logistic regression
DESCRIPTION
Logistic regression. Recall the simple linear regression model: y = b 0 + b 1 x + e. where we are trying to predict a continuous dependent variable y from a continuous independent variable x. This model can be extended to Multiple linear regression model: - PowerPoint PPT PresentationTRANSCRIPT
-
Logistic regression
-
Recall the simple linear regression model:y = b0 + b1x + ewhere we are trying to predict a continuous dependent variable y from a continuous independent variable x.This model can be extended to Multiple linear regression model:y = b0 + b1x1 + b2x2 + + + bpxp + eHere we are trying to predict a continuous dependent variable y from a several continuous dependent variables x1 , x2 , , xp .
-
Now suppose the dependent variable y is binary. It takes on two values Success (1) or Failure (0)This is the situation in which Logistic Regression is usedWe are interested in predicting a y from a continuous dependent variable x.
-
ExampleWe are interested how the success (y) of a new antibiotic cream is curing acne problems and how it depends on the amount (x) that is applied daily.The values of y are 1 (Success) or 0 (Failure).The values of x range over a continuum
-
The logisitic Regression ModelLet p denote P[y = 1] = P[Success].This quantity will increase with the value of x. The ratio: is called the odds ratio This quantity will also increase with the value of x, ranging from zero to infinity. The quantity: is called the log odds ratio
-
Example: odds ratio, log odds ratioSuppose a die is rolled:Success = roll a six, p = 1/6The odds ratio The log odds ratio
-
The logisitic Regression Modeli. e. : In terms of the odds ratio Assumes the log odds ratio is linearly related to x.
-
The logisitic Regression Modelor Solving for p in terms x.
-
Interpretation of the parameter b0 (determines the intercept)p x
Chart3
0.119202922
0.1418510649
0.1679816149
0.1978161114
0.2314752165
0.2689414214
0.3100255189
0.3543436938
0.4013123399
0.4501660027
0.5
0.5498339973
0.5986876601
0.6456563062
0.6899744811
0.7310585786
0.7685247835
0.8021838886
0.8320183851
0.8581489351
0.880797078
0.9002495109
0.9168273035
0.9308615797
0.9426758241
0.9525741268
0.9608342772
0.9677045353
0.9734030064
0.9781187291
0.98201379
0.9852259683
0.987871565
0.9900481981
0.9918374288
0.9933071491
0.9945137011
0.9955037268
0.9963157601
0.9969815837
0.9975273768
0.9979746796
0.9983411989
0.99864148
0.998887464
0.9990889488
0.9992539712
0.9993891206
0.9994997989
0.9995904328
0.9996646499
Sheet1
b0-2
b11
00.119202922
0.20.1418510649
0.40.1679816149
0.60.1978161114
0.80.2314752165
10.2689414214
1.20.3100255189
1.40.3543436938
1.60.4013123399
1.80.4501660027
20.5
2.20.5498339973
2.40.5986876601
2.60.6456563062
2.80.6899744811
30.7310585786
3.20.7685247835
3.40.8021838886
3.60.8320183851
3.80.8581489351
40.880797078
4.20.9002495109
4.40.9168273035
4.60.9308615797
4.80.9426758241
50.9525741268
5.20.9608342772
5.40.9677045353
5.60.9734030064
5.80.9781187291
60.98201379
6.20.9852259683
6.40.987871565
6.60.9900481981
6.80.9918374288
70.9933071491
7.20.9945137011
7.40.9955037268
7.60.9963157601
7.80.9969815837
80.9975273768
8.20.9979746796
8.40.9983411989
8.60.99864148
8.80.998887464
90.9990889488
9.20.9992539712
9.40.9993891206
9.60.9994997989
9.80.9995904328
100.9996646499
Sheet1
Sheet2
Sheet3
-
Interpretation of the parameter b1 (determines when p is 0.50 (along with b0))p x when
Chart3
0.119202922
0.1418510649
0.1679816149
0.1978161114
0.2314752165
0.2689414214
0.3100255189
0.3543436938
0.4013123399
0.4501660027
0.5
0.5498339973
0.5986876601
0.6456563062
0.6899744811
0.7310585786
0.7685247835
0.8021838886
0.8320183851
0.8581489351
0.880797078
0.9002495109
0.9168273035
0.9308615797
0.9426758241
0.9525741268
0.9608342772
0.9677045353
0.9734030064
0.9781187291
0.98201379
0.9852259683
0.987871565
0.9900481981
0.9918374288
0.9933071491
0.9945137011
0.9955037268
0.9963157601
0.9969815837
0.9975273768
0.9979746796
0.9983411989
0.99864148
0.998887464
0.9990889488
0.9992539712
0.9993891206
0.9994997989
0.9995904328
0.9996646499
Sheet1
b0-2
b11
00.119202922
0.20.1418510649
0.40.1679816149
0.60.1978161114
0.80.2314752165
10.2689414214
1.20.3100255189
1.40.3543436938
1.60.4013123399
1.80.4501660027
20.5
2.20.5498339973
2.40.5986876601
2.60.6456563062
2.80.6899744811
30.7310585786
3.20.7685247835
3.40.8021838886
3.60.8320183851
3.80.8581489351
40.880797078
4.20.9002495109
4.40.9168273035
4.60.9308615797
4.80.9426758241
50.9525741268
5.20.9608342772
5.40.9677045353
5.60.9734030064
5.80.9781187291
60.98201379
6.20.9852259683
6.40.987871565
6.60.9900481981
6.80.9918374288
70.9933071491
7.20.9945137011
7.40.9955037268
7.60.9963157601
7.80.9969815837
80.9975273768
8.20.9979746796
8.40.9983411989
8.60.99864148
8.80.998887464
90.9990889488
9.20.9992539712
9.40.9993891206
9.60.9994997989
9.80.9995904328
100.9996646499
Sheet1
Sheet2
Sheet3
-
Alsowhenis the rate of increase in p with respect to x when p = 0.50
-
Interpretation of the parameter b1 (determines slope when p is 0.50 )p x
Chart3
0.119202922
0.1418510649
0.1679816149
0.1978161114
0.2314752165
0.2689414214
0.3100255189
0.3543436938
0.4013123399
0.4501660027
0.5
0.5498339973
0.5986876601
0.6456563062
0.6899744811
0.7310585786
0.7685247835
0.8021838886
0.8320183851
0.8581489351
0.880797078
0.9002495109
0.9168273035
0.9308615797
0.9426758241
0.9525741268
0.9608342772
0.9677045353
0.9734030064
0.9781187291
0.98201379
0.9852259683
0.987871565
0.9900481981
0.9918374288
0.9933071491
0.9945137011
0.9955037268
0.9963157601
0.9969815837
0.9975273768
0.9979746796
0.9983411989
0.99864148
0.998887464
0.9990889488
0.9992539712
0.9993891206
0.9994997989
0.9995904328
0.9996646499
Sheet1
b0-2
b11
00.119202922
0.20.1418510649
0.40.1679816149
0.60.1978161114
0.80.2314752165
10.2689414214
1.20.3100255189
1.40.3543436938
1.60.4013123399
1.80.4501660027
20.5
2.20.5498339973
2.40.5986876601
2.60.6456563062
2.80.6899744811
30.7310585786
3.20.7685247835
3.40.8021838886
3.60.8320183851
3.80.8581489351
40.880797078
4.20.9002495109
4.40.9168273035
4.60.9308615797
4.80.9426758241
50.9525741268
5.20.9608342772
5.40.9677045353
5.60.9734030064
5.80.9781187291
60.98201379
6.20.9852259683
6.40.987871565
6.60.9900481981
6.80.9918374288
70.9933071491
7.20.9945137011
7.40.9955037268
7.60.9963157601
7.80.9969815837
80.9975273768
8.20.9979746796
8.40.9983411989
8.60.99864148
8.80.998887464
90.9990889488
9.20.9992539712
9.40.9993891206
9.60.9994997989
9.80.9995904328
100.9996646499
Sheet1
Sheet2
Sheet3
-
The dataThe data will for each case consist ofa value for x, the continuous independent variablea value for y (1 or 0) (Success or Failure)Total of n = 250 cases
-
Sheet2 (2)
-2
1
casexy
10.800.2314752165
22.310.5744425168
32.500.6224593312
42.810.6899744811
53.510.8175744762
64.410.9168273035
70.500.1824255238
84.510.92414182
94.410.9168273035
100.900.2497398944
113.310.785834983
121.100.2890504974
132.510.6224593312
140.310.1544652651
154.510.92414182
161.800.4501660027
172.410.5986876601
181.600.4013123399
191.910.4750208125
204.610.9308615797
212.610.6456563062
223.500.8175744762
230.900.2497398944
242.300.5744425168
252.000.5
263.110.7502601056
272.700.6681877722
281.600.4013123399
294.310.908877039
302.100.5249791875
313.100.7502601056
322.210.5498339973
331.410.3543436938
341.000.2689414214
353.310.785834983
363.210.7685247835
373.610.8320183851
380.300.1544652651
395.000.9525741268
400.410.1679816149
410.100.1301084744
424.210.9002495109
431.510.3775406688
442.100.5249791875
452.310.5744425168
461.000.2689414214
471.000.2689414214
483.310.785834983
491.500.3775406688
502.310.5744425168
514.610.9308615797
522.710.6681877722
533.810.8581489351
543.810.8581489351
550.500.1824255238
564.010.880797078
574.010.880797078
584.710.9370266439
593.410.8021838886
603.610.8320183851
614.410.9168273035
623.510.8175744762
633.400.8021838886
640.710.214165017
652.200.5498339973
662.210.5498339973
672.510.6224593312
684.710.9370266439
690.500.1824255238
700.500.1824255238
712.500.6224593312
721.800.4501660027
730.600.1978161114
741.800.4501660027
751.910.4750208125
762.810.6899744811
772.510.6224593312
782.010.5
790.700.214165017
803.110.7502601056
810.000.119202922
823.910.8698915256
831.810.4501660027
844.910.9478464369
853.010.7310585786
861.700.4255574832
871.100.2890504974
882.000.5
893.110.7502601056
900.700.214165017
910.110.1301084744
921.910.4750208125
932.100.5249791875
940.310.1544652651
952.400.5986876601
962.310.5744425168
973.610.8320183851
984.110.8909031788
993.310.785834983
1003.000.7310585786
1013.410.8021838886
1020.700.214165017
1032.100.5249791875
1044.710.9370266439
1053.610.8320183851
1060.900.2497398944
1073.610.8320183851
1082.610.6456563062
1094.110.8909031788
1104.000.880797078
1112.510.6224593312
1120.810.2314752165
1134.410.9168273035
1143.510.8175744762
1150.500.1824255238
1164.100.8909031788
1174.510.92414182
1184.910.9478464369
1193.510.8175744762
1203.710.8455347349
1210.200.1418510649
1224.710.9370266439
1231.610.4013123399
1242.810.6899744811
1253.510.8175744762
1262.910.7109495026
1271.800.4501660027
1284.510.92414182
1292.300.5744425168
1302.710.6681877722
1312.810.6899744811
1324.310.908877039
1333.310.785834983
1341.810.4501660027
1352.810.6899744811
1362.300.5744425168
1370.000.119202922
1380.400.1679816149
1392.200.5498339973
1402.510.6224593312
1410.500.1824255238
1421.910.4750208125
1431.300.3318122278
1442.110.5249791875
1452.110.5249791875
1463.110.7502601056
1472.510.6224593312
1480.800.2314752165
1490.400.1679816149
1502.800.6899744811
1510.300.1544652651
1524.010.880797078
1532.800.6899744811
1540.800.2314752165
1552.810.6899744811
1560.100.1301084744
1572.700.6681877722
1584.510.92414182
1590.210.1418510649
1602.100.5249791875
1613.610.8320183851
1624.110.8909031788
1631.410.3543436938
1640.900.2497398944
1651.900.4750208125
1662.900.7109495026
1674.010.880797078
1684.910.9478464369
1692.510.6224593312
1702.500.6224593312
1710.200.1418510649
1720.900.2497398944
1731.000.2689414214
1742.400.5986876601
1752.510.6224593312
1764.710.9370266439
1770.600.1978161114
1781.710.4255574832
1791.110.2890504974
1801.310.3318122278
1814.010.880797078
1822.710.6681877722
1832.800.6899744811
1843.810.8581489351
1854.410.9168273035
1863.610.8320183851
1872.710.6681877722
1882.700.6681877722
1892.110.5249791875
1900.410.1679816149
1910.300.1544652651
1923.200.7685247835
1931.000.2689414214
1941.300.3318122278
1954.010.880797078
1960.000.119202922
1971.210.3100255189
1980.400.1679816149
1993.910.8698915256
2003.710.8455347349
2012.210.5498339973
2021.310.3318122278
2030.500.1824255238
2040.500.1824255238
2050.300.1544652651
2061.900.4750208125
2073.510.8175744762
2083.510.8175744762
2094.310.908877039
2100.110.1301084744
2113.010.7310585786
2121.010.2689414214
2130.200.1418510649
2144.510.92414182
2150.500.1824255238
2162.600.6456563062
2173.410.8021838886
2183.510.8175744762
2190.500.1824255238
2204.410.9168273035
2213.410.8021838886
2222.500.6224593312
2233.910.8698915256
2241.800.4501660027
2254.610.9308615797
2262.010.5
2270.400.1679816149
2284.510.92414182
2292.200.5498339973
2304.710.9370266439
2310.300.1544652651
2321.400.3543436938
2334.510.92414182
2341.410.3543436938
2354.510.92414182
2363.900.8698915256
2370.000.119202922
2384.310.908877039
2391.000.2689414214
2403.910.8698915256
2411.100.2890504974
2423.410.8021838886
2430.600.1978161114
2441.600.4013123399
2453.900.8698915256
2460.200.1418510649
2472.500.6224593312
2484.110.8909031788
2494.210.9002495109
2504.910.9478464369
Sheet1
b0-2
b11
00.119202922
0.20.1418510649
0.40.1679816149
0.60.1978161114
0.80.2314752165
10.2689414214
1.20.3100255189
1.40.3543436938
1.60.4013123399
1.80.4501660027
20.5
2.20.5498339973
2.40.5986876601
2.60.6456563062
2.80.6899744811
30.7310585786
3.20.7685247835
3.40.8021838886
3.60.8320183851
3.80.8581489351
40.880797078
4.20.9002495109
4.40.9168273035
4.60.9308615797
4.80.9426758241
50.9525741268
5.20.9608342772
5.40.9677045353
5.60.9734030064
5.80.9781187291
60.98201379
6.20.9852259683
6.40.987871565
6.60.9900481981
6.80.9918374288
70.9933071491
7.20.9945137011
7.40.9955037268
7.60.9963157601
7.80.9969815837
80.9975273768
8.20.9979746796
8.40.9983411989
8.60.99864148
8.80.998887464
90.9990889488
9.20.9992539712
9.40.9993891206
9.60.9994997989
9.80.9995904328
100.9996646499
Sheet1
Sheet2
-2
1
casexy
13.010.7310585786
21.200.3100255189
30.600.1978161114
40.800.2314752165
53.210.7685247835
63.810.8581489351
72.700.6681877722
82.100.5249791875
90.910.2497398944
104.910.9478464369
111.500.3775406688
123.410.8021838886
133.310.785834983
144.810.9426758241
152.010.5
162.100.5249791875
174.410.9168273035
184.510.92414182
192.910.7109495026
202.400.5986876601
211.800.4501660027
223.810.8581489351
233.710.8455347349
242.910.7109495026
250.410.1679816149
263.500.8175744762
271.810.4501660027
283.200.7685247835
294.010.880797078
303.310.785834983
313.110.7502601056
322.000.5
334.510.92414182
344.110.8909031788
351.110.2890504974
363.000.7310585786
371.100.2890504974
380.200.1418510649
394.210.9002495109
400.810.2314752165
411.310.3318122278
424.010.880797078
431.100.2890504974
442.410.5986876601
454.510.92414182
463.110.7502601056
473.610.8320183851
481.700.4255574832
492.210.5498339973
502.610.6456563062
512.600.6456563062
524.210.9002495109
534.400.9168273035
543.100.7502601056
552.910.7109495026
561.600.4013123399
572.110.5249791875
581.600.4013123399
592.500.6224593312
604.310.908877039
611.610.4013123399
623.610.8320183851
632.210.5498339973
643.300.785834983
651.800.4501660027
664.110.8909031788
674.700.9370266439
684.710.9370266439
690.800.2314752165
701.800.4501660027
714.710.9370266439
721.600.4013123399
733.110.7502601056
740.100.1301084744
751.500.3775406688
764.910.9478464369
770.500.1824255238
781.100.2890504974
791.710.4255574832
801.600.4013123399
813.700.8455347349
824.410.9168273035
830.410.1679816149
840.200.1418510649
853.010.7310585786
860.910.2497398944
874.610.9308615797
881.600.4013123399
891.200.3100255189
903.010.7310585786
911.400.3543436938
922.410.5986876601
933.310.785834983
942.510.6224593312
953.800.8581489351
960.300.1544652651
971.400.3543436938
980.200.1418510649
993.300.785834983
1002.810.6899744811
1010.810.2314752165
1024.710.9370266439
1034.310.908877039
1041.710.4255574832
1052.910.7109495026
1061.210.3100255189
1073.810.8581489351
1082.810.6899744811
1092.110.5249791875
1100.910.2497398944
1113.010.7310585786
1122.910.7109495026
1131.800.4501660027
1143.410.8021838886
1152.510.6224593312
1161.110.2890504974
1170.000.119202922
1180.500.1824255238
1191.910.4750208125
1203.110.7502601056
1211.000.2689414214
1223.510.8175744762
1232.400.5986876601
1244.510.92414182
1251.200.3100255189
1260.500.1824255238
1270.800.2314752165
1284.300.908877039
1294.910.9478464369
1301.500.3775406688
1310.100.1301084744
1321.310.3318122278
1331.900.4750208125
1340.100.1301084744
1353.910.8698915256
1360.800.2314752165
1374.810.9426758241
1382.800.6899744811
1390.110.1301084744
1404.310.908877039
1411.700.4255574832
1420.700.214165017
1430.700.214165017
1443.300.785834983
1454.210.9002495109
1462.910.7109495026
1470.200.1418510649
1482.010.5
1494.510.92414182
1500.600.1978161114
1510.800.2314752165
1524.210.9002495109
1531.000.2689414214
1541.810.4501660027
1553.710.8455347349
1563.110.7502601056
1571.100.2890504974
1583.600.8320183851
1593.410.8021838886
1600.110.1301084744
1611.500.3775406688
1621.210.3100255189
1630.100.1301084744
1643.310.785834983
1655.010.9525741268
1663.810.8581489351
1671.810.4501660027
1683.310.785834983
1694.110.8909031788
1701.000.2689414214
1714.310.908877039
1722.810.6899744811
1730.700.214165017
1745.010.9525741268
1754.710.9370266439
1763.610.8320183851
1771.300.3318122278
1783.810.8581489351
1790.700.214165017
1802.010.5
1811.610.4013123399
1820.200.1418510649
1834.210.9002495109
1842.110.5249791875
1850.300.1544652651
1861.300.3318122278
1871.300.3318122278
1884.310.908877039
1890.600.1978161114
1903.110.7502601056
1914.010.880797078
1924.510.92414182
1931.000.2689414214
1944.910.9478464369
1951.600.4013123399
1963.610.8320183851
1970.910.2497398944
1982.510.6224593312
1991.010.2689414214
2001.810.4501660027
2011.110.2890504974
2021.000.2689414214
2030.300.1544652651
2044.810.9426758241
2050.210.1418510649
2061.910.4750208125
2070.600.1978161114
2080.400.1679816149
2092.510.6224593312
2104.910.9478464369
2114.410.9168273035
2124.710.9370266439
2134.210.9002495109
2142.400.5986876601
2151.010.2689414214
2164.410.9168273035
2174.710.9370266439
2180.600.1978161114
2191.200.3100255189
2201.000.2689414214
2215.010.9525741268
2220.200.1418510649
2230.200.1418510649
2242.810.6899744811
2254.710.9370266439
2260.910.2497398944
2273.710.8455347349
2282.200.5498339973
2292.810.6899744811
2301.400.3543436938
2313.710.8455347349
2321.100.2890504974
2330.100.1301084744
2340.000.119202922
2351.900.4750208125
2360.510.1824255238
2370.610.1978161114
2384.510.92414182
2391.310.3318122278
2400.300.1544652651
2412.710.6681877722
2422.600.6456563062
2431.700.4255574832
2443.110.7502601056
2452.000.5
2461.010.2689414214
2474.310.908877039
2484.710.9370266439
2490.100.1301084744
2504.110.8909031788
Sheet3
Sheet2 (2)
-2
1
casexy
10.800.2314752165
22.310.5744425168
32.500.6224593312
42.810.6899744811
53.510.8175744762
64.410.9168273035
70.500.1824255238
84.510.92414182
94.410.9168273035
100.900.2497398944
113.310.785834983
121.100.2890504974
132.510.6224593312
140.310.1544652651
154.510.92414182
161.800.4501660027
172.410.5986876601
181.600.4013123399
191.910.4750208125
204.610.9308615797
212.610.6456563062
223.500.8175744762
230.900.2497398944
242.300.5744425168
252.000.5
263.110.7502601056
272.700.6681877722
281.600.4013123399
294.310.908877039
302.100.5249791875
313.100.7502601056
322.210.5498339973
331.410.3543436938
341.000.2689414214
353.310.785834983
363.210.7685247835
373.610.8320183851
380.300.1544652651
395.000.9525741268
400.410.1679816149
410.100.1301084744
424.210.9002495109
431.510.3775406688
442.100.5249791875
452.310.5744425168
461.000.2689414214
471.000.2689414214
483.310.785834983
491.500.3775406688
502.310.5744425168
514.610.9308615797
522.710.6681877722
533.810.8581489351
543.810.8581489351
550.500.1824255238
564.010.880797078
574.010.880797078
584.710.9370266439
593.410.8021838886
603.610.8320183851
614.410.9168273035
623.510.8175744762
633.400.8021838886
640.710.214165017
652.200.5498339973
662.210.5498339973
672.510.6224593312
684.710.9370266439
690.500.1824255238
700.500.1824255238
712.500.6224593312
721.800.4501660027
730.600.1978161114
741.800.4501660027
751.910.4750208125
762.810.6899744811
772.510.6224593312
782.010.5
790.700.214165017
803.110.7502601056
810.000.119202922
823.910.8698915256
831.810.4501660027
844.910.9478464369
853.010.7310585786
861.700.4255574832
871.100.2890504974
882.000.5
893.110.7502601056
900.700.214165017
910.110.1301084744
921.910.4750208125
932.100.5249791875
940.310.1544652651
952.400.5986876601
962.310.5744425168
973.610.8320183851
984.110.8909031788
993.310.785834983
1003.000.7310585786
1013.410.8021838886
1020.700.214165017
1032.100.5249791875
1044.710.9370266439
1053.610.8320183851
1060.900.2497398944
1073.610.8320183851
1082.610.6456563062
1094.110.8909031788
1104.000.880797078
1112.510.6224593312
1120.810.2314752165
1134.410.9168273035
1143.510.8175744762
1150.500.1824255238
1164.100.8909031788
1174.510.92414182
1184.910.9478464369
1193.510.8175744762
1203.710.8455347349
1210.200.1418510649
1224.710.9370266439
1231.610.4013123399
1242.810.6899744811
1253.510.8175744762
1262.910.7109495026
1271.800.4501660027
1284.510.92414182
1292.300.5744425168
1302.710.6681877722
1312.810.6899744811
1324.310.908877039
1333.310.785834983
1341.810.4501660027
1352.810.6899744811
1362.300.5744425168
1370.000.119202922
1380.400.1679816149
1392.200.5498339973
1402.510.6224593312
1410.500.1824255238
1421.910.4750208125
1431.300.3318122278
1442.110.5249791875
1452.110.5249791875
1463.110.7502601056
1472.510.6224593312
1480.800.2314752165
1490.400.1679816149
1502.800.6899744811
1510.300.1544652651
1524.010.880797078
1532.800.6899744811
1540.800.2314752165
1552.810.6899744811
1560.100.1301084744
1572.700.6681877722
1584.510.92414182
1590.210.1418510649
1602.100.5249791875
1613.610.8320183851
1624.110.8909031788
1631.410.3543436938
1640.900.2497398944
1651.900.4750208125
1662.900.7109495026
1674.010.880797078
1684.910.9478464369
1692.510.6224593312
1702.500.6224593312
1710.200.1418510649
1720.900.2497398944
1731.000.2689414214
1742.400.5986876601
1752.510.6224593312
1764.710.9370266439
1770.600.1978161114
1781.710.4255574832
1791.110.2890504974
1801.310.3318122278
1814.010.880797078
1822.710.6681877722
1832.800.6899744811
1843.810.8581489351
1854.410.9168273035
1863.610.8320183851
1872.710.6681877722
1882.700.6681877722
1892.110.5249791875
1900.410.1679816149
1910.300.1544652651
1923.200.7685247835
1931.000.2689414214
1941.300.3318122278
1954.010.880797078
1960.000.119202922
1971.210.3100255189
1980.400.1679816149
1993.910.8698915256
2003.710.8455347349
2012.210.5498339973
2021.310.3318122278
2030.500.1824255238
2040.500.1824255238
2050.300.1544652651
2061.900.4750208125
2073.510.8175744762
2083.510.8175744762
2094.310.908877039
2100.110.1301084744
2113.010.7310585786
2121.010.2689414214
2130.200.1418510649
2144.510.92414182
2150.500.1824255238
2162.600.6456563062
2173.410.8021838886
2183.510.8175744762
2190.500.1824255238
2204.410.9168273035
2213.410.8021838886
2222.500.6224593312
2233.910.8698915256
2241.800.4501660027
2254.610.9308615797
2262.010.5
2270.400.1679816149
2284.510.92414182
2292.200.5498339973casexy
2304.710.93702664392304.71
2310.300.15446526512310.30
2321.400.35434369382321.40
2334.510.924141822334.51
2341.410.35434369382341.41
2354.510.924141822354.51
2363.900.86989152562363.90
2370.000.1192029222370.00
2384.310.9088770392384.31
2391.000.26894142142391.00
2403.910.86989152562403.91
2411.100.28905049742411.10
2423.410.80218388862423.41
2430.600.19781611142430.60
2441.600.40131233992441.60
2453.900.86989152562453.90
2460.200.14185106492460.20
2472.500.62245933122472.50
2484.110.89090317882484.11
2494.210.90024951092494.21
2504.910.94784643692504.91
Sheet1
b0-2
b11
00.119202922
0.20.1418510649
0.40.1679816149
0.60.1978161114
0.80.2314752165
10.2689414214
1.20.3100255189
1.40.3543436938
1.60.4013123399
1.80.4501660027
20.5
2.20.5498339973
2.40.5986876601
2.60.6456563062
2.80.6899744811
30.7310585786
3.20.7685247835
3.40.8021838886
3.60.8320183851
3.80.8581489351
40.880797078
4.20.9002495109
4.40.9168273035
4.60.9308615797
4.80.9426758241
50.9525741268
5.20.9608342772
5.40.9677045353
5.60.9734030064
5.80.9781187291
60.98201379
6.20.9852259683
6.40.987871565
6.60.9900481981
6.80.9918374288
70.9933071491
7.20.9945137011
7.40.9955037268
7.60.9963157601
7.80.9969815837
80.9975273768
8.20.9979746796
8.40.9983411989
8.60.99864148
8.80.998887464
90.9990889488
9.20.9992539712
9.40.9993891206
9.60.9994997989
9.80.9995904328
100.9996646499
Sheet1
Sheet2
-2
1
casexy
10.200.1418510649
20.400.1679816149
33.510.8175744762
41.910.4750208125
54.210.9002495109
63.210.7685247835
72.210.5498339973
81.800.4501660027
90.300.1544652651
104.310.908877039
113.410.8021838886
121.410.3543436938
134.610.9308615797
141.810.4501660027
153.100.7502601056
163.510.8175744762
171.700.4255574832
180.700.214165017
193.110.7502601056
204.910.9478464369
210.300.1544652651
220.400.1679816149
233.310.785834983
240.200.1418510649
250.700.214165017
260.300.1544652651
274.710.9370266439
281.310.3318122278
293.500.8175744762
304.510.92414182
312.610.6456563062
322.810.6899744811
330.900.2497398944
342.110.5249791875
354.910.9478464369
362.310.5744425168
373.610.8320183851
384.010.880797078
394.410.9168273035
401.210.3100255189
411.900.4750208125
424.610.9308615797
434.710.9370266439
443.110.7502601056
450.710.214165017
464.810.9426758241
474.600.9308615797
480.300.1544652651
490.500.1824255238
501.910.4750208125
511.610.4013123399
524.110.8909031788
535.010.9525741268
540.100.1301084744
552.110.5249791875
564.310.908877039
573.410.8021838886
581.400.3543436938
590.100.1301084744
602.500.6224593312
612.910.7109495026
624.010.880797078
631.400.3543436938
640.800.2314752165
653.810.8581489351
660.000.119202922
673.810.8581489351
680.600.1978161114
691.300.3318122278
701.510.3775406688
711.200.3100255189
723.110.7502601056
734.410.9168273035
741.900.4750208125
753.400.8021838886
761.100.2890504974
773.300.785834983
780.610.1978161114
792.100.5249791875
803.710.8455347349
811.100.2890504974
824.910.9478464369
834.310.908877039
841.510.3775406688
853.110.7502601056
860.710.214165017
871.210.3100255189
882.110.5249791875
892.310.5744425168
903.100.7502601056
914.400.9168273035
922.910.7109495026
934.010.880797078
944.410.9168273035
953.710.8455347349
961.100.2890504974
971.100.2890504974
984.110.8909031788
993.610.8320183851
1001.800.4501660027
1011.010.2689414214
1021.200.3100255189
1030.500.1824255238
1042.100.5249791875
1051.400.3543436938
1061.200.3100255189
1073.500.8175744762
1081.700.4255574832
1093.500.8175744762
1100.300.1544652651
1112.100.5249791875
1120.200.1418510649
1132.010.5
1141.200.3100255189
1151.010.2689414214
1164.310.908877039
1171.900.4750208125
1182.300.5744425168
1195.010.9525741268
1200.100.1301084744
1211.100.2890504974
1224.910.9478464369
1230.110.1301084744
1241.200.3100255189
1250.800.2314752165
1263.410.8021838886
1271.510.3775406688
1284.510.92414182
1292.900.7109495026
1300.200.1418510649
1313.310.785834983
1321.810.4501660027
1330.400.1679816149
1340.010.119202922
1353.100.7502601056
1362.100.5249791875
1375.010.9525741268
1382.410.5986876601
1390.900.2497398944
1400.800.2314752165
1413.610.8320183851
1421.500.3775406688
1434.610.9308615797
1441.400.3543436938
1450.200.1418510649
1462.610.6456563062
1473.210.7685247835
1480.210.1418510649
1492.610.6456563062
1502.200.5498339973
1513.410.8021838886
1524.710.9370266439
1531.800.4501660027
1544.310.908877039
1553.410.8021838886
1562.010.5
1573.910.8698915256
1584.010.880797078
1592.110.5249791875
1601.400.3543436938
1614.510.92414182
1624.210.9002495109
1630.510.1824255238
1642.300.5744425168
1650.300.1544652651
1661.100.2890504974
1672.810.6899744811
1681.600.4013123399
1692.600.6456563062
1701.310.3318122278
1712.510.6224593312
1723.910.8698915256
1732.600.6456563062
1742.300.5744425168
1750.010.119202922
1763.410.8021838886
1772.910.7109495026
1781.300.3318122278
1790.000.119202922
1804.710.9370266439
1810.710.214165017
1821.210.3100255189
1833.110.7502601056
1842.410.5986876601
1851.900.4750208125
1863.210.7685247835
1870.500.1824255238
1883.110.7502601056
1890.300.1544652651
1903.610.8320183851
1913.910.8698915256
1920.210.1418510649
1930.710.214165017
1944.110.8909031788
1954.210.9002495109
1964.010.880797078
1974.800.9426758241
1984.710.9370266439
1990.400.1679816149
2004.910.9478464369
2010.400.1679816149
2022.500.6224593312
2034.510.92414182
2040.400.1679816149
2054.010.880797078
2060.900.2497398944
2070.700.214165017
2080.900.2497398944
2090.300.1544652651
2102.010.5
2114.510.92414182
2121.500.3775406688
2133.510.8175744762
2142.510.6224593312
2150.500.1824255238
2164.210.9002495109
2170.300.1544652651
2184.210.9002495109
2194.310.908877039
2200.000.119202922
2213.400.8021838886
2221.710.4255574832
2230.400.1679816149
2242.810.6899744811
2253.900.8698915256
2264.210.9002495109
2273.610.8320183851
2282.010.5
2294.710.9370266439
2302.410.5986876601
2313.310.785834983
2323.510.8175744762
2330.610.1978161114
2340.700.214165017
2352.910.7109495026
2361.300.3318122278
2372.600.6456563062
2382.500.6224593312
2390.100.1301084744
2403.010.7310585786
2411.300.3318122278
2421.010.2689414214
2430.010.119202922
2442.000.5
2453.810.8581489351
2460.300.1544652651
2474.010.880797078
2480.000.119202922
2494.310.908877039
2504.510.92414182
Sheet3
-
Estimation of the parametersThe parameters are estimated by Maximum Likelihood estimation and require a statistical package such as SPSS
-
Using SPSS to perform Logistic regressionOpen the data file:
-
Choose from the menu: Analyze -> Regression -> Binary Logistic
-
The following dialogue box appearsSelect the dependent variable (y) and the independent variable (x) (covariate).Press OK.
-
Here is the outputThe Estimates and their S.E.
-
The parameter Estimates
Sheet1
bSE
X1.03090.1334
Constant-2.04750.332
Sheet2
Sheet3
Sheet1
bSE
X1.03090.1334
Constant-2.04750.332
1.0309
-2.0475
Sheet2
Sheet3
-
Interpretation of the parameter b0 (determines the intercept)Interpretation of the parameter b1 (determines when p is 0.50 (along with b0))
-
Another interpretation of the parameter b1is the rate of increase in p with respect to x when p = 0.50
-
Nonparametric Statistical Methods
-
DefinitionWhen the data is generated from process (model) that is known except for finite number of unknown parameters the model is called a parametric model.
Otherwise, the model is called a non-parametric modelStatistical techniques that assume a non-parametric model are called non-parametric.
-
Example Parametric modelNormal distribution known except for the two parameters m and s.
sm
Chart1
0.000514093
0.0006405993
0.0007946961
0.000981489
0.0012068121
0.0014772828
0.0018003521
0.002184344
0.0026384839
0.0031729092
0.003798662
0.0045276564
0.0053726194
0.0063469998
0.0074648434
0.0087406297
0.0101890699
0.0118248643
0.0136624187
0.0157155222
0.0179969888
0.0205182671
0.0232890254
0.0263167194
0.0296061537
0.0331590463
0.0369736116
0.041044174
0.0453608275
0.0499091552
0.0546700249
0.0596194722
0.064728685
0.0699640987
0.0752876092
0.0806569082
0.086025942
0.0913454891
0.0965638509
0.1016276423
0.1064826685
0.1110748676
0.1153512977
0.1192611431
0.1227567134
0.1257944092
0.1283356249
0.1303475647
0.131803947
0.1326855754
0.1329807601
0.1326855754
0.131803947
0.1303475647
0.1283356249
0.1257944092
0.1227567134
0.1192611431
0.1153512977
0.1110748676
0.1064826685
0.1016276423
0.0965638509
0.0913454891
0.086025942
0.0806569082
0.0752876092
0.0699640987
0.064728685
0.0596194722
0.0546700249
0.0499091552
0.0453608275
0.041044174
0.0369736116
0.0331590463
0.0296061537
0.0263167194
0.0232890254
0.0205182671
0.0179969888
0.0157155222
0.0136624187
0.0118248643
0.0101890699
0.0087406297
0.0074648434
0.0063469998
0.0053726194
0.0045276564
0.003798662
0.0031729092
0.0026384839
0.002184344
0.0018003521
0.0014772828
0.0012068121
0.000981489
0.0007946961
0.0006405993
0.000514093
Sheet1
00.00051409300.0011349982
0.20.00064059930.00226209350.0012294051
0.40.00079469610.00818730750.0013313929
0.60.0009814890.016668410.001441541
0.80.00120681210.02681280180.0015604701
10.00147728280.03790816620.0016888444
1.20.00180035210.04939304730.0018273746
1.40.0021843440.06083169970.0019768204
1.60.00263848390.07189263430.0021379937
1.80.00317290920.08233035610.0023117614
20.0037986620.09196986030.0024990485
2.20.00452765640.10069350280.0027008417
2.40.00537261940.10842991630.0029181923
2.60.00634699980.11514468260.00315222
2.80.00746484340.12083251230.0034041161
30.00874062970.12551071510.0036751474
3.20.01018906990.12921377150.0039666596
3.40.01182486430.13198884610.0042800811
3.60.01366241870.13389209950.0046169265
3.80.01571552220.13498567890.0049788007
40.01799698880.13533528330.005367402
4.20.02051826710.13500821220.0057845261
4.40.02328902540.13407182160.006232069
4.60.02631671940.13259232080.0067120309
4.80.02960615370.13063385280.0072265187
50.03315904630.12825781040.007777749
5.20.03697361160.12552234720.0083680503
5.40.0410441740.1224820470.0089998653
5.60.04536082750.11918772280.0096757521
5.80.04990915520.11568632020.0103983848
60.05467002490.11202090380.0111705541
6.20.05961947220.10823070880.0119951661
6.40.0647286850.10435124220.0128752407
6.60.06996409870.10041442330.0138139088
6.80.07528760920.09644875020.0148144079
70.08065690820.09247948670.0158800765
7.20.0860259420.08852886070.0170143465
7.40.09134548910.08461626940.0182207342
7.60.09656385090.08075848640.0195028284
7.80.10162764230.07696986830.020864277
80.10648266850.07326255560.0223087696
8.20.11107486760.06964666840.0238400185
8.40.11535129770.06613049380.0254617354
8.60.11926114310.0627206640.0271776044
8.80.12275671340.05942232510.0289912515
90.12579440920.0562392950.030906209
9.20.12833562490.05317421110.0329258751
9.40.13034756470.05022866780.0350534682
9.60.1318039470.0474033430.0372919755
9.80.13268557540.04469811490.0396440945
100.13298076010.04211216870.0421121687
10.20.13268557540.03964409450.0446981149
10.40.1318039470.03729197550.047403343
10.60.13034756470.03505346820.0502286678
10.80.12833562490.03292587510.0531742111
110.12579440920.0309062090.056239295
11.20.12275671340.02899125150.0594223251
11.40.11926114310.02717760440.062720664
11.60.11535129770.02546173540.0661304938
11.80.11107486760.02384001850.0696466684
120.10648266850.02230876960.0732625556
12.20.10162764230.0208642770.0769698683
12.40.09656385090.01950282840.0807584864
12.60.09134548910.01822073420.0846162694
12.80.0860259420.01701434650.0885288607
130.08065690820.01588007650.0924794867
13.20.07528760920.01481440790.0964487502
13.40.06996409870.01381390880.1004144233
13.60.0647286850.01287524070.1043512422
13.80.05961947220.01199516610.1082307088
140.05467002490.01117055410.1120209038
14.20.04990915520.01039838480.1156863202
14.40.04536082750.00967575210.1191877228
14.60.0410441740.00899986530.122482047
14.80.03697361160.00836805030.1255223472
150.03315904630.0077777490.1282578104
15.20.02960615370.00722651870.1306338528
15.40.02631671940.00671203090.1325923208
15.60.02328902540.0062320690.1340718216
15.80.02051826710.00578452610.1350082122
160.01799698880.0053674020.1353352833
16.20.01571552220.00497880070.1349856789
16.40.01366241870.00461692650.1338920995
16.60.01182486430.00428008110.1319888461
16.80.01018906990.00396665960.1292137715
170.00874062970.00367514740.1255107151
17.20.00746484340.00340411610.1208325123
17.40.00634699980.003152220.1151446826
17.60.00537261940.00291819230.1084299163
17.80.00452765640.00270084170.1006935028
180.0037986620.00249904850.0919698603
18.20.00317290920.00231176140.0823303561
18.40.00263848390.00213799370.0718926343
18.60.0021843440.00197682040.0608316997
18.80.00180035210.00182737460.0493930473
190.00147728280.00168884440.0379081662
19.20.00120681210.00156047010.0268128018
19.40.0009814890.0014415410.01666841
19.60.00079469610.00133139290.0081873075
19.80.00064059930.00122940510.0022620935
200.0005140930.00113499820
Sheet1
Sheet2
Sheet3
-
Example Non parametric modelNo assumptions are made about the distribution could be normal, skewed bimodal etc
-
The sign testA nonparametric test for the central location of a distribution
-
We want to test:H0: median = m0HA: median m0against(or against a one-sided alternative)
-
The Sign test:S = the number of observations that exceed m0Comment: If H0: median = m0 is true we would expect 50% of the observations to be above m0, and 50% of the observations to be below m0, The test statistic:
-
If H 0 is true then S will have a binomial distribution with p = 0.50, n = sample size.50%50%median = m0
Chart5
0.0048681898
0.0085315612
0.0144442537
0.0235865744
0.0371080024
0.0562068907
0.0819264245
0.1148785605
0.154937933
0.2009775396
0.2507348748
0.3008888179
0.3473881218
0.3860074367
0.413034492
0.4259377637
0.4238506014
0.4077452195
0.3802491416
0.3451519701
0.3067295224
0.2690497878
0.2522307747
0
0.2522307747
0.2354117617
0.2080138622
0.1878762468
0.1749763288
0.1685172178
0.1672403067
0.1697099359
0.174527753
0.1804645118
0.1865191584
0.1919261269
0.1961332489
0.1987682207
0.1996049704
0.1985352168
0.1955462895
0.1907040486
0.1841390315
0.1760341501
0.1666128375
0.1561271524
0.1448458382
0.1330426447
0.1209853683
0.1089260903
0.097093028
0.0856842962
0.0748637329
0.0647587978
0.0554604173
0.0470245387
0.0394750792
0.0328079074
0.0269954833
0.021991798
0.0177372964
0.0141635189
0.0111972651
0.0087641502
0.0067914846
0.0052104674
0.0039577258
0.0029762662
0.0022159242
0.0016334095
0.0011920441
0.0008612845
0.0006161096
0.0004363413
Bimodal
00.00486818980.0007913203msw
0.10.00853156120.00144608573.510.5
0.20.01444425370.00257235221.50.50.5
0.30.02358657440.0044423655
0.40.03710800240.0074355351
0.50.05620689070.0120500146
0.60.08192642450.0188980728
0.70.11487856050.02867724
0.80.1549379330.0421118672
0.90.20097753960.0598654768
10.25073487480.0824324698
1.10.30088881790.1100264314
1.20.34738812180.142488573
1.30.38600743670.1792408512
1.40.4130344920.2193023351
1.50.42593776370.2613750309
1.60.42385060140.30398809
1.70.40774521950.3456759811
1.80.38024914160.3851561834
1.90.34515197010.4214719778
20.30672952240.4540759845
2.10.26904978780.4828434898
2.150.2522307747
2.150
2.150.2522307747
2.20.23541176170.5080219191
2.30.20801386220.5301352211
2.40.18787624680.549867918
2.50.17497632880.5679525989
2.60.16851721780.5850783464
2.70.16724030670.6018289025
2.80.16970993590.6186511784
2.90.1745277530.6358489371
30.18046451180.6535937827
3.10.18651915840.6719455507
3.20.19192612690.6908758309
3.30.19613324890.7102905836
3.40.19876822070.730049866
3.50.19960497040.7499841569
3.60.19853521680.7699072707
3.70.19554628950.7896271351
3.80.19070404860.8089546214
3.90.18413903150.8277104511
40.17603415010.8457310901
4.10.16661283750.8628734176
4.20.15612715240.8790181941
4.30.14484583820.8940723277
4.40.13304264470.9079699525
4.50.12098536830.9206723696
4.60.10892609030.9321669491
4.70.0970930280.9424651341
4.80.08568429620.9515997252
4.90.07486373290.9596216444
50.06475879780.9665963856
5.10.0554604173
5.20.0470245387
5.30.0394750792
5.40.0328079074
5.50.0269954833
5.60.021991798
5.70.0177372964
5.80.0141635189
5.90.0111972651
60.0087641502
6.10.0067914846
6.20.0052104674
6.30.0039577258
6.40.0029762662
6.50.0022159242
6.60.0016334095
6.70.0011920441
6.80.0008612845
6.90.0006161096
70.0004363413
Bimodal
Positively Skewed
00088
0.050.1572514170.99809254
0.10.150.36528311740.9823673983
0.20.250.54112728590.9458390865
0.30.350.65023111490.8917263579
0.40.450.69724257820.8267032464
0.50.550.69863640890.7569789886
0.60.650.67076831950.6871153477
0.70.750.62631923520.6200385158
0.80.850.57406592220.5574065923
0.90.950.51965391140.5
11.050.46648743270.4480346089
1.11.150.41646425490.4013858656
1.21.250.37051285510.3597394402
1.31.350.32896154620.3226881546
1.41.450.29178271080.289792
1.51.550.25875001440.260613729
1.61.650.22953717480.2347387275
1.71.750.20377840430.21178501
1.81.850.18110416040.1914071696
1.91.950.16116124230.1732967536
22.050.1436231250.1571806293
2.12.150.1281943340.1428183168
2.22.250.11461128630.1299988834
2.32.350.10264112730.1185377548
2.42.450.0920795190.1082736421
2.52.550.0827479610.0990656902
2.62.650.07449099320.0907908941
2.72.750.06717347730.0833417948
2.82.850.06067806390.076624447
2.92.950.05490289330.0705566406
33.050.04975954550.0650663513
3.13.150.04517123310.0600903967
3.23.250.04107122320.0555732734
3.33.350.03740146620.0514661511
3.43.450.03411140970.0477260045
3.53.550.0311569760.0443148635
3.63.650.02849968090.0411991659
3.73.750.02610587680.0383491978
3.83.850.02394610160.0357386102
3.93.950.02199451920.033344
44.050.02022843890.0311445481
4.14.150.01862790130.0291217042
4.24.250.01717532250.0272589141
4.34.350.01585518740.0255413818
4.44.450.01465378530.0239558631
4.54.550.01355898120.0224904845
4.64.650.01256001850.0211345864
4.74.750.01164734810.0198785846
4.84.850.01081248030.0187138498
4.90.0176326017
5
Positively Skewed
Negaitively Skewed
00000.010812480388
10.050.1572514170.01164734810.99809254
20.10.150.36528311740.01256001850.9823673983
30.20.250.54112728590.01355898120.9458390865
40.30.350.65023111490.01465378530.8917263579
50.40.450.69724257820.01585518740.8267032464
60.50.550.69863640890.01717532250.7569789886
70.60.650.67076831950.01862790130.6871153477
80.70.750.62631923520.02022843890.6200385158
90.80.850.57406592220.02199451920.5574065923
100.90.950.51965391140.02394610160.5
1111.050.46648743270.02610587680.4480346089
121.11.150.41646425490.02849968090.4013858656
131.21.250.37051285510.0311569760.3597394402
141.31.350.32896154620.03411140970.3226881546
151.41.450.29178271080.03740146620.289792
161.51.550.25875001440.04107122320.260613729
171.61.650.22953717480.04517123310.2347387275
181.71.750.20377840430.04975954550.21178501
191.81.850.18110416040.05490289330.1914071696
201.91.950.16116124230.06067806390.1732967536
2122.050.1436231250.06717347730.1571806293
222.12.150.1281943340.07449099320.1428183168
232.22.250.11461128630.0827479610.1299988834
242.32.350.10264112730.0920795190.1185377548
252.42.450.0920795190.10264112730.1082736421
262.52.550.0827479610.11461128630.0990656902
272.62.650.07449099320.1281943340.0907908941
282.72.750.06717347730.1436231250.0833417948
292.82.850.06067806390.16116124230.076624447
302.92.950.05490289330.18110416040.0705566406
3133.050.04975954550.20377840430.0650663513
323.13.150.04517123310.22953717480.0600903967
333.23.250.04107122320.25875001440.0555732734
343.33.350.03740146620.29178271080.0514661511
353.43.450.03411140970.32896154620.0477260045
363.53.550.0311569760.37051285510.0443148635
373.63.650.02849968090.41646425490.0411991659
383.73.750.02610587680.46648743270.0383491978
393.83.850.02394610160.51965391140.0357386102
403.93.950.02199451920.57406592220.033344
4144.050.02022843890.62631923520.0311445481
424.14.150.01862790130.67076831950.0291217042
434.24.250.01717532250.69863640890.0272589141
444.34.350.01585518740.69724257820.0255413818
454.44.450.01465378530.65023111490.0239558631
464.54.550.01355898120.54112728590.0224904845
474.64.650.01256001850.36528311740.0211345864
484.74.750.01164734810.1572514170.0198785846
494.84.850.010812480300.0187138498
4.90.0176326017
5
Negaitively Skewed
0.0108124803
0.0116473481
0.0125600185
0.0135589812
0.0146537853
0.0158551874
0.0171753225
0.0186279013
0.0202284389
0.0219945192
0.0239461016
0.0261058768
0.0284996809
0.031156976
0.0341114097
0.0374014662
0.0410712232
0.0451712331
0.0497595455
0.0549028933
0.0606780639
0.0671734773
0.0744909932
0.082747961
0.092079519
0.1026411273
0.1146112863
0.128194334
0.143623125
0.1611612423
0.1811041604
0.2037784043
0.2295371748
0.2587500144
0.2917827108
0.3289615462
0.3705128551
0.4164642549
0.4664874327
0.5196539114
0.5740659222
0.6263192352
0.6707683195
0.6986364089
0.6972425782
0.6502311149
0.5411272859
0.3652831174
0.157251417
0
Normal
00.0175283005msw
0.10.02239453032.510.5
0.20.02832703772.510.5
0.30.0354745928
0.40.043983596
0.50.0539909665
0.60.0656158148
0.70.0789501583
0.80.0940490774
0.90.1109208347
10.1295175957
1.10.1497274656
1.20.171368592
1.30.194186055
1.40.217852177
1.50.2419707245
1.60.2660852499
1.70.2896915528
1.80.3122539334
1.90.3332246029
20.3520653268
2.10.3682701403
2.20.3813878155
2.30.391042694
2.40.3969525475
2.50.3989422804
2.60.3969525475
2.70.391042694
2.80.3813878155
2.90.3682701403
30.3520653268
3.10.3332246029
3.20.3122539334
3.30.2896915528
3.40.2660852499
3.50.2419707245
3.60.217852177
3.70.194186055
3.80.171368592
3.90.1497274656
40.1295175957
4.10.1109208347
4.20.0940490774
4.30.0789501583
4.40.0656158148
4.50.0539909665
4.60.043983596
4.70.0354745928
4.80.0283270377
4.90.0223945303
50.0175283005
Normal
-
If H 0 is not true then S will still have a binomial distribution. However p will not be equal to 0.50.medianm0pm0 > median p < 0.50
-
medianm0pm0 < median p > 0.50p = the probability that an observation is greater than m0.
-
Summarizing: If H 0 is true then S will have a binomial distribution with p = 0.50, n = sample size.n = 10
Chart3
0.0009765625
0.009765625
0.0439453125
0.1171875
0.205078125
0.24609375
0.205078125
0.1171875
0.0439453125
0.009765625
0.0009765625
Sheet1
00.00097656250.5
10.009765625
20.0439453125
30.1171875
40.205078125
50.24609375
60.205078125
70.1171875
80.0439453125
90.009765625
100.0009765625
Sheet1
Bimodal
00.00486818980.0007913203msw
0.10.00853156120.00144608573.510.5
0.20.01444425370.00257235221.50.50.5
0.30.02358657440.0044423655
0.40.03710800240.0074355351
0.50.05620689070.0120500146
0.60.08192642450.0188980728
0.70.11487856050.02867724
0.70
0.70.1148785605
0.80.1549379330.0421118672
0.90.20097753960.0598654768
10.25073487480.0824324698
1.10.30088881790.1100264314
1.20.34738812180.142488573
1.30.38600743670.1792408512
1.40.4130344920.2193023351
1.50.42593776370.2613750309
1.60.42385060140.30398809
1.70.40774521950.3456759811
1.80.38024914160.3851561834
1.90.34515197010.4214719778
20.30672952240.4540759845
2.10.26904978780.4828434898
2.150.2522307747
2.150
2.150.2522307747
2.20.23541176170.5080219191
2.30.20801386220.5301352211
2.40.18787624680.549867918
2.50.17497632880.5679525989
2.60.16851721780.5850783464
2.70.16724030670.6018289025
2.80.16970993590.6186511784
2.90.1745277530.6358489371
30.18046451180.6535937827
3.10.18651915840.6719455507
3.20.19192612690.6908758309
3.30.19613324890.7102905836
3.40.19876822070.730049866
3.50.19960497040.7499841569
3.60.19853521680.7699072707
3.70.19554628950.7896271351
3.80.19070404860.8089546214
3.90.18413903150.8277104511
40.17603415010.8457310901
4.10.16661283750.8628734176
4.20.15612715240.8790181941
4.30.14484583820.8940723277
4.40.13304264470.9079699525
4.50.12098536830.9206723696
4.60.10892609030.9321669491
4.70.0970930280.9424651341
4.80.08568429620.9515997252
4.90.07486373290.9596216444
50.06475879780.9665963856
5.10.0554604173
5.20.0470245387
5.30.0394750792
5.40.0328079074
5.50.0269954833
5.60.021991798
5.70.0177372964
5.80.0141635189
5.90.0111972651
60.0087641502
6.10.0067914846
6.20.0052104674
6.30.0039577258
6.40.0029762662
6.50.0022159242
6.60.0016334095
6.70.0011920441
6.80.0008612845
6.90.0006161096
70.0004363413
Bimodal
Positively Skewed
00088
0.050.1572514170.99809254
0.10.150.36528311740.9823673983
0.20.250.54112728590.9458390865
0.30.350.65023111490.8917263579
0.40.450.69724257820.8267032464
0.50.550.69863640890.7569789886
0.60.650.67076831950.6871153477
0.70.750.62631923520.6200385158
0.80.850.57406592220.5574065923
0.90.950.51965391140.5
11.050.46648743270.4480346089
1.11.150.41646425490.4013858656
1.21.250.37051285510.3597394402
1.31.350.32896154620.3226881546
1.41.450.29178271080.289792
1.51.550.25875001440.260613729
1.61.650.22953717480.2347387275
1.71.750.20377840430.21178501
1.81.850.18110416040.1914071696
1.91.950.16116124230.1732967536
22.050.1436231250.1571806293
2.12.150.1281943340.1428183168
2.22.250.11461128630.1299988834
2.32.350.10264112730.1185377548
2.42.450.0920795190.1082736421
2.52.550.0827479610.0990656902
2.62.650.07449099320.0907908941
2.72.750.06717347730.0833417948
2.82.850.06067806390.076624447
2.92.950.05490289330.0705566406
33.050.04975954550.0650663513
3.13.150.04517123310.0600903967
3.23.250.04107122320.0555732734
3.33.350.03740146620.0514661511
3.43.450.03411140970.0477260045
3.53.550.0311569760.0443148635
3.63.650.02849968090.0411991659
3.73.750.02610587680.0383491978
3.83.850.02394610160.0357386102
3.93.950.02199451920.033344
44.050.02022843890.0311445481
4.14.150.01862790130.0291217042
4.24.250.01717532250.0272589141
4.34.350.01585518740.0255413818
4.44.450.01465378530.0239558631
4.54.550.01355898120.0224904845
4.64.650.01256001850.0211345864
4.74.750.01164734810.0198785846
4.84.850.01081248030.0187138498
4.90.0176326017
5
Positively Skewed
0
0.157251417
0.3652831174
0.5411272859
0.6502311149
0.6972425782
0.6986364089
0.6707683195
0.6263192352
0.5740659222
0.5196539114
0.4664874327
0.4164642549
0.3705128551
0.3289615462
0.2917827108
0.2587500144
0.2295371748
0.2037784043
0.1811041604
0.1611612423
0.143623125
0.128194334
0.1146112863
0.1026411273
0.092079519
0.082747961
0.0744909932
0.0671734773
0.0606780639
0.0549028933
0.0497595455
0.0451712331
0.0410712232
0.0374014662
0.0341114097
0.031156976
0.0284996809
0.0261058768
0.0239461016
0.0219945192
0.0202284389
0.0186279013
0.0171753225
0.0158551874
0.0146537853
0.0135589812
0.0125600185
0.0116473481
0.0108124803
Negaitively Skewed
00000.010812480388
10.050.1572514170.01164734810.99809254
20.10.150.36528311740.01256001850.9823673983
30.20.250.54112728590.01355898120.9458390865
40.30.350.65023111490.01465378530.8917263579
50.40.450.69724257820.01585518740.8267032464
60.50.550.69863640890.01717532250.7569789886
70.60.650.67076831950.01862790130.6871153477
80.70.750.62631923520.02022843890.6200385158
90.80.850.57406592220.02199451920.5574065923
100.90.950.51965391140.02394610160.5
1111.050.46648743270.02610587680.4480346089
121.11.150.41646425490.02849968090.4013858656
131.21.250.37051285510.0311569760.3597394402
141.31.350.32896154620.03411140970.3226881546
151.41.450.29178271080.03740146620.289792
161.51.550.25875001440.04107122320.260613729
171.61.650.22953717480.04517123310.2347387275
181.71.750.20377840430.04975954550.21178501
191.81.850.18110416040.05490289330.1914071696
201.91.950.16116124230.06067806390.1732967536
2122.050.1436231250.06717347730.1571806293
222.12.150.1281943340.07449099320.1428183168
232.22.250.11461128630.0827479610.1299988834
242.32.350.10264112730.0920795190.1185377548
252.42.450.0920795190.10264112730.1082736421
262.52.550.0827479610.11461128630.0990656902
272.62.650.07449099320.1281943340.0907908941
282.72.750.06717347730.1436231250.0833417948
292.82.850.06067806390.16116124230.076624447
302.92.950.05490289330.18110416040.0705566406
3133.050.04975954550.20377840430.0650663513
323.13.150.04517123310.22953717480.0600903967
333.23.250.04107122320.25875001440.0555732734
343.33.350.03740146620.29178271080.0514661511
353.43.450.03411140970.32896154620.0477260045
363.53.550.0311569760.37051285510.0443148635
373.63.650.02849968090.41646425490.0411991659
383.73.750.02610587680.46648743270.0383491978
393.83.850.02394610160.51965391140.0357386102
403.93.950.02199451920.57406592220.033344
4144.050.02022843890.62631923520.0311445481
424.14.150.01862790130.67076831950.0291217042
434.24.250.01717532250.69863640890.0272589141
444.34.350.01585518740.69724257820.0255413818
454.44.450.01465378530.65023111490.0239558631
464.54.550.01355898120.54112728590.0224904845
474.64.650.01256001850.36528311740.0211345864
484.74.750.01164734810.1572514170.0198785846
494.84.850.010812480300.0187138498
4.90.0176326017
5
Negaitively Skewed
0.0108124803
0.0116473481
0.0125600185
0.0135589812
0.0146537853
0.0158551874
0.0171753225
0.0186279013
0.0202284389
0.0219945192
0.0239461016
0.0261058768
0.0284996809
0.031156976
0.0341114097
0.0374014662
0.0410712232
0.0451712331
0.0497595455
0.0549028933
0.0606780639
0.0671734773
0.0744909932
0.082747961
0.092079519
0.1026411273
0.1146112863
0.128194334
0.143623125
0.1611612423
0.1811041604
0.2037784043
0.2295371748
0.2587500144
0.2917827108
0.3289615462
0.3705128551
0.4164642549
0.4664874327
0.5196539114
0.5740659222
0.6263192352
0.6707683195
0.6986364089
0.6972425782
0.6502311149
0.5411272859
0.3652831174
0.157251417
0
Normal
00.0175283005msw
0.10.02239453032.510.5
0.20.02832703772.510.5
0.30.0354745928
0.40.043983596
0.50.0539909665
0.60.0656158148
0.70.0789501583
0.80.0940490774
0.90.1109208347
10.1295175957
1.10.1497274656
1.20.171368592
1.30.194186055
1.40.217852177
1.50.2419707245
1.60.2660852499
1.70.2896915528
1.80.3122539334
1.90.3332246029
20.3520653268
2.10.3682701403
2.20.3813878155
2.30.391042694
2.40.3969525475
2.50.3989422804
2.50
2.50.3989422804
2.60.3969525475
2.70.391042694
2.80.3813878155
2.90.3682701403
30.3520653268
3.10.3332246029
3.20.3122539334
3.30.2896915528
3.40.2660852499
3.50.2419707245
3.60.217852177
3.70.194186055
3.80.171368592
3.90.1497274656
40.1295175957
4.10.1109208347
4.20.0940490774
4.30.0789501583
4.40.0656158148
4.50.0539909665
4.60.043983596
4.70.0354745928
4.80.0283270377
4.90.0223945303
50.0175283005
Normal
0.0175283005
0.0223945303
0.0283270377
0.0354745928
0.043983596
0.0539909665
0.0656158148
0.0789501583
0.0940490774
0.1109208347
0.1295175957
0.1497274656
0.171368592
0.194186055
0.217852177
0.2419707245
0.2660852499
0.2896915528
0.3122539334
0.3332246029
0.3520653268
0.3682701403
0.3813878155
0.391042694
0.3969525475
0.3989422804
0
0.3989422804
0.3969525475
0.391042694
0.3813878155
0.3682701403
0.3520653268
0.3332246029
0.3122539334
0.2896915528
0.2660852499
0.2419707245
0.217852177
0.194186055
0.171368592
0.1497274656
0.1295175957
0.1109208347
0.0940490774
0.0789501583
0.0656158148
0.0539909665
0.043983596
0.0354745928
0.0283270377
0.0223945303
0.0175283005
Sheet1
0.5
xp(x)
00.0010
10.0098
20.0439
30.1172
40.2051
50.2461
60.2051
70.1172
80.0439
90.0098
100.0010
Sheet1
Bimodal
00.00486818980.0007913203msw
0.10.00853156120.00144608573.510.5
0.20.01444425370.00257235221.50.50.5
0.30.02358657440.0044423655
0.40.03710800240.0074355351
0.50.05620689070.0120500146
0.60.08192642450.0188980728
0.70.11487856050.02867724
0.70
0.70.1148785605
0.80.1549379330.0421118672
0.90.20097753960.0598654768
10.25073487480.0824324698
1.10.30088881790.1100264314
1.20.34738812180.142488573
1.30.38600743670.1792408512
1.40.4130344920.2193023351
1.50.42593776370.2613750309
1.60.42385060140.30398809
1.70.40774521950.3456759811
1.80.38024914160.3851561834
1.90.34515197010.4214719778
20.30672952240.4540759845
2.10.26904978780.4828434898
2.150.2522307747
2.150
2.150.2522307747
2.20.23541176170.5080219191
2.30.20801386220.5301352211
2.40.18787624680.549867918
2.50.17497632880.5679525989
2.60.16851721780.5850783464
2.70.16724030670.6018289025
2.80.16970993590.6186511784
2.90.1745277530.6358489371
30.18046451180.6535937827
3.10.18651915840.6719455507
3.20.19192612690.6908758309
3.30.19613324890.7102905836
3.40.19876822070.730049866
3.50.19960497040.7499841569
3.60.19853521680.7699072707
3.70.19554628950.7896271351
3.80.19070404860.8089546214
3.90.18413903150.8277104511
40.17603415010.8457310901
4.10.16661283750.8628734176
4.20.15612715240.8790181941
4.30.14484583820.8940723277
4.40.13304264470.9079699525
4.50.12098536830.9206723696
4.60.10892609030.9321669491
4.70.0970930280.9424651341
4.80.08568429620.9515997252
4.90.07486373290.9596216444
50.06475879780.9665963856
5.10.0554604173
5.20.0470245387
5.30.0394750792
5.40.0328079074
5.50.0269954833
5.60.021991798
5.70.0177372964
5.80.0141635189
5.90.0111972651
60.0087641502
6.10.0067914846
6.20.0052104674
6.30.0039577258
6.40.0029762662
6.50.0022159242
6.60.0016334095
6.70.0011920441
6.80.0008612845
6.90.0006161096
70.0004363413
Bimodal
Positively Skewed
00088
0.050.1572514170.99809254
0.10.150.36528311740.9823673983
0.20.250.54112728590.9458390865
0.30.350.65023111490.8917263579
0.40.450.69724257820.8267032464
0.50.550.69863640890.7569789886
0.60.650.67076831950.6871153477
0.70.750.62631923520.6200385158
0.80.850.57406592220.5574065923
0.90.950.51965391140.5
11.050.46648743270.4480346089
1.11.150.41646425490.4013858656
1.21.250.37051285510.3597394402
1.31.350.32896154620.3226881546
1.41.450.29178271080.289792
1.51.550.25875001440.260613729
1.61.650.22953717480.2347387275
1.71.750.20377840430.21178501
1.81.850.18110416040.1914071696
1.91.950.16116124230.1732967536
22.050.1436231250.1571806293
2.12.150.1281943340.1428183168
2.22.250.11461128630.1299988834
2.32.350.10264112730.1185377548
2.42.450.0920795190.1082736421
2.52.550.0827479610.0990656902
2.62.650.07449099320.0907908941
2.72.750.06717347730.0833417948
2.82.850.06067806390.076624447
2.92.950.05490289330.0705566406
33.050.04975954550.0650663513
3.13.150.04517123310.0600903967
3.23.250.04107122320.0555732734
3.33.350.03740146620.0514661511
3.43.450.03411140970.0477260045
3.53.550.0311569760.0443148635
3.63.650.02849968090.0411991659
3.73.750.02610587680.0383491978
3.83.850.02394610160.0357386102
3.93.950.02199451920.033344
44.050.02022843890.0311445481
4.14.150.01862790130.0291217042
4.24.250.01717532250.0272589141
4.34.350.01585518740.0255413818
4.44.450.01465378530.0239558631
4.54.550.01355898120.0224904845
4.64.650.01256001850.0211345864
4.74.750.01164734810.0198785846
4.84.850.01081248030.0187138498
4.90.0176326017
5
Positively Skewed
0
0.157251417
0.3652831174
0.5411272859
0.6502311149
0.6972425782
0.6986364089
0.6707683195
0.6263192352
0.5740659222
0.5196539114
0.4664874327
0.4164642549
0.3705128551
0.3289615462
0.2917827108
0.2587500144
0.2295371748
0.2037784043
0.1811041604
0.1611612423
0.143623125
0.128194334
0.1146112863
0.1026411273
0.092079519
0.082747961
0.0744909932
0.0671734773
0.0606780639
0.0549028933
0.0497595455
0.0451712331
0.0410712232
0.0374014662
0.0341114097
0.031156976
0.0284996809
0.0261058768
0.0239461016
0.0219945192
0.0202284389
0.0186279013
0.0171753225
0.0158551874
0.0146537853
0.0135589812
0.0125600185
0.0116473481
0.0108124803
Negaitively Skewed
00000.010812480388
10.050.1572514170.01164734810.99809254
20.10.150.36528311740.01256001850.9823673983
30.20.250.54112728590.01355898120.9458390865
40.30.350.65023111490.01465378530.8917263579
50.40.450.69724257820.01585518740.8267032464
60.50.550.69863640890.01717532250.7569789886
70.60.650.67076831950.01862790130.6871153477
80.70.750.62631923520.02022843890.6200385158
90.80.850.57406592220.02199451920.5574065923
100.90.950.51965391140.02394610160.5
1111.050.46648743270.02610587680.4480346089
121.11.150.41646425490.02849968090.4013858656
131.21.250.37051285510.0311569760.3597394402
141.31.350.32896154620.03411140970.3226881546
151.41.450.29178271080.03740146620.289792
161.51.550.25875001440.04107122320.260613729
171.61.650.22953717480.04517123310.2347387275
181.71.750.20377840430.04975954550.21178501
191.81.850.18110416040.05490289330.1914071696
201.91.950.16116124230.06067806390.1732967536
2122.050.1436231250.06717347730.1571806293
222.12.150.1281943340.07449099320.1428183168
232.22.250.11461128630.0827479610.1299988834
242.32.350.10264112730.0920795190.1185377548
252.42.450.0920795190.10264112730.1082736421
262.52.550.0827479610.11461128630.0990656902
272.62.650.07449099320.1281943340.0907908941
282.72.750.06717347730.1436231250.0833417948
292.82.850.06067806390.16116124230.076624447
302.92.950.05490289330.18110416040.0705566406
3133.050.04975954550.20377840430.0650663513
323.13.150.04517123310.22953717480.0600903967
333.23.250.04107122320.25875001440.0555732734
343.33.350.03740146620.29178271080.0514661511
353.43.450.03411140970.32896154620.0477260045
363.53.550.0311569760.37051285510.0443148635
373.63.650.02849968090.41646425490.0411991659
383.73.750.02610587680.46648743270.0383491978
393.83.850.02394610160.51965391140.0357386102
403.93.950.02199451920.57406592220.033344
4144.050.02022843890.62631923520.0311445481
424.14.150.01862790130.67076831950.0291217042
434.24.250.01717532250.69863640890.0272589141
444.34.350.01585518740.69724257820.0255413818
454.44.450.01465378530.65023111490.0239558631
464.54.550.01355898120.54112728590.0224904845
474.64.650.01256001850.36528311740.0211345864
484.74.750.01164734810.1572514170.0198785846
494.84.850.010812480300.0187138498
4.90.0176326017
5
Negaitively Skewed
0.0108124803
0.0116473481
0.0125600185
0.0135589812
0.0146537853
0.0158551874
0.0171753225
0.0186279013
0.0202284389
0.0219945192
0.0239461016
0.0261058768
0.0284996809
0.031156976
0.0341114097
0.0374014662
0.0410712232
0.0451712331
0.0497595455
0.0549028933
0.0606780639
0.0671734773
0.0744909932
0.082747961
0.092079519
0.1026411273
0.1146112863
0.128194334
0.143623125
0.1611612423
0.1811041604
0.2037784043
0.2295371748
0.2587500144
0.2917827108
0.3289615462
0.3705128551
0.4164642549
0.4664874327
0.5196539114
0.5740659222
0.6263192352
0.6707683195
0.6986364089
0.6972425782
0.6502311149
0.5411272859
0.3652831174
0.157251417
0
Normal
00.0175283005msw
0.10.02239453032.510.5
0.20.02832703772.510.5
0.30.0354745928
0.40.043983596
0.50.0539909665
0.60.0656158148
0.70.0789501583
0.80.0940490774
0.90.1109208347
10.1295175957
1.10.1497274656
1.20.171368592
1.30.194186055
1.40.217852177
1.50.2419707245
1.60.2660852499
1.70.2896915528
1.80.3122539334
1.90.3332246029
20.3520653268
2.10.3682701403
2.20.3813878155
2.30.391042694
2.40.3969525475
2.50.3989422804
2.50
2.50.3989422804
2.60.3969525475
2.70.391042694
2.80.3813878155
2.90.3682701403
30.3520653268
3.10.3332246029
3.20.3122539334
3.30.2896915528
3.40.2660852499
3.50.2419707245
3.60.217852177
3.70.194186055
3.80.171368592
3.90.1497274656
40.1295175957
4.10.1109208347
4.20.0940490774
4.30.0789501583
4.40.0656158148
4.50.0539909665
4.60.043983596
4.70.0354745928
4.80.0283270377
4.90.0223945303
50.0175283005
Normal
0.0175283005
0.0223945303
0.0283270377
0.0354745928
0.043983596
0.0539909665
0.0656158148
0.0789501583
0.0940490774
0.1109208347
0.1295175957
0.1497274656
0.171368592
0.194186055
0.217852177
0.2419707245
0.2660852499
0.2896915528
0.3122539334
0.3332246029
0.3520653268
0.3682701403
0.3813878155
0.391042694
0.3969525475
0.3989422804
0
0.3989422804
0.3969525475
0.391042694
0.3813878155
0.3682701403
0.3520653268
0.3332246029
0.3122539334
0.2896915528
0.2660852499
0.2419707245
0.217852177
0.194186055
0.171368592
0.1497274656
0.1295175957
0.1109208347
0.0940490774
0.0789501583
0.0656158148
0.0539909665
0.043983596
0.0354745928
0.0283270377
0.0223945303
0.0175283005
-
The critical and acceptance region:n = 10Choose the critical region so that a is close to 0.05 or 0.01.e. g. If critical region is {0,1,9,10} then a = .0010 + .0098 + .0098 +.0010 = .0216
Sheet1
0.5
xp(x)
00.0010
10.0098
20.0439
30.1172
40.2051
50.2461
60.2051
70.1172
80.0439
90.0098
100.0010
Sheet1
Bimodal
00.00486818980.0007913203msw
0.10.00853156120.00144608573.510.5
0.20.01444425370.00257235221.50.50.5
0.30.02358657440.0044423655
0.40.03710800240.0074355351
0.50.05620689070.0120500146
0.60.08192642450.0188980728
0.70.11487856050.02867724
0.70
0.70.1148785605
0.80.1549379330.0421118672
0.90.20097753960.0598654768
10.25073487480.0824324698
1.10.30088881790.1100264314
1.20.34738812180.142488573
1.30.38600743670.1792408512
1.40.4130344920.2193023351
1.50.42593776370.2613750309
1.60.42385060140.30398809
1.70.40774521950.3456759811
1.80.38024914160.3851561834
1.90.34515197010.4214719778
20.30672952240.4540759845
2.10.26904978780.4828434898
2.150.2522307747
2.150
2.150.2522307747
2.20.23541176170.5080219191
2.30.20801386220.5301352211
2.40.18787624680.549867918
2.50.17497632880.5679525989
2.60.16851721780.5850783464
2.70.16724030670.6018289025
2.80.16970993590.6186511784
2.90.1745277530.6358489371
30.18046451180.6535937827
3.10.18651915840.6719455507
3.20.19192612690.6908758309
3.30.19613324890.7102905836
3.40.19876822070.730049866
3.50.19960497040.7499841569
3.60.19853521680.7699072707
3.70.19554628950.7896271351
3.80.19070404860.8089546214
3.90.18413903150.8277104511
40.17603415010.8457310901
4.10.16661283750.8628734176
4.20.15612715240.8790181941
4.30.14484583820.8940723277
4.40.13304264470.9079699525
4.50.12098536830.9206723696
4.60.10892609030.9321669491
4.70.0970930280.9424651341
4.80.08568429620.9515997252
4.90.07486373290.9596216444
50.06475879780.9665963856
5.10.0554604173
5.20.0470245387
5.30.0394750792
5.40.0328079074
5.50.0269954833
5.60.021991798
5.70.0177372964
5.80.0141635189
5.90.0111972651
60.0087641502
6.10.0067914846
6.20.0052104674
6.30.0039577258
6.40.0029762662
6.50.0022159242
6.60.0016334095
6.70.0011920441
6.80.0008612845
6.90.0006161096
70.0004363413
Bimodal
Positively Skewed
00088
0.050.1572514170.99809254
0.10.150.36528311740.9823673983
0.20.250.54112728590.9458390865
0.30.350.65023111490.8917263579
0.40.450.69724257820.8267032464
0.50.550.69863640890.7569789886
0.60.650.67076831950.6871153477
0.70.750.62631923520.6200385158
0.80.850.57406592220.5574065923
0.90.950.51965391140.5
11.050.46648743270.4480346089
1.11.150.41646425490.4013858656
1.21.250.37051285510.3597394402
1.31.350.32896154620.3226881546
1.41.450.29178271080.289792
1.51.550.25875001440.260613729
1.61.650.22953717480.2347387275
1.71.750.20377840430.21178501
1.81.850.18110416040.1914071696
1.91.950.16116124230.1732967536
22.050.1436231250.1571806293
2.12.150.1281943340.1428183168
2.22.250.11461128630.1299988834
2.32.350.10264112730.1185377548
2.42.450.0920795190.1082736421
2.52.550.0827479610.0990656902
2.62.650.07449099320.0907908941
2.72.750.06717347730.0833417948
2.82.850.06067806390.076624447
2.92.950.05490289330.0705566406
33.050.04975954550.0650663513
3.13.150.04517123310.0600903967
3.23.250.04107122320.0555732734
3.33.350.03740146620.0514661511
3.43.450.03411140970.0477260045
3.53.550.0311569760.0443148635
3.63.650.02849968090.0411991659
3.73.750.02610587680.0383491978
3.83.850.02394610160.0357386102
3.93.950.02199451920.033344
44.050.02022843890.0311445481
4.14.150.01862790130.0291217042
4.24.250.01717532250.0272589141
4.34.350.01585518740.0255413818
4.44.450.01465378530.0239558631
4.54.550.01355898120.0224904845
4.64.650.01256001850.0211345864
4.74.750.01164734810.0198785846
4.84.850.01081248030.0187138498
4.90.0176326017
5
Positively Skewed
0
0.157251417
0.3652831174
0.5411272859
0.6502311149
0.6972425782
0.6986364089
0.6707683195
0.6263192352
0.5740659222
0.5196539114
0.4664874327
0.4164642549
0.3705128551
0.3289615462
0.2917827108
0.2587500144
0.2295371748
0.2037784043
0.1811041604
0.1611612423
0.143623125
0.128194334
0.1146112863
0.1026411273
0.092079519
0.082747961
0.0744909932
0.0671734773
0.0606780639
0.0549028933
0.0497595455
0.0451712331
0.0410712232
0.0374014662
0.0341114097
0.031156976
0.0284996809
0.0261058768
0.0239461016
0.0219945192
0.0202284389
0.0186279013
0.0171753225
0.0158551874
0.0146537853
0.0135589812
0.0125600185
0.0116473481
0.0108124803
Negaitively Skewed
00000.010812480388
10.050.1572514170.01164734810.99809254
20.10.150.36528311740.01256001850.9823673983
30.20.250.54112728590.01355898120.9458390865
40.30.350.65023111490.01465378530.8917263579
50.40.450.69724257820.01585518740.8267032464
60.50.550.69863640890.01717532250.7569789886
70.60.650.67076831950.01862790130.6871153477
80.70.750.62631923520.02022843890.6200385158
90.80.850.57406592220.02199451920.5574065923
100.90.950.51965391140.02394610160.5
1111.050.46648743270.02610587680.4480346089
121.11.150.41646425490.02849968090.4013858656
131.21.250.37051285510.0311569760.3597394402
141.31.350.32896154620.03411140970.3226881546
151.41.450.29178271080.03740146620.289792
161.51.550.25875001440.04107122320.260613729
171.61.650.22953717480.04517123310.2347387275
181.71.750.20377840430.04975954550.21178501
191.81.850.18110416040.05490289330.1914071696
201.91.950.16116124230.06067806390.1732967536
2122.050.1436231250.06717347730.1571806293
222.12.150.1281943340.07449099320.1428183168
232.22.250.11461128630.0827479610.1299988834
242.32.350.10264112730.0920795190.1185377548
252.42.450.0920795190.10264112730.1082736421
262.52.550.0827479610.11461128630.0990656902
272.62.650.07449099320.1281943340.0907908941
282.72.750.06717347730.1436231250.0833417948
292.82.850.06067806390.16116124230.076624447
302.92.950.05490289330.18110416040.0705566406
3133.050.04975954550.20377840430.0650663513
323.13.150.04517123310.22953717480.0600903967
333.23.250.04107122320.25875001440.0555732734
343.33.350.03740146620.29178271080.0514661511
353.43.450.03411140970.32896154620.0477260045
363.53.550.0311569760.37051285510.0443148635
373.63.650.02849968090.41646425490.0411991659
383.73.750.02610587680.46648743270.0383491978
393.83.850.02394610160.51965391140.0357386102
403.93.950.02199451920.57406592220.033344
4144.050.02022843890.62631923520.0311445481
424.14.150.01862790130.67076831950.0291217042
434.24.250.01717532250.69863640890.0272589141
444.34.350.01585518740.69724257820.0255413818
454.44.450.01465378530.65023111490.0239558631
464.54.550.01355898120.54112728590.0224904845
474.64.650.01256001850.36528311740.0211345864
484.74.750.01164734810.1572514170.0198785846
494.84.850.010812480300.0187138498
4.90.0176326017
5
Negaitively Skewed
0.0108124803
0.0116473481
0.0125600185
0.0135589812
0.0146537853
0.0158551874
0.0171753225
0.0186279013
0.0202284389
0.0219945192
0.0239461016
0.0261058768
0.0284996809
0.031156976
0.0341114097
0.0374014662
0.0410712232
0.0451712331
0.0497595455
0.0549028933
0.0606780639
0.0671734773
0.0744909932
0.082747961
0.092079519
0.1026411273
0.1146112863
0.128194334
0.143623125
0.1611612423
0.1811041604
0.2037784043
0.2295371748
0.2587500144
0.2917827108
0.3289615462
0.3705128551
0.4164642549
0.4664874327
0.5196539114
0.5740659222
0.6263192352
0.6707683195
0.6986364089
0.6972425782
0.6502311149
0.5411272859
0.3652831174
0.157251417
0
Normal
00.0175283005msw
0.10.02239453032.510.5
0.20.02832703772.510.5
0.30.0354745928
0.40.043983596
0.50.0539909665
0.60.0656158148
0.70.0789501583
0.80.0940490774
0.90.1109208347
10.1295175957
1.10.1497274656
1.20.171368592
1.30.194186055
1.40.217852177
1.50.2419707245
1.60.2660852499
1.70.2896915528
1.80.3122539334
1.90.3332246029
20.3520653268
2.10.3682701403
2.20.3813878155
2.30.391042694
2.40.3969525475
2.50.3989422804
2.50
2.50.3989422804
2.60.3969525475
2.70.391042694
2.80.3813878155
2.90.3682701403
30.3520653268
3.10.3332246029
3.20.3122539334
3.30.2896915528
3.40.2660852499
3.50.2419707245
3.60.217852177
3.70.194186055
3.80.171368592
3.90.1497274656
40.1295175957
4.10.1109208347
4.20.0940490774
4.30.0789501583
4.40.0656158148
4.50.0539909665
4.60.043983596
4.70.0354745928
4.80.0283270377
4.90.0223945303
50.0175283005
Normal
0.0175283005
0.0223945303
0.0283270377
0.0354745928
0.043983596
0.0539909665
0.0656158148
0.0789501583
0.0940490774
0.1109208347
0.1295175957
0.1497274656
0.171368592
0.194186055
0.217852177
0.2419707245
0.2660852499
0.2896915528
0.3122539334
0.3332246029
0.3520653268
0.3682701403
0.3813878155
0.391042694
0.3969525475
0.3989422804
0
0.3989422804
0.3969525475
0.391042694
0.3813878155
0.3682701403
0.3520653268
0.3332246029
0.3122539334
0.2896915528
0.2660852499
0.2419707245
0.217852177
0.194186055
0.171368592
0.1497274656
0.1295175957
0.1109208347
0.0940490774
0.0789501583
0.0656158148
0.0539909665
0.043983596
0.0354745928
0.0283270377
0.0223945303
0.0175283005
Chart4
0.0009765625
0.009765625
0.0439453125
0.1171875
0.205078125
0.24609375
0.205078125
0.1171875
0.0439453125
0.009765625
0.0009765625
Sheet1
0.5
xp(x)
00.0010
10.0098
20.0439
30.1172
40.2051
50.2461
60.2051
70.1172
80.0439
90.0098
100.0010
Sheet1
Bimodal
00.00486818980.0007913203msw
0.10.00853156120.00144608573.510.5
0.20.01444425370.00257235221.50.50.5
0.30.02358657440.0044423655
0.40.03710800240.0074355351
0.50.05620689070.0120500146
0.60.08192642450.0188980728
0.70.11487856050.02867724
0.70
0.70.1148785605
0.80.1549379330.042