E 2E  0D 0D C1 53 20 CE 20 28  |   S /N....S . (|
000002F0: 54 48 45 20 53 41 4D 50  4C 45 20 53 49 5A 45 29  |THE SAMPLE SIZE)|
00000300: 20 49 4E 43 52 45 41 53  45 53 2C 20 54 48 45 0D  | INCREASES, THE.|
00000310: 0D 44 49 53 54 52 49 42  55 54 49 4F 4E 20 57 49  |.DISTRIBUTION WI|
00000320: 4C 4C 20 42 45 43 4F 4D  45 20 49 4E 43 52 45 41  |LL BECOME INCREA|
00000330: 53 49 4E 47 4C 59 0D 0D  4E 4F 52 4D 41 4C 2E 0D  |SINGLY..NORMAL..|
00000340: 0D 20 20 D4 48 49 53 20  52 45 53 55 4C 54 20 46  |.  .HIS RESULT F|
00000350: 4F 4C 4C 4F 57 53 20 46  52 4F 4D 20 41 20 4D 4F  |OLLOWS FROM A MO|
00000360: 52 45 0D 0D 47 45 4E 45  52 41 4C 20 54 48 45 4F  |RE..GENERAL THEO|
00000370: 52 45 4D 20 4B 4E 4F 57  4E 20 41 53 20 54 48 45  |REM KNOWN AS THE|
00000380: 20 43 45 4E 54 52 41 4C  0D 0D 4C 49 4D 49 54 20  | CENTRAL..LIMIT |
00000390: 54 48 45 4F 52 45 4D 2E  20 20 CD 4F 53 54 20 4F  |THEOREM.  .OST O|
000003A0: 46 20 54 48 45 20 52 45  53 55 4C 54 53 20 49 4E  |F THE RESULTS IN|
000003B0: 0D 0D 54 48 45 20 46 49  45 4C 44 20 4F 46 20 49  |..THE FIELD OF I|
000003C0: 4E 46 45 52 45 4E 54 49  41 4C 20 53 54 41 54 49  |NFERENTIAL STATI|
000003D0: 53 54 49 43 53 0D 0D 52  45 53 54 20 49 4E 20 53  |STICS..REST IN S|
000003E0: 4F 4D 45 20 57 41 59 20  4F 4E 20 54 48 49 53 20  |OME WAY ON THIS |
000003F0: 54 48 45 4F 52 45 4D 2E  0D 0D 20 20 CF 55 52 20  |THEOREM...  .UR |
00000400: 41 50 50 4C 49 43 41 54  49 4F 4E 20 57 49 4C 4C  |APPLICATION WILL|
00000410: 20 42 45 20 54 4F 20 54  41 4B 45 20 54 48 45 0D  | BE TO TAKE THE.|
00000420: 0D 55 4E 49 46 4F 52 4D  20 44 49 53 54 52 49 42  |.UNIFORM DISTRIB|
00000430: 55 54 49 4F 4E 20 47 45  4E 45 52 41 54 45 44 20  |UTION GENERATED |
00000440: 42 59 20 54 48 45 0D 0D  C3 2D 36 34 27 53 20 52  |BY THE...-64'S R|
00000450: 41 4E 44 4F 4D 20 4E 55  4D 42 45 52 20 47 45 4E  |ANDOM NUMBER GEN|
00000460: 45 52 41 54 4F 52 2C 20  54 41 4B 45 0D 0D 41 56  |ERATOR, TAKE..AV|
00000470: 45 52 41 47 45 53 20 4F  46 20 CE 20 52 41 4E 44  |ERAGES OF . RAND|
00000480: 4F 4D 20 4E 55 4D 42 45  52 53 20 41 4E 44 20 57  |OM NUMBERS AND W|
00000490: 41 54 43 48 0D 0D 54 48  45 20 44 49 53 54 52 49  |ATCH..THE DISTRI|
000004A0: 42 55 54 49 4F 4E 20 4F  46 20 54 48 4F 53 45 20  |BUTION OF THOSE |
000004B0: 41 56 45 52 41 47 45 53  20 41 53 0D 0D 54 48 45  |AVERAGES AS..THE|
000004C0: 59 20 41 52 45 20 50 4C  4F 54 54 45 44 2E 20 20  |Y ARE PLOTTED.  |
000004D0: C8 4F 50 45 46 55 4C 4C  59 2C 20 49 4E 53 54 45  |.OPEFULLY, INSTE|
000004E0: 41 44 0D 0D 4F 46 20 4C  4F 4F 4B 49 4E 47 20 55  |AD..OF LOOKING U|
000004F0: 4E 49 46 4F 52 4D 2C 20  54 48 49 53 20 44 49 53  |NIFORM, THIS DIS|
00000500: 54 52 49 42 55 54 49 4F  4E 0D 0D 57 49 4C 4C 20  |TRIBUTION..WILL |
00000510: 4C 4F 4F 4B 20 4E 4F 52  4D 41 4C 2E 0D 0D 20 20  |LOOK NORMAL...  |
00000520: D7 48 45 4E 20 59 4F 55  20 52 55 4E 20 54 48 45  |.HEN YOU RUN THE|
00000530: 20 50 52 4F 47 52 41 4D  2C 0D 0D 43 48 4F 4F 53  | PROGRAM,..CHOOS|
00000540: 45 20 54 48 45 20 43 45  4E 54 52 41 4C 20 4C 49  |E THE CENTRAL LI|
00000550: 4D 49 54 20 54 48 45 4F  52 45 4D 0D 0D 4F 50 54  |MIT THEOREM..OPT|
00000560: 49 4F 4E 20 57 49 54 48  2C 20 53 41 59 2C 20 32  |ION WITH, SAY, 2|
00000570: 35 20 53 55 42 49 4E 54  45 52 56 41 4C 53 2C 0D  |5 SUBINTERVALS,.|
00000580: 0D 41 56 45 52 41 47 49  4E 47 20 31 30 20 52 41  |.AVERAGING 10 RA|
00000590: 4E 44 4F 4D 20 4E 55 4D  42 45 52 53 2C 20 41 4E  |NDOM NUMBERS, AN|
000005A0: 44 20 33 30 30 0D 0D 49  54 45 52 41 54 49 4F 4E  |D 300..ITERATION|
000005B0: 53 2E 20 20 D4 48 45 4E  20 54 52 59 20 4F 54 48  |S.  .HEN TRY OTH|
000005C0: 45 52 0D 0D 43 4F 4D 42  49 4E 41 54 49 4F 4E 53  |ER..COMBINATIONS|
000005D0: 2E 20 20 CE 4F 54 49 43  45 20 41 53 20 59 4F 55  |.  .OTICE AS YOU|
000005E0: 20 41 56 45 52 41 47 45  0D 0D 4C 41 52 47 45 52  | AVERAGE..LARGER|
000005F0: 20 41 4E 44 20 4C 41 52  47 45 52 20 53 41 4D 50  | AND LARGER SAMP|
00000600: 4C 45 53 2C 20 54 48 45  0D 0D 44 49 53 54 52 49  |LES, THE..DISTRI|
00000610: 42 55 54 49 4F 4E 20 42  45 43 4F 4D 45 53 20 4C  |BUTION BECOMES L|
00000620: 45 53 53 20 53 50 52 45  41 44 20 4F 55 54 2E 0D  |ESS SPREAD OUT..|
00000630: 0D D4 48 49 53 20 49 53  20 42 45 43 41 55 53 45  |..HIS IS BECAUSE|
00000640: 20 54 48 45 20 56 41 52  49 41 4E 43 45 20 49 53  | THE VARIANCE IS|
00000650: 20 42 45 49 4E 47 0D 0D  52 45 44 55 43 45 44 20  | BEING..REDUCED |
00000660: 41 53 20 54 48 45 20 54  48 45 4F 52 45 4D 20 43  |AS THE THEOREM C|
00000670: 4C 41 49 4D 53 20 49 54  20 57 49 4C 4C 2E 0D 0D  |LAIMS IT WILL...|
00000680: 2D 2D 2D 2D 2D 2D 2D 2D  2D 3C 20 45 4E 44 20 4F  |---------< END O|
00000690: 46 20 41 52 54 49 43 4C  45 20 3E 2D 2D 2D 2D 2D  |F ARTICLE >-----|
000006A0: 2D 2D 2D 2D 2D 2D 0D 0D                           |------..        |

.     THE CENTRAL LIMIT THEOREM...  IF S
AMPLES OF N NUMBERS ARE..REPEATEDLY CHOS
EN IN A RANDOM FASHION..FROM A GIVEN POP
ULATION WITH MEAN M..AND VARIANCE.      
        2.             S..AND IF FROM EA
CH SAMPLE THE MEAN IS..FORMED (THAT IS, 
THE NUMBERS IN THE..SAMPLE ARE ADDED AND
 DIVIDED BY N),..THEN THESE MEANS FORM A
 NEW..DISTRIBUTION...  THERE IS A THEORE
M IN MATHEMATICAL..STATISTICS WHICH STAT
ES THAT THE NEW..DISTRIBUTION OF SAMPLE 
MEANS WILL BE..APPROXIMATELY NORMAL (BEL
L-SHAPED)..REGARDLESS OF THE SHAPE OF TH
E..ORIGINAL DISTRIBUTION FROM WHICH THE.
.SAMPLES WERE DRAWN.  FURTHERMORE, THE..
DISTRIBUTION OF SAMPLE MEANS WILL HAVE..
THE SAME MEAN, M, AS THE ORIGINAL..DISTR
IBUTION, AND HAVE VARIANCE EQUAL..TO:.  
        2..        S /N...AS N (THE SAMP
LE SIZE) INCREASES, THE..DISTRIBUTION WI
LL BECOME INCREASINGLY..NORMAL...  THIS 
RESULT FOLLOWS FROM A MORE..GENERAL THEO
REM KNOWN AS THE CENTRAL..LIMIT THEOREM.
  MOST OF THE RESULTS IN..THE FIELD OF I
NFERENTIAL STATISTICS..REST IN SOME WAY 
ON THIS THEOREM...  OUR APPLICATION WILL
 BE TO TAKE THE..UNIFORM DISTRIBUTION GE
NERATED BY THE..C-64'S RANDOM NUMBER GEN
ERATOR, TAKE..AVERAGES OF N RANDOM NUMBE
RS AND WATCH..THE DISTRIBUTION OF THOSE 
AVERAGES AS..THEY ARE PLOTTED.  HOPEFULL
Y, INSTEAD..OF LOOKING UNIFORM, THIS DIS
TRIBUTION..WILL LOOK NORMAL...  WHEN YOU
 RUN THE PROGRAM,..CHOOSE THE CENTRAL LI
MIT THEOREM..OPTION WITH, SAY, 25 SUBINT
ERVALS,..AVERAGING 10 RANDOM NUMBERS, AN
D 300..ITERATIONS.  THEN TRY OTHER..COMB
INATIONS.  NOTICE AS YOU AVERAGE..LARGER
 AND LARGER SAMPLES, THE..DISTRIBUTION B
ECOMES LESS SPREAD OUT...THIS IS BECAUSE
 THE VARIANCE IS BEING..REDUCED AS THE T
HEOREM CLAIMS IT WILL...---------< END O
F ARTICLE >-----------..

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