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