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MATH 204: PRINCIPLES OF STATISTICS 2USING THE FISHER-F TABLES
Tables VIII to XI in McClave and Sincich (pages 899 to 906) contain information on the 1− α probabilitypoints for the Fisher-F distribution for α = 0.1, 0.05, 0.025 and 0.01 respectively, and for different valuesof the degrees of freedom parameters. The values in the body of the table are the numbers x which solvethe equation
Pr[F > x] = α
when the statistic F has a Fisher-F distribution with ν1 and ν2 degrees of freedom, written
F ∼ Fisher-F(ν1, ν2)
where ν1 and ν2 are whole numbers greater than zero.
The table on the reverse of this sheet is the Fisher-F table for α = 0.05, equivalent to the table on pages901-902 of McClave and Sincich. We read ν1 from the column and ν2 from the row. For example,
• if ν1 = 10 and ν2 = 4, we know from the table that
Pr[ F > 5.96 ] = 0.05
• if ν1 = 6 and ν2 = 18, we know from the table that
Pr[ F > 2.66 ] = 0.05
• if ν1 = 20 and ν2 = 20, we know from the table that
Pr[ F > 2.12 ] = 0.05
The Fisher-F distribution is a non-symmetric probability distribution with a specific property that allowsthe tables in McClave and Sincich to tabulate only the right-hand tail of the distribution. If we need tolook up the left-hand tail, we can use the fact that if F ∼ Fisher-F(ν1, ν2), and 0 < p < 1
Pr[ F > x ] = p =⇒ Pr[ 1/F ≤ 1/x ] = p
so thatPr[ 1/F > 1/x ] = 1− p.
But it transpires that
F ∼ Fisher-F(ν1, ν2) =⇒ 1F∼ Fisher-F(ν2, ν1).
Therefore to look up the left-tail α probability point for the F ∼ Fisher-F(ν1, ν2) distribution, we look upthe right-tail 1−α probability point for the Fisher-F(ν2, ν1) distribution, and then take the reciprocal. Forexample,
• if F ∼ Fisher-F(10, 4), we use tables to discover that as
F0.05(4, 10) = 3.48
it follows thatPr[ F ≤ 1/3.48 ] = Pr[ F ≤ 0.29 ] = 0.05
giving the α = 0.05 probability point of the Fisher-F(10, 4) distribution as 0.29.
1
Tabl
eof
the
Fish
er-F
dist
ribu
tion
Entr
ies
inta
ble
are
the
α=
0.05
tail
quan
tile
ofFi
sher
-F(ν
1,ν
2)
dist
ribu
tion
ν 1gi
ven
inco
lum
ns,ν
2gi
ven
inro
ws.
ν 2\ν
11
23
45
67
89
1011
1213
1415
1617
1819
2021
2223
2425
116
1.45
199.
5021
5.71
224.
5823
0.16
233.
9923
6.77
238.
8824
0.54
241.
8824
2.98
243.
9124
4.69
245.
3624
5.95
246.
4624
6.92
247.
3224
7.69
248.
0124
8.31
248.
5824
8.83
249.
0524
9.26
218
.51
19.0
019
.16
19.2
519
.30
19.3
319
.35
19.3
719
.38
19.4
019
.40
19.4
119
.42
19.4
219
.43
19.4
319
.44
19.4
419
.44
19.4
519
.45
19.4
519
.45
19.4
519
.46
310
.13
9.55
9.28
9.12
9.01
8.94
8.89
8.85
8.81
8.79
8.76
8.74
8.73
8.71
8.70
8.69
8.68
8.67
8.67
8.66
8.65
8.65
8.64
8.64
8.63
47.
716.
946.
596.
396.
266.
166.
096.
046.
005.
965.
945.
915.
895.
875.
865.
845.
835.
825.
815.
805.
795.
795.
785.
775.
775
6.61
5.79
5.41
5.19
5.05
4.95
4.88
4.82
4.77
4.74
4.70
4.68
4.66
4.64
4.62
4.60
4.59
4.58
4.57
4.56
4.55
4.54
4.53
4.53
4.52
65.
995.
144.
764.
534.
394.
284.
214.
154.
104.
064.
034.
003.
983.
963.
943.
923.
913.
903.
883.
873.
863.
863.
853.
843.
837
5.59
4.74
4.35
4.12
3.97
3.87
3.79
3.73
3.68
3.64
3.60
3.57
3.55
3.53
3.51
3.49
3.48
3.47
3.46
3.44
3.43
3.43
3.42
3.41
3.40
85.
324.
464.
073.
843.
693.
583.
503.
443.
393.
353.
313.
283.
263.
243.
223.
203.
193.
173.
163.
153.
143.
133.
123.
123.
119
5.12
4.26
3.86
3.63
3.48
3.37
3.29
3.23
3.18
3.14
3.10
3.07
3.05
3.03
3.01
2.99
2.97
2.96
2.95
2.94
2.93
2.92
2.91
2.90
2.89
104.
964.
103.
713.
483.
333.
223.
143.
073.
022.
982.
942.
912.
892.
862.
852.
832.
812.
802.
792.
772.
762.
752.
752.
742.
7311
4.84
3.98
3.59
3.36
3.20
3.09
3.01
2.95
2.90
2.85
2.82
2.79
2.76
2.74
2.72
2.70
2.69
2.67
2.66
2.65
2.64
2.63
2.62
2.61
2.60
124.
753.
893.
493.
263.
113.
002.
912.
852.
802.
752.
722.
692.
662.
642.
622.
602.
582.
572.
562.
542.
532.
522.
512.
512.
5013
4.67
3.81
3.41
3.18
3.03
2.92
2.83
2.77
2.71
2.67
2.63
2.60
2.58
2.55
2.53
2.51
2.50
2.48
2.47
2.46
2.45
2.44
2.43
2.42
2.41
144.
603.
743.
343.
112.
962.
852.
762.
702.
652.
602.
572.
532.
512.
482.
462.
442.
432.
412.
402.
392.
382.
372.
362.
352.
3415
4.54
3.68
3.29
3.06
2.90
2.79
2.71
2.64
2.59
2.54
2.51
2.48
2.45
2.42
2.40
2.38
2.37
2.35
2.34
2.33
2.32
2.31
2.30
2.29
2.28
164.
493.
633.
243.
012.
852.
742.
662.
592.
542.
492.
462.
422.
402.
372.
352.
332.
322.
302.
292.
282.
262.
252.
242.
242.
2317
4.45
3.59
3.20
2.96
2.81
2.70
2.61
2.55
2.49
2.45
2.41
2.38
2.35
2.33
2.31
2.29
2.27
2.26
2.24
2.23
2.22
2.21
2.20
2.19
2.18
184.
413.
553.
162.
932.
772.
662.
582.
512.
462.
412.
372.
342.
312.
292.
272.
252.
232.
222.
202.
192.
182.
172.
162.
152.
1419
4.38
3.52
3.13
2.90
2.74
2.63
2.54
2.48
2.42
2.38
2.34
2.31
2.28
2.26
2.23
2.21
2.20
2.18
2.17
2.16
2.14
2.13
2.12
2.11
2.11
204.
353.
493.
102.
872.
712.
602.
512.
452.
392.
352.
312.
282.
252.
222.
202.
182.
172.
152.
142.
122.
112.
102.
092.
082.
0721
4.32
3.47
3.07
2.84
2.68
2.57
2.49
2.42
2.37
2.32
2.28
2.25
2.22
2.20
2.18
2.16
2.14
2.12
2.11
2.10
2.08
2.07
2.06
2.05
2.05
224.
303.
443.
052.
822.
662.
552.
462.
402.
342.
302.
262.
232.
202.
172.
152.
132.
112.
102.
082.
072.
062.
052.
042.
032.
0223
4.28
3.42
3.03
2.80
2.64
2.53
2.44
2.37
2.32
2.27
2.24
2.20
2.18
2.15
2.13
2.11
2.09
2.08
2.06
2.05
2.04
2.02
2.01
2.01
2.00
244.
263.
403.
012.
782.
622.
512.
422.
362.
302.
252.
222.
182.
152.
132.
112.
092.
072.
052.
042.
032.
012.
001.
991.
981.
9725
4.24
3.39
2.99
2.76
2.60
2.49
2.40
2.34
2.28
2.24
2.20
2.16
2.14
2.11
2.09
2.07
2.05
2.04
2.02
2.01
2.00
1.98
1.97
1.96
1.96
264.
233.
372.
982.
742.
592.
472.
392.
322.
272.
222.
182.
152.
122.
092.
072.
052.
032.
022.
001.
991.
981.
971.
961.
951.
9427
4.21
3.35
2.96
2.73
2.57
2.46
2.37
2.31
2.25
2.20
2.17
2.13
2.10
2.08
2.06
2.04
2.02
2.00
1.99
1.97
1.96
1.95
1.94
1.93
1.92
284.
203.
342.
952.
712.
562.
452.
362.
292.
242.
192.
152.
122.
092.
062.
042.
022.
001.
991.
971.
961.
951.
931.
921.
911.
9129
4.18
3.33
2.93
2.70
2.55
2.43
2.35
2.28
2.22
2.18
2.14
2.10
2.08
2.05
2.03
2.01
1.99
1.97
1.96
1.94
1.93
1.92
1.91
1.90
1.89
304.
173.
322.
922.
692.
532.
422.
332.
272.
212.
162.
132.
092.
062.
042.
011.
991.
981.
961.
951.
931.
921.
911.
901.
891.
8831
4.16
3.30
2.91
2.68
2.52
2.41
2.32
2.25
2.20
2.15
2.11
2.08
2.05
2.03
2.00
1.98
1.96
1.95
1.93
1.92
1.91
1.90
1.88
1.88
1.87
324.
153.
292.
902.
672.
512.
402.
312.
242.
192.
142.
102.
072.
042.
011.
991.
971.
951.
941.
921.
911.
901.
881.
871.
861.
85