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Text File | 1994-07-12 | 12.0 KB | 535 lines

.SET #8(*1, *) .LET (*#34=#34+#8*) TEXT .LET (*#21=#21+#6*) @LAB:SCREEN10 .SCR 0 .SNA(*(.INF 39)0*) .COL 7,0 .LOC 2,46(*10*) .COL 0,3 .LOC 5,11(* Contingency Tables *) .PAU(**)#4 .COL 14,0 .LOC 8,10(*Another important use of the X² distribution is to investigate*) .LOC 10,16(*a population which has been split into pre-defined*) .LOC 12,14(*categories. It is*) .COL 12,0 .LOC 12,33(*ESSENTIAL*) .COL 14,0 .LOC 12,43(*that these categories are*) .COL 11,0 .LOC 15,32(*MUTUALLY EXCLUSIVE*) .PAU(**)#4 .COL 10,0 .LOC 18,15(*Our Null Hypothesis, in this case, is always that the*) .COL 12,0 .LOC 20,21(*categories are statistically independent.*) .SPE PLAY (*AM140810.SPE*) WAIT .PAU(**)#4 .COL 0,3 .LOC 24,64(*SPACE >>*) .PAU(**) @LAB:SCREEN11 .SCR 0 .SNA(*(.INF 39)0*) .COL 7,0 .LOC 2,46(*11*) .COL 14,0 .LOC 5,13(*We shall use the X² test in precisely the same way as in*) .LOC 7,36(*Module 14.7*) .PAU(**)#4 .COL 12,0 .LOC 9,13(*An example:*) .COL 10,0 .LOC 11,15(*A population of 1000 schoolchildren was split up according*) .LOC 13,15(*to the colour of their hair and the colour of their eyes.*) .LOC 15,15(*The results were...*) .COL 14,0 .LOC 15,43(*[ fictitious data ]*) .PAU(**)#4 .COL 0,3 .LOC 24,64(*SPACE >>*) .PAU(**) .LOC 24,64(* *) .COL 7,0 .WIP (4,2)-(23,79) #6 .LOC 2,46(*11a*) .COL 14,0 .LOC 11,38(*Hair Colour*) .LOC 12,29(*Black Brown Blond Red*) .LOC 15,12(*Eye*) .LOC 17,12(*Colour*) .LOC 14,21(*Blue*) .LOC 16,21(*Brown*) .LOC 18,21(*Other*) .COL 13,0 .LOC 11,27(*║*) .LOC 11,58(*║*) .LOC 12,27(*║*) .LOC 12,58(*║*) .LOC 13,21(*──────╫──────────────────────────────╫───────*) .LOC 14,27(*║ ║*) .LOC 15,27(*║ ║*) .LOC 16,27(*║ ║*) .LOC 17,27(*║ ║*) .LOC 18,27(*║ ║*) .LOC 19,21(*──────╫──────────────────────────────╫───────*) .LOC 20,27(*║ ║*) .COL 14,0 .LOC 20,21(*Totals*) .LOC 12,60(*Totals*) .COL 10,0 .LOC 14,30(*110 70 130 40*) .LOC 14,62(*350*) .LOC 16,31(*20 270 20 40*) .LOC 16,62(*350*) .LOC 18,31(*50 170 50 30*) .LOC 18,62(*300*) .LOC 20,30(*180 510 200 110*) .LOC 20,61(*1000*) .SPE PLAY (*AM140811.SPE*) WAIT .PAU(**)#4 .COL 12,0 .LOC 22,15(*NULL HYPOTHESIS:*) .PAU(**)#4 .COL 10,0 .LOC 22,32(*hair colour is independent of eye colour*) .PAU(**)#4 .COL 0,3 .LOC 24,64(*SPACE >>*) .PAU(**) .LOC 24,64(* *) .COL 7,0 .WIP (4,2)-(23,79) #6 .LOC 2,46(*11b*) .PAU(**)#2 .COL 10,0 .LOC 18,15(*Each individual entry in the table is called a*) .COL 11,0 .LOC 18,62(*CELL*) .PAU(**)#4 .COL 0,3 .LOC 8,30(*110*) .PAU(**)#2 .COL 10,0 .LOC 8,30(*110*) .COL 0,3 .LOC 10,47(*20*) .PAU(**)#2 .COL 10,0 .LOC 10,47(*20*) .COL 0,3 .LOC 12,47(*50*) .PAU(**)#2 .COL 10,0 .LOC 12,47(*50*) .COL 0,3 .LOC 12,38(*170*) .PAU(**)#2 .COL 10,0 .LOC 12,38(*170*) .PAU(**)#4 .LOC 20,15(*This Contingency Table has*) .COL 11,0 .LOC 20,43(*3 rows*) .COL 0,3 .LOC 8,70(*«*) .LOC 10,70(*«*) .LOC 12,70(*«*) .PAU(**)#2 .COL 10,0 .LOC 8,70(* *) .LOC 10,70(* *) .LOC 12,70(* *) .PAU(**)#2 .COL 10,0 .LOC 20,51(*and*) .COL 11,0 .LOC 20,56(*4 columns*) .COL 0,3 .LOC 15,31(*^*) .LOC 15,39(*^*) .LOC 15,47(*^*) .LOC 15,54(*^*) .PAU(**)#2 .COL 10,0 .LOC 15,30(* *) .SPE PLAY (*AM140812.SPE*) WAIT .PAU(**)#4 .COL 0,3 .LOC 24,64(*SPACE >>*) .PAU(**) .LOC 24,64(* *) .COL 7,0 .WIP (15,2)-(23,79) .LOC 2,46(*12 *) .PAU(**)#2 .COL 14,0 .LOC 16,11(*From the table, assuming that the cells are independent*) .PAU(**)#4 .COL 10,0 .LOC 18,18(*the probability of blue eyes is 350 out of 1000*) .COL 0,3 .LOC 8,61(* 350 *) .LOC 14,60(* 1000 *) .PAU(**)#4 .COL 14,0 .LOC 20,26(*»*) .COL 10,0 .LOC 20,33(*p( blue eyes ) = 0.35*) .COL 10,0 .LOC 8,61(* 350 *) .LOC 14,60(* 1000 *) .PAU(**)#4 .LOC 22,20(*p( brown eyes ) = 35 out of 1000 = 0.35*) .COL 0,3 .LOC 10,61(* 350 *) .LOC 14,60(* 1000 *) .PAU(**)#4 .COL 10,0 .LOC 10,61(* 350 *) .LOC 14,60(* 1000 *) .COL 0,3 .LOC 24,64(*SPACE >>*) .PAU(**) .LOC 24,64(* *) .COL 7,0 .WIP (15,2)-(23,79) #4 .LOC 2,46(*12a*) .PAU(**)#2 .COL 10,0 .LOC 20,19(*p( black hair ) = 180 out of 1000 = 0.18*) .COL 0,3 .LOC 14,29(* 180 *) .LOC 14,60(* 1000 *) .PAU(**)#4 .COL 10,0 .LOC 14,29(* 180 *) .LOC 14,60(* 1000 *) .LOC 22,21(*p( red hair ) = 110 out of 1000 = 0.11*) .COL 0,3 .LOC 14,52(* 110 *) .LOC 14,60(* 1000 *) .PAU(**)#4 .COL 10,0 .LOC 14,52(* 110 *) .LOC 14,60(* 1000 *) .PAU(**)#4 .COL 0,3 .LOC 24,64(*SPACE >>*) .PAU(**) .LOC 24,64(* *) .COL 7,0 .WIP (4,2)-(23,79) #11 .LOC 2,46(*13 *) .PAU(**)#2 .COL 14,0 .LOC 13,11(*Using Probability Law 4 for statistically independent events:*) .PAU(**)#4 .COL 10,0 .LOC 15,11(*p( brown eyes*) .COL 12,0 .LOC 15,25(*AND*) .COL 10,0 .LOC 15,29(*red hair ) = p( brown eyes )*) .COL 12,0 .LOC 15,60(*x*) .COL 10,0 .LOC 15,62(*p( red hair )*) .SPE PLAY (*AM140813.SPE*) WAIT .PAU(**)#4 .LOC 17,41(*= 0.35 x 0.11*) .PAU(**)#4 .LOC 17,61(*= 0.0385 (1)*) .PAU(**)#4 .LOC 19,11(*Expected frequency for the cell [ brown eyes and red hair ]*) .LOC 21,41(*= 1000 x 0.0385*) .PAU(**)#4 .LOC 21,61(*= 38.5 (2)*) .COL 0,3 .LOC 24,64(*C / R >>*) @SIN(*"C"=L:SCREEN14,"R"=L:-SCREEN10,"c"=L:SCREEN14,"r"=L:-SCREEN10*) @LAB:SCREEN14 .SNA(*(.INF 39)0*) .COL 7,0 .LOC 2,46(*14*) .COL 10,0 .LOC 7,11(*Note how line (1) could be written as...*) .PAU(**)#4 .COL 14,0 .LOC 9,34(*350 110*) .LOC 10,33(*──── x ────*) .LOC 11,33(*1000 1000*) .PAU(**)#4 .COL 10,0 .LOC 13,11(*and line (2) as...*) .PAU(**)#4 .COL 14,0 .LOC 15,36(*350 110 350 x 110*) .LOC 16,28(*1000 x ──── x ──── = ─────────*) .LOC 17,35(*1000 1000 1000*) .PAU(**)#4 .COL 10,0 .LOC 19,38(*Row Total x Column Total*) .LOC 20,17(*Expected frequency = ────────────────────────*) .LOC 21,41(*Grand Total*) .SPE PLAY (*AM140814.SPE*) WAIT .PAU(**)#4 .WIP (4,2)-(23,79)#14 .PAU(**)#2 .LOC 9,11(*You may like to verify for yourself algebraically that this*) .LOC 11,27(*expression is a general rule.*) .SPE PLAY (*AM140814.SPE*) WAIT .PAU(**)#4 .COL 14,0 .LOC 15,13(*We can re-draw the table in our example and add expected*) .LOC 17,22(*frequencies based on these assumptions.*) .PAU(**)#4 .COL 15,0 .LOC 20,15(*You may find it convenient to use two colours of pen.*) .PAU(**)#4 .COL 0,3 .LOC 24,64(*SPACE >>*) .PAU(**) @LAB:SCREEN15 .SNA(*(.INF 39)0*) .COL 7,0 .LOC 2,46(*15*) .COL 14,0 .LOC 5,22(*Black Brown Blond Red Totals*) .COL 11,0 .LOC 6,21(*O E O E O E O E O E*) .COL 14,0 .LOC 8,11(*Blue*) .LOC 10,11(*Brown*) .LOC 12,11(*Other*) .LOC 14,11(*Totals*) .COL 13,0 .LOC 5,18(*║*) .LOC 5,30(*║*) .LOC 5,42(*║*) .LOC 5,54(*║*) .LOC 5,66(*║*) .LOC 6,18(*║*) .LOC 6,30(*║*) .LOC 6,42(*║*) .LOC 6,54(*║*) .LOC 6,66(*║*) .LOC 6,24(*│*) .LOC 6,36(*│*) .LOC 6,48(*│*) .LOC 6,60(*│*) .LOC 6,72(*│*) .LOC 7,11(*───────╫─────┼─────╫─────┼─────╫─────┼─────╫─────┼─────╫─────┼─────*) .LOC 8,18(*║ │ ║ │ ║ │ ║ │ ║ │*) .LOC 9,18(*║ │ ║ │ ║ │ ║ │ ║ │*) .LOC 10,18(*║ │ ║ │ ║ │ ║ │ ║ │*) .LOC 11,18(*║ │ ║ │ ║ │ ║ │ ║ │*) .LOC 12,18(*║ │ ║ │ ║ │ ║ │ ║ │*) .LOC 13,11(*───────╫─────┼─────╫─────┼─────╫─────┼─────╫─────┼─────╫─────┼─────*) .LOC 14,18(*║ │ ║ │ ║ │ ║ │ ║ │*) .COL 10,0 .LOC 8,20(*110*) .LOC 8,33(*70*) .LOC 8,44(*130*) .LOC 8,57(*40*) .LOC 8,68(*350*) .LOC 10,21(*20*) .LOC 10,32(*270*) .LOC 10,45(*20*) .LOC 10,57(*40*) .LOC 10,68(*350*) .LOC 12,21(*50*) .LOC 12,32(*170*) .LOC 12,45(*50*) .LOC 12,57(*30*) .LOC 12,68(*300*) .LOC 14,20(*180*) .LOC 14,32(*510*) .LOC 14,44(*200*) .LOC 14,56(*110*) .LOC 14,67(*1000*) .PAU(**)#4 .COL 12,0 .LOC 8,27(*63*) .PAU(**)#2 .LOC 10,27(*63*) .PAU(**)#2 .LOC 12,27(*54*) .PAU(**)#2 .LOC 14,26(*180*) .PAU(**)#4 .LOC 8,37(*178.5*) .PAU(**)#2 .LOC 10,37(*178.5*) .PAU(**)#2 .LOC 12,38(*153*) .PAU(**)#2 .LOC 14,38(*510*) .PAU(**)#4 .LOC 8,50(*70*) .PAU(**)#2 .LOC 10,50(*70*) .PAU(**)#2 .LOC 12,50(*60*) .PAU(**)#2 .LOC 14,49(*200*) .PAU(**)#4 .LOC 8,62(*38.5*) .PAU(**)#2 .LOC 10,62(*38.5*) .PAU(**)#2 .LOC 12,62(*33*) .PAU(**)#2 .LOC 14,62(*110*) .PAU(**)# .LOC 8,74(*350*) .PAU(**)#2 .LOC 10,74(*350*) .PAU(**)#2 .LOC 12,74(*300*) .PAU(**)#2 .LOC 14,73(*1000*) .PAU(**)#4 .COL 14,0 .LOC 17,9(*In this particular example, the sums of the expected frequencies*) .LOC 19,11(*agree with those of the observed ones. Rounding errors can *) .LOC 21,14(*prevent this but any differences should be very small.*) .PAU(**)#4 .COL 0,3 .LOC 24,64(*SPACE >>*) .PAU(**) .LOC 24,64(* *) .COL 7,0 .WIP (16,2)-(23,79) .LOC 2,46(*15a*) .PAU(**)#2 .COL 14,0 .LOC 17,11(*Now we can calculate X² by adding up the*) .COL 12,0 .LOC 16,52(*( O - E )²*) .LOC 17,53(*───────*) .LOC 18,56(*E*) .COL 14,0 .LOC 17,63(*for each cell.*) .SPE PLAY (*AM140815.SPE*) WAIT .PAU(**)#4 .COL 10,0 .LOC 21,14(*X² =*) .LOC 20,21(*(110 - 63)²*) .LOC 21,22(*────────*) .LOC 22,25(*63*) .PAU(**)#4 .LOC 21,33(*+*) .LOC 20,35(*(70 - 178.5)²*) .LOC 21,36(*──────────*) .LOC 22,39(*178.5*) .PAU(**)#4 .LOC 21,49(*+*) .LOC 20,51(*(130 - 70)²*) .LOC 21,52(*────────*) .LOC 22,56(*70*) .PAU(**)#4 .LOC 21,63(*+*) .COL 12,0 .LOC 20,66(*for all*) .LOC 22,66(*12 cells*) .PAU(**)#4 .COL 0,3 .LOC 24,64(*SPACE >>*) .PAU(**) .LOC 24,64(* *) .COL 7,0 .WIP (15,2)-(23,79) #4 .LOC 2,46(*15b*) .PAU(**)#2 .COL 10,0 .LOC 20,18(*= 35.06 + 65.95 + 51.43 + 0.06 + 29.35 + 46.90 + 35.71*) .LOC 22,23(*+ 0.06 + 0.30 + 1.89 + 1.67 + 0.27*) .PAU(**)#4 .LOC 22,59(*= 268.65*) .COL 0,3 .LOC 24,64(*C / R >>*) @SIN(*"C"=L:SCREEN16,"R"=L:-SCREEN14,"c"=L:SCREEN16,"r"=L:-SCREEN14*) @LAB:SCREEN16 .SNA(*(.INF 39)0*) .COL 7,0 .LOC 2,46(*16*) .COL 14,0 .LOC 5,15(*Now we need to look at the Degrees of Freedom to use.*) .PAU(**)#4 .COL 12,0 .LOC 8,11(*For the rows:*) .PAU(**)#4 .COL 10,0 .LOC 8,28(*we made the totals of the rows agree so*) .LOC 10,28(*we could choose all but one of them*) .PAU(**)#4 .COL 11,0 .LOC 12,28(*number of rows - 1 constraint*) .PAU(**)#4 .COL 14,0 .LOC 12,62(*»*) .COL 11,0 .LOC 12,66(*r - 1*) .PAU(**)#4 .COL 12,0 .LOC 14,11(*and columns:*) .PAU(**)#4 .COL 10,0 .LOC 14,28(*we made the totals of the columns agree so*) .LOC 16,28(*we could choose all but one of them*) .PAU(**)#4 .COL 11,0 .LOC 18,28(*number of columns - 1 constraint*) .PAU(**)#4 .COL 14,0 .LOC 18,62(*»*) .COL 11,0 .LOC 18,66(*c - 1*) .PAU(**)#4 .COL 10,0 .LOC 20,18(*Degrees of Freedom = ( r - 1 ) x ( c - 1 )*) .LOC 22,38(*= ( 3 - 1 ) x ( 4 - 1 ) = 6*) .PAU(**)#4 .COL 0,3 .LOC 24,64(*SPACE >>*) .PAU(**) @LAB:SCREEN17 .SCR 2 .SNA(*(.INF 39)2*) .LOC 2,46(*17*) .PUT 183,29 PART=6 .LOC 5,27(*= 268.65*) .PUT 344,31 PART=7 .LOC 5,47(*= 6*) .PAU(**)#4 .LOC 7,15(*Enter your*) .PUT 200,45 PART=6 .LOC 7,29(*table at*) .PUT 304,47 PART=7 .LOC 7,41(*= 6, level 5 %*) .PAU(**)#4 .LOC 7,60(*» 12.59*) .PAU(**)#4 .LOC 9,16(*268.65 IS ≥ 12.59 so the result is SIGNIFICANT*) .PAU(**)#4 .LOC 12,9(*In fact, it is still highly significant even at the 0.1 % level !*) .PAU(**)#4 .LOC 15,11(*We reject the Null Hypothesis and the Alternate Hypothesis*) .LOC 17,12(*that the categories are NOT INDEPENDENT would be adopted.*) .PAU(**)#4 .USE PROG=PAUSE.USE .WIP (4,2)-(23,79) #3 .LOC 2,46(*17a*) .PAU(**)#2 .LOC 17,11(*The result is as you should expect:*) .PAU(**)#4 .LOC 19,17(*brown hair is usually associated with brown eyes*) .LOC 21,17(*blond hair is usually associated with blue eyes.*) .PUT 500,187 PART=2 PSET @SIN(*"C"=L:SCREEN17Z,"R"=L:-SCREEN11,"c"=L:SCREEN17Z,"r"=L:-SCREEN11*) @LAB:SCREEN17Z @GOB:SCREEN18 PROG=AM1408B1.CHA