Appunti Statistica

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Quantili della distribuzione t-Student

0,85 0,900 0,925 0,95 0,975 0,99 0,995 0,9975 0,999

1 1,376382 1,962612 3,077685 4,165304 6,313749 12,70615 31 63,6559

,82096 318,2888

127,3211

ι

2 1 ,386206

,06066 1,885619 2,919987

2,281931 4,302656 6,964547 9,924988 14,08916 22,32846

ι

3 0,978472 ,249778 1

,637745 1,924318 2,353363 3,182449 4,540707 5,840848 7,4532 10,21428

4 0,940964 1,189567 1

,533206 1,778192 2,131846 4,60408 5,59754

2,776451 7,17293

3,746936

5 0,919543 1,155768 1 ,475885 2,015049

1 ,699364 2,570578 4,032117 4,773319

3,36493 5,893526

6 0,905703 1,134157 1,650174

1 ,439755 4,316826 5,207548

2,446914 3,142668 3,707428

1,943181

7 0,89603 1,119159 1,414924 4,029353 4,785252

1,894578

1,61659 2,364623 2,997949 3,499481

8 0,88889 ι

1,108145 1,396816 4,500762

,859548 3,832538

1 ,592221 2,306006 2,896468 3,355381

9 0,883404 1,099716 4,29689

1,573735

1,383029 1,833114 3,689638

3,249843

2,262159 2,821434

10 0,879057 ι

1 ,093058 4,143658

1,372184 ,559235 3,581372

1,812462 2,228139 2,763772 3,169262

ι

11 0,87553 ,087667 4,024769

1 ,36343 3,496607

1,795884

1,54756 2,200986 3,105815

2,718079

12 0,872609 ι

1,083212 3,929599

,537956

1,356218 1,782287 2,178813 3,054538 3,428431

2,68099

ι

13 0,870151 ,079469 3,852037

1,350172 1 3,372479

,52992 2,160368 3,012283

1,770932 2,650304

14 0,868055 1,07628 3,787427

2,144789 3,325695

1,345031 1,523094 2,624492 2,976849

1,761309

15 0,866245 1,073531 3,732857

1,340605 1,517228 3,286041

2,602483 2,946726

1,753051 2,131451

16 0,864667 1,071137 3,686146

1,336757 1,745884 3,251989

1,51213 2,119905 2,920788

2,583492

17 0,863279 1

,069034 3,645764

1

,333379 3,222449

1,50766 2,109819 2,898232

1,739606 2,56694

18 ι ι

0,862049 1,067169 ,330391 3,610476

,503709 3,196583

2,100924 2,878442

1,734063 2,552379

19 0,86095 1

,065507 3,579335

1

,327728 3,1737

1,500189 2,093025 2,539482 2,860943

1,729131

20 0,859965 ι

1

,064016 ,497035 3,551831

3,1534

1 ,325341 2,085962 2,527977 2,845336

1,724718

21 0,859075 1 ,06267 3,527093

1,323187 1,720744 2,079614

1,494193 2,517645 2,831366 3,13521

22 0,858266 ,Ο61449

1 1,321237 3,504974

1,717144 2,073875 3,118839

1,491619 2,508323 2,818761

23 0,85753 1 ,060337 3,484965

1,319461 1 2,068655

,489277 2,807337 3,103996

1,71387 2,499874

24 0,856855 1,059319 1,317835 3,466776

1 ,487135 2,063898

1,710882 3,090536

2,796951

2,492161

25 0,856236 1

,058385 1,316346 2,059537 3,450186

1,485171 1,70814 2,787438 3,078203

2,485103

26 0,855665 1

1,057523 1,705616

1,314972 ,483363 3,43498

2,778725 3,066889

2,055531 2,478628

ι

27 0,855138 ,056727 1,313704 1,481692 2,051829

1,703288 2,770685

2,472661 3,056521 3,42101

28 0,854648 1 ,055989 1,480144

1,312526 2,048409

1,70113 2,763263 3,046953 3,408204

2,467141

ι

29 0,854192 1,055303 ,478704

1,311435 1,699127 2,756387 3,03804 3,396271

2,045231 2,46202

ι

30 0,853768 ,054663 1,310416 1,477365 1,69726 2,457264 2,749985

2,04227 3,029782 3,385212

31 0,85337 1,054065 1,476112

1

,309463 2,039515

1,695519 2,452825 2,744036 3,022105 3,37488

Ι

ι ι

32 0,852998 ,474941

,053504 ,308573 1,693888 2,036932 2,738489

2,448678 3,014939 3,365276

ι ι

33 0,852649 ,052979 ,307737 2,444795

2,034517

1

,473843 1,69236 2,733286 3,008245 3,356326

34 0,852322 1,052485 2,032243 2,441147

1,306951 1 ,472811 1,690923 2,728393 3,001951 3,347959

35 0,852012 1,052019 1,306212 1,471838 1 ,689573 2,437719 2,723809

2,03011 2,996057 3,340028

Ι ι

36 0,85172 1,305514 ,688297

,470921

1,051581 2,434499

2,028091 2,71948 2,990491 3,332607

ι

0,851444

37 1,051164 1

,304854 1,470055 ,687094 2,02619 2,431443 2,715406 2,985253 3,325622

38 0,851182 1,050772 1

,469234 2,024394

1

,30423 1 2,428569

,685953 2,711568 2,980305 3,319001

ι ι

ι

39 0,850935 ,050399 ,468459

,303638 1

,684875 2,022689 2,425841 2,975612 3,312744

2,707911

ι

40 0,850699 ,050046 1,303076 1 ,46772 1 2,021075

,683852 2,423258 2,704455 2,971174 3,306923

ι ,047295

50 0,848869 1,461995 1,675905 2,00856

1,298713 2,403267 2,677789 2,936977 3,261375

0,847652

60 1

,045469 1,670649

1,295821 1 ,458202 2,000297 2,390116 2,660272 2,914567 3,231689

70 0,846786 1,455505

1,044168 1,666915 1

,994435

1,293763 2,380802 2,647903 2,898742 3,210807

ι ι

80 0,846137 ,453488

1,292224 1,664125 ,990065

1,043195 2,373872 2,638699 2,886954 3,195237

0,845633

90 1

,04244 1,291029 1,451924 1,986673

1,661961 2,368497 2,631568 2,877896 3,183231

100 0,845231 1,290075 1

,450676

1,041835 1

,660235 1

,983972 2,364213 2,625893 2,870656 3,173773

ι

0,844901

110 1

,289295 1

,449655 ,658823

1

,041342 1,981766 2,621273

2,360721 2,864763 3,165987

Ι ι

ι 448805

120 0,844627 ,040931 1,288646 ,65765 1,979929 2,357829 2,617417 2,859852 3,159512

Tavola della funzione di ripartizione della v.a. Normale Standard

x

1 -t /

e 2 dt

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.00

0.0 0.5120 0.5160 0.5199 0.5239 0.3319

().3279

0.1 ().3398 ().3438 ().3478 ().3517 ().5537 0.5596 ().3636 0.3714

().3675 ().3753

0.2 0.7>703 ().3832 ().3871 ().5910 0.5987

().5948 ().6026 0.6103

().6064 ().6141

0.3 0.61 0.6217

79 0.6293

().6255 443

().6331 ().6368 ().6406 ().6480 ().6517

().6391 0.6664

().6628 ().6844

0.6772

().6736 ().6870

().6808

0.5 ().6913 ().6085 0.7088 0.7224

0.7123 0.7157 ().7100

0.6 ().7237 ().7291 ().7324 0.7357 ().7540

().7454

0.7389 0.7422 o. 7486 ().7317

0.7 ().7380 0.7611 0.7673

().7642 7764

0.7734

().77()4 0.7823

0.7794

O.

0.8 0.7881 0,7910 0.7967

().7939 0.8133

0.8106

0.8023 0.8078

().7993 ().8031

0.9 0.81 0.8186

39 0.8212 0.8389

0.8264

().8238 ().8365

().8340

().8289 ().8313

0.8413 ().8438 ().8309

().8461 ().8485 ().8577

().8531 ().8621

().8534

1.1 ().8643 ().8665 ().8686 0.8729 0.8770

().8740 ().8790

().8840

1.2 ().8860 0.8888 ().8944 0.8997

().8007 0.8962

1.3 0.9177

().9082 ().0162

0.91 15 ().9147

().9131

1.4 0.9192 ().9207 0.9222 0.9319

().9236 ().9251 ().9263 ().9279 ().9292

1.5 ().9332 ().9345 ().9357 ().9441

0.9406 0.9418

0.93 70 ().9382 ().9429

().9394

1.6 ().9432 ().9545

().9474 ().9484

().9463 ().9493 ().9335

().9313 ().9325

1.7 ().9354 ().0590

().9364 ().0373 ().9633

().9582 ().0625

().9591 ().9616

().9608

1.8 ().9641 ().9640 0.9636 0.0664 0.9678 0.9693

().0671 ().9690

().9686

1.0 0.9713 ().0710 ().0726 0.9732 ().0767

0.9738 0.9744 ().9761

2.0 ().9772 0.9778 00783 0.9788 0.9817

0.0798 0.9812

().0793 ().9803

2.1 ().9821 ().9826 0.9857

0.9838

().9834 0.9842 ().9854

().9846 ().9830

2.2 0.9861 ().9864 0.9875

().9868 0.9884 ().9887

().9871 ().9878 0.9881

2.3 ().9893 0.989G ().9898 0.9916

0.9901 0.9913

().9906 ().9909 ().991 1

2.4 0.9918 ().0934

().9032

().0922 ().9936

().9925 ().9927 ().9929 ().9031

2.5 ().9943

().0041 0.9948

().0945 ().9052

().9046 ().9040 ().0051

995 R

(i

2.6 ().9035 ().0056 ().9059 ().9064

().9957 0.9961 ().9963

().0062

2.7 ().9074

0.99G6 ().0067 ().0068 ().0960 ().9971 ().0973

().9072

2.8 ().9974 ().9975 0.9977 0.9977

0.997G 0.9979

().9978 ().9979 ().9981

2.9 ().9981 ().9982 0.9984

().0982 ().9983 0.9986

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().9984 ().9983 ().9985

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3.0 ().9089

00988

().9987 ().9087 ().0087 ().9988 ().9989

3.1 ().9091 0.00!) 1 ().9991 ().9992 ().9992 ().9902 ().0093

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3.2 ().0093 ().9003 ().0004 00095

().9094 ().9904

().0904 ().9904 ().9095

3.3 0.00')3 ().9996

().0095 ().0006 ().9996 ().0906 ().9007

().9006

Quantili della distribuzione Chi-Quadrato

m 0,001 0,025

0,01 0,05 0,5 0,95 0,975 0,99 0,999

I 1,57E-06 0,000157 0,000982 0,003932 0,015791 0,454936 2,705543 3,841459 5,023886 6,634897 10,82757

2 0,002001 0,020101 ı

0,050636 0,102587 0,210721 4,60517

,386294 5,991465 7,377759 9,21034 13,81551

3 0,024298 0,114832 0,215795 0,351846 0,584374 2,365974 6,251389 7,814728 11,34487

9,348404 16,26624

4 0,090804 0,484419

0,297109 0,710723 1,063623 7,77944

3,356694 9,487729 11,14329 13,2767 18,46683

5 0,210213 0,554298 0,831212 1,145476 1,610308 4,35146 9,236357 11,0705 12,8325 15,08627 20,51501

6 0,381067 0,87209 ı

,237344

1 ,635383 14,44938

5,348121

2,204131 10,64464 12,59159 22,45774

16,81189

7 ı

0,598494 1,239042 ,689869 2,16735 2,833107 12,01704 14,06714

6,345811 16,01276 18,47531 24,32189

ı

8 0,857105 ,646497 2,179731 2,732637 3,489539 13,36157 15,50731 20,09024 26,12448

7,344121 17,53455

9 1,15195 2,087901 2,700389 3,325113 4,168159 8,342833 14,68366 27,87716

16,91898 19,02277 21

,66599

10 1,478743 2,558212 3,246973 3,940299 4,865182 9,341818 23,20925 29,5883

15,98718 18,30704 20,48318

ı

II ,833853 3,053484 4,574813

3,815748 5,577785 17,27501 24,72497 31,26413

19,67514 21,92005

10,341

12 2,214209 3,570569 4,403789 5,226029 6,303796 11,34032 18,54935 32,90949

26,21697

21,02607 23,33666

13 2,617218 4,106915 5,008751 5,891864 7,041505 34,52818

12,33976 27,68825

19,81193 22,36203 24,7356

14 3,040673 4,660425 5,628726 6,570631 7,789534 13,33927 21,06414 23,68479 29,14124 36,12327

26,11895

15 3,482684

5,229349 6,262138 7,260944 8,546756 14,33886 37,6973

22,30713 24,99579 27,48839 30,57791

16 3,941628

5,812212 6,907664 7,961646 15,3385 31,99993

9,312236 39,25235

26,