UNIVERSITT OF SVAZILLLND

RE-SIT EXAIVIINATION PAPER 2018

TITLE OF PAPER

COURSE CODE

TIME ALLOCATED

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INSTRUCTION

STATISTiCAL INFERENCE 1

STA220ST220

2 (TWO) HOURS

STATISTICAL TABLES AND CALCULATOR

ANSWER ANY THREE QUESTIONS THE QUESTIONS CARRY THE

MARKS AS INDICATED WITHIN THE PARENTHESIS

THIS PAPER IS NOT TO BE OPENED FNTlL PER1vHSSION HAS BEE GRANTED BY THE

INvIGILATOR

I Let ~ and B be events in a space S with =040 P(B I =02 and A B) =(110 Find (lch of the

1 P( A) (2) ll P(AuB) (31

Ill P(A nB)

n peA nB) (4 ) P(BiA) (3)

VI Are A and B independent) (4 )

[20 marks]

2a An engineer measured the Brinell hardness of 25 of ductile ron that were annealed The resulting dat~1 were

167 174

163 ]56

The engineer hypothesized that the mean Brinell bardness of all such ductile iron pieces is than 170 Test his claim at 5 level of (J 0)

2b NCHS report indicated that in 2002 the of cigarette smoking among American adults was 211 Data on prevalent smoking in n=3536 participants who attended the seventh examination of the Offspring in the Framingham Heart Study indicated that 4823536 136co of the respondents were currently smoking at the tiri1e of the exam Suppose we vant to assess vhether the prevalence smoking is lower in the Framingham Offspnng sample given the focus on cardiovascular health in that community Is there evidence of a statistically lower prevalence of in the Framingham Offspring study as

to the prevalence among all Americans at of significance (l 0)

120 marks]

3a Males and females were asked about what they vould do if they received a S 100 bill by to their neighbor but wrongly delivered to them Would they return it to their

Of the 69 males sampled 52 said yes and of 31 females sampled 120 said

Does the data indicate that the proportions that said are for male and female at a level significance (12)

3b Give a 99 confidence interval for the difference behveen the two proportions(81

[20 marks]

mail

A biologist assumes that there ie a linear relationship beteen the amount supplied to tomato plants and the subsequent yield tomatoes obtained Eight tomato plants of the same yariety vere selected at random and treated weekly with a solution which x grams of fertilizer was dissolved in a fixed quantity of water The yield y kilograms of tomatoes was recorded

L Calculate the equation the least squares regression line yon x 11 Estimate the yield of the plant treated weekly vith 32 grams of fertilizer (2) Ill Calculate the coefficient of determination and the coefficient of variation and

interpret them (4) IV Indicate vhy it may not be appropriate to use your equation to predict the yield of

a plant treated weekly with 20grams of fertilizer (2)

4b A random sample of 395 people were surveyed and each person was asked to report the highest education level they obtained The data that resulted from the survey is summarized in the following table

High School i Bachelors Masters PhD Total i

i Ibull i

Female 46 i 41 I 201 i60 54 i

5~) 57i Male 40 44 194 1

i Total 100 98 98 395 i

Question Are gender and education Jevel dependent at 5 level of significance (10) [20 marks

5 20 young pigs are assigned at random among 4 experimental groups Each group is fed a different diet The data are the pigs weight in kg after being raised on these diets for 10 months Here are the data that were obtained from the four experimental groups

i iIFeed 1 2 i Feed 3 i Feed 4

975 i 858

903699 i 989i 618

i 608 683 i 1026 - 1 ) I 677 1 1022 847

i 650 i 740 1005 8~ )i ~-

i -8-) 1 663

Are the mean pig weights the same for all 4 diets Test at 5 level of significance [20 marks]

ATISTICAl TBlES Cumulative normal distribution

Critical values of thet distribution

Critical values of the F distribution

Critical values of the chi-squared distribution

copyC These abies nave been co-nputed to acompany lne text C Introiuction to Press OxforC) ray be reorooued freely Jroyjjed that tillS retained

STATISTICAL TABLES

TABLE 41

Cumulative Standardized Normal Distribution

A(z) is the cf the standardized normal distribution from GO to z other the area under the c11i~e to the left It gives tbe

of a normal random variable not z deviatio1S above its

4~

n fj000 001 003 0C6 0G7 008 0[9~ vlt

00 05000 05040 05080 05120 05199 05279 05319 05359 05398 05438 05478 05517 05557 05596 05636 05675 05714 05753

02 05793 05832 05871 05910 05948 05987 06026 06064 e6103 06141 OJ 06179 Dmiddot6255 06331 06368 06406 06443 0648C 0651 04 06554 06591 06628 06664 0670~ 06772 06808 06844 06879 05 06915 06950 06985 07Cl9 O705~ C7088 0712~ 07157 07190 07224 OG 0 07291 0 07357 07389 07422 07454 07486 0517 0749

f0 O 07611 07642 07673 O Q7734 0776middot 07794 07823 07852 08 07881 07910 07939 07967 o1-CC 08C22 08051 08078 O81C)6 081 09 08 59 08186 08212 08238 08264 08289 08315 08340 08365 08389 10 08413 08438 08461 08485 08508 08554 08577 08middot599 08621 1 08643 08665 08686 08 7 08 O872S 08770 08790 08810 08830 12 08849 08869 08888 08907 08925 08944 08962 08980 08997 09015 l3 09032 09049 09082 09099 0911 09131 09147 09162 09177 14 09 09207 09236 09251 (19265 09279 09292 09306 9 15 09332 09345 09370 C9394 09406 09418 09429 09441 16 O9~ 52 09463 09~74 09484 (9505 G9515 09525 09535 09545

7 09554 09564 09573 09582 09599 09608 09616 09625 0963 0964 09649 09656 09664 0967 0908 09686 09693 09699 09706

19 0973 0979 09726 09732 (938 (94 09750 09756 09767 20 09772 09778 09788 0979 Q9803 09808 Q981 09817 21 09821 09826 09830 09834 O983S 09846 09850 09854 09857 22 09861 09864 09868 09871 0987- O9nS 09881 09884 09887 09890 23 09893 09896 09898 0990 J 89904 09906 09909 0991 i 09913 099 i 6 14 0991 S 09920 09922 09925 09927 09929 09931 09932 09934 09936 25 09938 09940 09941 09943 09945 09946 09948 09949 09951 09952

COQltq26 09953 09955 09956 09957 09960 09961 09962 09964 -

~ ~-r 09965 09966 09967 09968 C969 09970 09971 09972 09973 09974 28 09974 099 7 5 09976 0997 0997 09978 09979 09979 09980 0998 i 29 09981 09982 09982 09983 09984 09984 09985 09985 09986 09986 30 09987 09987 09987 09908 C9988 09989 09989 09989 09990 09990 31 J9990 09991 09991 0999i 09992 Q9992 09992 09992 O999~

32 09993 09993 09994 09994 0999 09994 09994 09995 09995 09995 i ~ 09995 09995 09995 09996 09996 G9996 (9996 09996 09996 09997 4 09997 09997 09997 09997 09997 09997 09997 09997 09998 35 09998 09998 09990 09998 09998 09998 09998 09998 26 09998 09998 09999

t Table tso t 75 t t[l t 55 lO tQ5 I 97E t 9S t 995 t999 t 9995

050 025 020 015 010 005 0025 001 0005 0001 oooos[ 100 050 040 030 020 010 005 002 001 0002 0001

08io

0741

076 -1061 CS7E 0941

~ 963 386 ~ 250 L~ to

37E

S8E 4632

O314 292G

127 4303 382 2J76

3~ S2 6965 454i

6366 8925 5641 46J4

3~ 83 22327 10215

63652 31599 12924 8610

I

-tablexs 7142007

3 STtTISl Ci TABLES

Tc3L A3

F Distribution Critical Values of F (5 significance level)

1 2 3 4 5 6 8 9 10 12 14 16 18 20

1 16145 199SD ~1571 22458 23016 ~339~l 77 23388 24054 24188 24391 24732 24801 2 lUI 190) 1916 192S1930 1933 1935 1938 1940 1~41 19t2 1943 1944 1945 3 1( 13 955 928 912 901 89~ 889 S81 879 87~ 8j 869 867 866 4 694 659 639 626 6 6 609 604600 596 91 587 581 582 580 5 66 579 54] 59 505 495 488 482 477 474 468 464 460 458 456

) ~o6 599 514 476 453 fi OS 421 ~ 15 410 406 400 396 392 390 387 f559 474 435 412 397 387 ) 37- 368 364 357 353 349 347 344

- bull lt8 531 446 407 384 369 35~ 35C 339 335 320 317 5 j_

9 512 386 363 348 3-9 38 303 296 294 10 496 371 348 333 Ii 02 298 291 286 280 277

11 484 398 359 336 32(1 309 301 290 285 279 274 270 267 265 12 475 349 326 311 300 29 285 280 264 260 257 254 13 467 318 303 292 283 i 271 267 260 255 251 248 246 14 46J 311 296 285 276 265 260 253 24R 24~ 241 239 15 451 3GS 329 306 259 254 248 242 238 235 23

16 U9 362 324 301 285 266 59 254 2A9 242 37 233 230 228 ~ cmiddot ~ ~o17 445 ) 296 281 2iO 261 25 249 245 238 i 229 226 22

- 4 18 AI 355 316 293 277 266 258 251 246 241 c) 229 225 222 219 19 438 352 313 290 274 263 254 248 242 238 231 226 221 218 216 20 435 349 310 287 271 260 251 2AS 239 235 228 222 218 215

21 432 347 307 284 268 2S7 249 242 237 232 225 220 216 212 2lC 22 430 344 305 282 266 255 246 240 234 230 223 217 213 210 207 23 428 303 280 264 253 2t-4 237 232 227 220 215 2 J1 208 205 24 426 40 301 278 262 51 242 236 230 225 218 213 209 205 203 25 424 299 276 260 249 228 224 216 211 204 20

26 422 337 298 274 259 247 39 232 227 222 25 209205 202 199 27 421 ) 296 273 257 246 237 231 225 220 21 208 204 200 197 28 420 334 295 271 256 245 236 229 219 212 206 202 99 ~ 96 29 418 333 293 270 255 243 235 228 218 210 205 201 197 194 30 417 332 292 269 253 242 233 227 221 216 209 204 199 196 193

35 4)2 327 287 264 249 237 229 216 211 204 199 194 191 188 40 408 323 284 261 245 234 225 218 212 208 200 195 190 L87 184 50 403 318 279 256 240 229 220 213 207 203 195 189 185 181 178

7 i ij60 400 315 276 2S3 237 225 217 204 199 192 J86 182 178 L7S 1

70 398 313 274 250 235 222 214 207 202 197 189 184 179 175 172

80 396 31 272 249 221 213 206 200 195 188 182 177 73 170 1190 395 310 L i 1 247 220 2 1 199 194 186 180 176 172 69

ro100 394 09 270 246 231 219 210 197 193 185 179 175 1 7 J 168 120 392 307 268 245229 218 209 202 196 191 183 178 173 169 166 150 390 306 266 243 227 216 207 200 194 189 182 l76 171 167 164

200 389 304 265 242 226 21 4 206 198 193 188 180 174 169 166 162 250 388 303 264 241 225 213 20- 198 192 187 179 173 168 65 61 300 387 303 263 240 224 213 204 197 191 L86 178 172 168 164 161 400 386 302 263 239 224 212 203 196 190 185 178 172 167 163 160 500 386 301 262 239 223 212 203 196 190 185 177 171 166 162 159

600 386 01 262 239 223 211 202 95 190 185 1 I 171 66 162 159 1 ~

750 385 01 262 238 223 211 202 95 189 1( i 170 66 162 158 1000 385 00 261 238 222 211 202 95 189 176 170 65 161 158

I Let ~ and B be events in a space S with =040 P(B I =02 and A B) =(110 Find (lch of the

1 P( A) (2) ll P(AuB) (31

Ill P(A nB)

n peA nB) (4 ) P(BiA) (3)

VI Are A and B independent) (4 )

[20 marks]

2a An engineer measured the Brinell hardness of 25 of ductile ron that were annealed The resulting dat~1 were

167 174

163 ]56

The engineer hypothesized that the mean Brinell bardness of all such ductile iron pieces is than 170 Test his claim at 5 level of (J 0)

2b NCHS report indicated that in 2002 the of cigarette smoking among American adults was 211 Data on prevalent smoking in n=3536 participants who attended the seventh examination of the Offspring in the Framingham Heart Study indicated that 4823536 136co of the respondents were currently smoking at the tiri1e of the exam Suppose we vant to assess vhether the prevalence smoking is lower in the Framingham Offspnng sample given the focus on cardiovascular health in that community Is there evidence of a statistically lower prevalence of in the Framingham Offspring study as

to the prevalence among all Americans at of significance (l 0)

120 marks]

3a Males and females were asked about what they vould do if they received a S 100 bill by to their neighbor but wrongly delivered to them Would they return it to their

Of the 69 males sampled 52 said yes and of 31 females sampled 120 said

Does the data indicate that the proportions that said are for male and female at a level significance (12)

3b Give a 99 confidence interval for the difference behveen the two proportions(81

[20 marks]

mail

A biologist assumes that there ie a linear relationship beteen the amount supplied to tomato plants and the subsequent yield tomatoes obtained Eight tomato plants of the same yariety vere selected at random and treated weekly with a solution which x grams of fertilizer was dissolved in a fixed quantity of water The yield y kilograms of tomatoes was recorded

L Calculate the equation the least squares regression line yon x 11 Estimate the yield of the plant treated weekly vith 32 grams of fertilizer (2) Ill Calculate the coefficient of determination and the coefficient of variation and

interpret them (4) IV Indicate vhy it may not be appropriate to use your equation to predict the yield of

a plant treated weekly with 20grams of fertilizer (2)

4b A random sample of 395 people were surveyed and each person was asked to report the highest education level they obtained The data that resulted from the survey is summarized in the following table

High School i Bachelors Masters PhD Total i

i Ibull i

Female 46 i 41 I 201 i60 54 i

5~) 57i Male 40 44 194 1

i Total 100 98 98 395 i

Question Are gender and education Jevel dependent at 5 level of significance (10) [20 marks

5 20 young pigs are assigned at random among 4 experimental groups Each group is fed a different diet The data are the pigs weight in kg after being raised on these diets for 10 months Here are the data that were obtained from the four experimental groups

i iIFeed 1 2 i Feed 3 i Feed 4

975 i 858

903699 i 989i 618

i 608 683 i 1026 - 1 ) I 677 1 1022 847

i 650 i 740 1005 8~ )i ~-

i -8-) 1 663

Are the mean pig weights the same for all 4 diets Test at 5 level of significance [20 marks]

ATISTICAl TBlES Cumulative normal distribution

Critical values of thet distribution

Critical values of the F distribution

Critical values of the chi-squared distribution

copyC These abies nave been co-nputed to acompany lne text C Introiuction to Press OxforC) ray be reorooued freely Jroyjjed that tillS retained

STATISTICAL TABLES

TABLE 41

Cumulative Standardized Normal Distribution

A(z) is the cf the standardized normal distribution from GO to z other the area under the c11i~e to the left It gives tbe

of a normal random variable not z deviatio1S above its

4~

n fj000 001 003 0C6 0G7 008 0[9~ vlt

00 05000 05040 05080 05120 05199 05279 05319 05359 05398 05438 05478 05517 05557 05596 05636 05675 05714 05753

02 05793 05832 05871 05910 05948 05987 06026 06064 e6103 06141 OJ 06179 Dmiddot6255 06331 06368 06406 06443 0648C 0651 04 06554 06591 06628 06664 0670~ 06772 06808 06844 06879 05 06915 06950 06985 07Cl9 O705~ C7088 0712~ 07157 07190 07224 OG 0 07291 0 07357 07389 07422 07454 07486 0517 0749

f0 O 07611 07642 07673 O Q7734 0776middot 07794 07823 07852 08 07881 07910 07939 07967 o1-CC 08C22 08051 08078 O81C)6 081 09 08 59 08186 08212 08238 08264 08289 08315 08340 08365 08389 10 08413 08438 08461 08485 08508 08554 08577 08middot599 08621 1 08643 08665 08686 08 7 08 O872S 08770 08790 08810 08830 12 08849 08869 08888 08907 08925 08944 08962 08980 08997 09015 l3 09032 09049 09082 09099 0911 09131 09147 09162 09177 14 09 09207 09236 09251 (19265 09279 09292 09306 9 15 09332 09345 09370 C9394 09406 09418 09429 09441 16 O9~ 52 09463 09~74 09484 (9505 G9515 09525 09535 09545

7 09554 09564 09573 09582 09599 09608 09616 09625 0963 0964 09649 09656 09664 0967 0908 09686 09693 09699 09706

19 0973 0979 09726 09732 (938 (94 09750 09756 09767 20 09772 09778 09788 0979 Q9803 09808 Q981 09817 21 09821 09826 09830 09834 O983S 09846 09850 09854 09857 22 09861 09864 09868 09871 0987- O9nS 09881 09884 09887 09890 23 09893 09896 09898 0990 J 89904 09906 09909 0991 i 09913 099 i 6 14 0991 S 09920 09922 09925 09927 09929 09931 09932 09934 09936 25 09938 09940 09941 09943 09945 09946 09948 09949 09951 09952

COQltq26 09953 09955 09956 09957 09960 09961 09962 09964 -

~ ~-r 09965 09966 09967 09968 C969 09970 09971 09972 09973 09974 28 09974 099 7 5 09976 0997 0997 09978 09979 09979 09980 0998 i 29 09981 09982 09982 09983 09984 09984 09985 09985 09986 09986 30 09987 09987 09987 09908 C9988 09989 09989 09989 09990 09990 31 J9990 09991 09991 0999i 09992 Q9992 09992 09992 O999~

32 09993 09993 09994 09994 0999 09994 09994 09995 09995 09995 i ~ 09995 09995 09995 09996 09996 G9996 (9996 09996 09996 09997 4 09997 09997 09997 09997 09997 09997 09997 09997 09998 35 09998 09998 09990 09998 09998 09998 09998 09998 26 09998 09998 09999

t Table tso t 75 t t[l t 55 lO tQ5 I 97E t 9S t 995 t999 t 9995

050 025 020 015 010 005 0025 001 0005 0001 oooos[ 100 050 040 030 020 010 005 002 001 0002 0001

08io

0741

076 -1061 CS7E 0941

~ 963 386 ~ 250 L~ to

37E

S8E 4632

O314 292G

127 4303 382 2J76

3~ S2 6965 454i

6366 8925 5641 46J4

3~ 83 22327 10215

63652 31599 12924 8610

I

-tablexs 7142007

3 STtTISl Ci TABLES

Tc3L A3

F Distribution Critical Values of F (5 significance level)

1 2 3 4 5 6 8 9 10 12 14 16 18 20

1 16145 199SD ~1571 22458 23016 ~339~l 77 23388 24054 24188 24391 24732 24801 2 lUI 190) 1916 192S1930 1933 1935 1938 1940 1~41 19t2 1943 1944 1945 3 1( 13 955 928 912 901 89~ 889 S81 879 87~ 8j 869 867 866 4 694 659 639 626 6 6 609 604600 596 91 587 581 582 580 5 66 579 54] 59 505 495 488 482 477 474 468 464 460 458 456

) ~o6 599 514 476 453 fi OS 421 ~ 15 410 406 400 396 392 390 387 f559 474 435 412 397 387 ) 37- 368 364 357 353 349 347 344

- bull lt8 531 446 407 384 369 35~ 35C 339 335 320 317 5 j_

9 512 386 363 348 3-9 38 303 296 294 10 496 371 348 333 Ii 02 298 291 286 280 277

11 484 398 359 336 32(1 309 301 290 285 279 274 270 267 265 12 475 349 326 311 300 29 285 280 264 260 257 254 13 467 318 303 292 283 i 271 267 260 255 251 248 246 14 46J 311 296 285 276 265 260 253 24R 24~ 241 239 15 451 3GS 329 306 259 254 248 242 238 235 23

16 U9 362 324 301 285 266 59 254 2A9 242 37 233 230 228 ~ cmiddot ~ ~o17 445 ) 296 281 2iO 261 25 249 245 238 i 229 226 22

- 4 18 AI 355 316 293 277 266 258 251 246 241 c) 229 225 222 219 19 438 352 313 290 274 263 254 248 242 238 231 226 221 218 216 20 435 349 310 287 271 260 251 2AS 239 235 228 222 218 215

21 432 347 307 284 268 2S7 249 242 237 232 225 220 216 212 2lC 22 430 344 305 282 266 255 246 240 234 230 223 217 213 210 207 23 428 303 280 264 253 2t-4 237 232 227 220 215 2 J1 208 205 24 426 40 301 278 262 51 242 236 230 225 218 213 209 205 203 25 424 299 276 260 249 228 224 216 211 204 20

26 422 337 298 274 259 247 39 232 227 222 25 209205 202 199 27 421 ) 296 273 257 246 237 231 225 220 21 208 204 200 197 28 420 334 295 271 256 245 236 229 219 212 206 202 99 ~ 96 29 418 333 293 270 255 243 235 228 218 210 205 201 197 194 30 417 332 292 269 253 242 233 227 221 216 209 204 199 196 193

35 4)2 327 287 264 249 237 229 216 211 204 199 194 191 188 40 408 323 284 261 245 234 225 218 212 208 200 195 190 L87 184 50 403 318 279 256 240 229 220 213 207 203 195 189 185 181 178

7 i ij60 400 315 276 2S3 237 225 217 204 199 192 J86 182 178 L7S 1

70 398 313 274 250 235 222 214 207 202 197 189 184 179 175 172

80 396 31 272 249 221 213 206 200 195 188 182 177 73 170 1190 395 310 L i 1 247 220 2 1 199 194 186 180 176 172 69

ro100 394 09 270 246 231 219 210 197 193 185 179 175 1 7 J 168 120 392 307 268 245229 218 209 202 196 191 183 178 173 169 166 150 390 306 266 243 227 216 207 200 194 189 182 l76 171 167 164

200 389 304 265 242 226 21 4 206 198 193 188 180 174 169 166 162 250 388 303 264 241 225 213 20- 198 192 187 179 173 168 65 61 300 387 303 263 240 224 213 204 197 191 L86 178 172 168 164 161 400 386 302 263 239 224 212 203 196 190 185 178 172 167 163 160 500 386 301 262 239 223 212 203 196 190 185 177 171 166 162 159

600 386 01 262 239 223 211 202 95 190 185 1 I 171 66 162 159 1 ~

750 385 01 262 238 223 211 202 95 189 1( i 170 66 162 158 1000 385 00 261 238 222 211 202 95 189 176 170 65 161 158

A biologist assumes that there ie a linear relationship beteen the amount supplied to tomato plants and the subsequent yield tomatoes obtained Eight tomato plants of the same yariety vere selected at random and treated weekly with a solution which x grams of fertilizer was dissolved in a fixed quantity of water The yield y kilograms of tomatoes was recorded

L Calculate the equation the least squares regression line yon x 11 Estimate the yield of the plant treated weekly vith 32 grams of fertilizer (2) Ill Calculate the coefficient of determination and the coefficient of variation and

interpret them (4) IV Indicate vhy it may not be appropriate to use your equation to predict the yield of

a plant treated weekly with 20grams of fertilizer (2)

4b A random sample of 395 people were surveyed and each person was asked to report the highest education level they obtained The data that resulted from the survey is summarized in the following table

High School i Bachelors Masters PhD Total i

i Ibull i

Female 46 i 41 I 201 i60 54 i

5~) 57i Male 40 44 194 1

i Total 100 98 98 395 i

Question Are gender and education Jevel dependent at 5 level of significance (10) [20 marks

5 20 young pigs are assigned at random among 4 experimental groups Each group is fed a different diet The data are the pigs weight in kg after being raised on these diets for 10 months Here are the data that were obtained from the four experimental groups

i iIFeed 1 2 i Feed 3 i Feed 4

975 i 858

903699 i 989i 618

i 608 683 i 1026 - 1 ) I 677 1 1022 847

i 650 i 740 1005 8~ )i ~-

i -8-) 1 663

Are the mean pig weights the same for all 4 diets Test at 5 level of significance [20 marks]

ATISTICAl TBlES Cumulative normal distribution

Critical values of thet distribution

Critical values of the F distribution

Critical values of the chi-squared distribution

copyC These abies nave been co-nputed to acompany lne text C Introiuction to Press OxforC) ray be reorooued freely Jroyjjed that tillS retained

STATISTICAL TABLES

TABLE 41

Cumulative Standardized Normal Distribution

A(z) is the cf the standardized normal distribution from GO to z other the area under the c11i~e to the left It gives tbe

of a normal random variable not z deviatio1S above its

4~

n fj000 001 003 0C6 0G7 008 0[9~ vlt

00 05000 05040 05080 05120 05199 05279 05319 05359 05398 05438 05478 05517 05557 05596 05636 05675 05714 05753

02 05793 05832 05871 05910 05948 05987 06026 06064 e6103 06141 OJ 06179 Dmiddot6255 06331 06368 06406 06443 0648C 0651 04 06554 06591 06628 06664 0670~ 06772 06808 06844 06879 05 06915 06950 06985 07Cl9 O705~ C7088 0712~ 07157 07190 07224 OG 0 07291 0 07357 07389 07422 07454 07486 0517 0749

f0 O 07611 07642 07673 O Q7734 0776middot 07794 07823 07852 08 07881 07910 07939 07967 o1-CC 08C22 08051 08078 O81C)6 081 09 08 59 08186 08212 08238 08264 08289 08315 08340 08365 08389 10 08413 08438 08461 08485 08508 08554 08577 08middot599 08621 1 08643 08665 08686 08 7 08 O872S 08770 08790 08810 08830 12 08849 08869 08888 08907 08925 08944 08962 08980 08997 09015 l3 09032 09049 09082 09099 0911 09131 09147 09162 09177 14 09 09207 09236 09251 (19265 09279 09292 09306 9 15 09332 09345 09370 C9394 09406 09418 09429 09441 16 O9~ 52 09463 09~74 09484 (9505 G9515 09525 09535 09545

7 09554 09564 09573 09582 09599 09608 09616 09625 0963 0964 09649 09656 09664 0967 0908 09686 09693 09699 09706

19 0973 0979 09726 09732 (938 (94 09750 09756 09767 20 09772 09778 09788 0979 Q9803 09808 Q981 09817 21 09821 09826 09830 09834 O983S 09846 09850 09854 09857 22 09861 09864 09868 09871 0987- O9nS 09881 09884 09887 09890 23 09893 09896 09898 0990 J 89904 09906 09909 0991 i 09913 099 i 6 14 0991 S 09920 09922 09925 09927 09929 09931 09932 09934 09936 25 09938 09940 09941 09943 09945 09946 09948 09949 09951 09952

COQltq26 09953 09955 09956 09957 09960 09961 09962 09964 -

~ ~-r 09965 09966 09967 09968 C969 09970 09971 09972 09973 09974 28 09974 099 7 5 09976 0997 0997 09978 09979 09979 09980 0998 i 29 09981 09982 09982 09983 09984 09984 09985 09985 09986 09986 30 09987 09987 09987 09908 C9988 09989 09989 09989 09990 09990 31 J9990 09991 09991 0999i 09992 Q9992 09992 09992 O999~

32 09993 09993 09994 09994 0999 09994 09994 09995 09995 09995 i ~ 09995 09995 09995 09996 09996 G9996 (9996 09996 09996 09997 4 09997 09997 09997 09997 09997 09997 09997 09997 09998 35 09998 09998 09990 09998 09998 09998 09998 09998 26 09998 09998 09999

t Table tso t 75 t t[l t 55 lO tQ5 I 97E t 9S t 995 t999 t 9995

050 025 020 015 010 005 0025 001 0005 0001 oooos[ 100 050 040 030 020 010 005 002 001 0002 0001

08io

0741

076 -1061 CS7E 0941

~ 963 386 ~ 250 L~ to

37E

S8E 4632

O314 292G

127 4303 382 2J76

3~ S2 6965 454i

6366 8925 5641 46J4

3~ 83 22327 10215

63652 31599 12924 8610

I

-tablexs 7142007

3 STtTISl Ci TABLES

Tc3L A3

F Distribution Critical Values of F (5 significance level)

1 2 3 4 5 6 8 9 10 12 14 16 18 20

1 16145 199SD ~1571 22458 23016 ~339~l 77 23388 24054 24188 24391 24732 24801 2 lUI 190) 1916 192S1930 1933 1935 1938 1940 1~41 19t2 1943 1944 1945 3 1( 13 955 928 912 901 89~ 889 S81 879 87~ 8j 869 867 866 4 694 659 639 626 6 6 609 604600 596 91 587 581 582 580 5 66 579 54] 59 505 495 488 482 477 474 468 464 460 458 456

) ~o6 599 514 476 453 fi OS 421 ~ 15 410 406 400 396 392 390 387 f559 474 435 412 397 387 ) 37- 368 364 357 353 349 347 344

- bull lt8 531 446 407 384 369 35~ 35C 339 335 320 317 5 j_

9 512 386 363 348 3-9 38 303 296 294 10 496 371 348 333 Ii 02 298 291 286 280 277

11 484 398 359 336 32(1 309 301 290 285 279 274 270 267 265 12 475 349 326 311 300 29 285 280 264 260 257 254 13 467 318 303 292 283 i 271 267 260 255 251 248 246 14 46J 311 296 285 276 265 260 253 24R 24~ 241 239 15 451 3GS 329 306 259 254 248 242 238 235 23

16 U9 362 324 301 285 266 59 254 2A9 242 37 233 230 228 ~ cmiddot ~ ~o17 445 ) 296 281 2iO 261 25 249 245 238 i 229 226 22

- 4 18 AI 355 316 293 277 266 258 251 246 241 c) 229 225 222 219 19 438 352 313 290 274 263 254 248 242 238 231 226 221 218 216 20 435 349 310 287 271 260 251 2AS 239 235 228 222 218 215

21 432 347 307 284 268 2S7 249 242 237 232 225 220 216 212 2lC 22 430 344 305 282 266 255 246 240 234 230 223 217 213 210 207 23 428 303 280 264 253 2t-4 237 232 227 220 215 2 J1 208 205 24 426 40 301 278 262 51 242 236 230 225 218 213 209 205 203 25 424 299 276 260 249 228 224 216 211 204 20

26 422 337 298 274 259 247 39 232 227 222 25 209205 202 199 27 421 ) 296 273 257 246 237 231 225 220 21 208 204 200 197 28 420 334 295 271 256 245 236 229 219 212 206 202 99 ~ 96 29 418 333 293 270 255 243 235 228 218 210 205 201 197 194 30 417 332 292 269 253 242 233 227 221 216 209 204 199 196 193

35 4)2 327 287 264 249 237 229 216 211 204 199 194 191 188 40 408 323 284 261 245 234 225 218 212 208 200 195 190 L87 184 50 403 318 279 256 240 229 220 213 207 203 195 189 185 181 178

7 i ij60 400 315 276 2S3 237 225 217 204 199 192 J86 182 178 L7S 1

70 398 313 274 250 235 222 214 207 202 197 189 184 179 175 172

80 396 31 272 249 221 213 206 200 195 188 182 177 73 170 1190 395 310 L i 1 247 220 2 1 199 194 186 180 176 172 69

ro100 394 09 270 246 231 219 210 197 193 185 179 175 1 7 J 168 120 392 307 268 245229 218 209 202 196 191 183 178 173 169 166 150 390 306 266 243 227 216 207 200 194 189 182 l76 171 167 164

200 389 304 265 242 226 21 4 206 198 193 188 180 174 169 166 162 250 388 303 264 241 225 213 20- 198 192 187 179 173 168 65 61 300 387 303 263 240 224 213 204 197 191 L86 178 172 168 164 161 400 386 302 263 239 224 212 203 196 190 185 178 172 167 163 160 500 386 301 262 239 223 212 203 196 190 185 177 171 166 162 159

600 386 01 262 239 223 211 202 95 190 185 1 I 171 66 162 159 1 ~

750 385 01 262 238 223 211 202 95 189 1( i 170 66 162 158 1000 385 00 261 238 222 211 202 95 189 176 170 65 161 158

ATISTICAl TBlES Cumulative normal distribution

Critical values of thet distribution

Critical values of the F distribution

Critical values of the chi-squared distribution

copyC These abies nave been co-nputed to acompany lne text C Introiuction to Press OxforC) ray be reorooued freely Jroyjjed that tillS retained

STATISTICAL TABLES

TABLE 41

Cumulative Standardized Normal Distribution

A(z) is the cf the standardized normal distribution from GO to z other the area under the c11i~e to the left It gives tbe

of a normal random variable not z deviatio1S above its

4~

n fj000 001 003 0C6 0G7 008 0[9~ vlt

00 05000 05040 05080 05120 05199 05279 05319 05359 05398 05438 05478 05517 05557 05596 05636 05675 05714 05753

02 05793 05832 05871 05910 05948 05987 06026 06064 e6103 06141 OJ 06179 Dmiddot6255 06331 06368 06406 06443 0648C 0651 04 06554 06591 06628 06664 0670~ 06772 06808 06844 06879 05 06915 06950 06985 07Cl9 O705~ C7088 0712~ 07157 07190 07224 OG 0 07291 0 07357 07389 07422 07454 07486 0517 0749

f0 O 07611 07642 07673 O Q7734 0776middot 07794 07823 07852 08 07881 07910 07939 07967 o1-CC 08C22 08051 08078 O81C)6 081 09 08 59 08186 08212 08238 08264 08289 08315 08340 08365 08389 10 08413 08438 08461 08485 08508 08554 08577 08middot599 08621 1 08643 08665 08686 08 7 08 O872S 08770 08790 08810 08830 12 08849 08869 08888 08907 08925 08944 08962 08980 08997 09015 l3 09032 09049 09082 09099 0911 09131 09147 09162 09177 14 09 09207 09236 09251 (19265 09279 09292 09306 9 15 09332 09345 09370 C9394 09406 09418 09429 09441 16 O9~ 52 09463 09~74 09484 (9505 G9515 09525 09535 09545

7 09554 09564 09573 09582 09599 09608 09616 09625 0963 0964 09649 09656 09664 0967 0908 09686 09693 09699 09706

19 0973 0979 09726 09732 (938 (94 09750 09756 09767 20 09772 09778 09788 0979 Q9803 09808 Q981 09817 21 09821 09826 09830 09834 O983S 09846 09850 09854 09857 22 09861 09864 09868 09871 0987- O9nS 09881 09884 09887 09890 23 09893 09896 09898 0990 J 89904 09906 09909 0991 i 09913 099 i 6 14 0991 S 09920 09922 09925 09927 09929 09931 09932 09934 09936 25 09938 09940 09941 09943 09945 09946 09948 09949 09951 09952

COQltq26 09953 09955 09956 09957 09960 09961 09962 09964 -

~ ~-r 09965 09966 09967 09968 C969 09970 09971 09972 09973 09974 28 09974 099 7 5 09976 0997 0997 09978 09979 09979 09980 0998 i 29 09981 09982 09982 09983 09984 09984 09985 09985 09986 09986 30 09987 09987 09987 09908 C9988 09989 09989 09989 09990 09990 31 J9990 09991 09991 0999i 09992 Q9992 09992 09992 O999~

32 09993 09993 09994 09994 0999 09994 09994 09995 09995 09995 i ~ 09995 09995 09995 09996 09996 G9996 (9996 09996 09996 09997 4 09997 09997 09997 09997 09997 09997 09997 09997 09998 35 09998 09998 09990 09998 09998 09998 09998 09998 26 09998 09998 09999

t Table tso t 75 t t[l t 55 lO tQ5 I 97E t 9S t 995 t999 t 9995

050 025 020 015 010 005 0025 001 0005 0001 oooos[ 100 050 040 030 020 010 005 002 001 0002 0001

08io

0741

076 -1061 CS7E 0941

~ 963 386 ~ 250 L~ to

37E

S8E 4632

O314 292G

127 4303 382 2J76

3~ S2 6965 454i

6366 8925 5641 46J4

3~ 83 22327 10215

63652 31599 12924 8610

I

-tablexs 7142007

3 STtTISl Ci TABLES

Tc3L A3

F Distribution Critical Values of F (5 significance level)

1 2 3 4 5 6 8 9 10 12 14 16 18 20

1 16145 199SD ~1571 22458 23016 ~339~l 77 23388 24054 24188 24391 24732 24801 2 lUI 190) 1916 192S1930 1933 1935 1938 1940 1~41 19t2 1943 1944 1945 3 1( 13 955 928 912 901 89~ 889 S81 879 87~ 8j 869 867 866 4 694 659 639 626 6 6 609 604600 596 91 587 581 582 580 5 66 579 54] 59 505 495 488 482 477 474 468 464 460 458 456

) ~o6 599 514 476 453 fi OS 421 ~ 15 410 406 400 396 392 390 387 f559 474 435 412 397 387 ) 37- 368 364 357 353 349 347 344

- bull lt8 531 446 407 384 369 35~ 35C 339 335 320 317 5 j_

9 512 386 363 348 3-9 38 303 296 294 10 496 371 348 333 Ii 02 298 291 286 280 277

11 484 398 359 336 32(1 309 301 290 285 279 274 270 267 265 12 475 349 326 311 300 29 285 280 264 260 257 254 13 467 318 303 292 283 i 271 267 260 255 251 248 246 14 46J 311 296 285 276 265 260 253 24R 24~ 241 239 15 451 3GS 329 306 259 254 248 242 238 235 23

16 U9 362 324 301 285 266 59 254 2A9 242 37 233 230 228 ~ cmiddot ~ ~o17 445 ) 296 281 2iO 261 25 249 245 238 i 229 226 22

- 4 18 AI 355 316 293 277 266 258 251 246 241 c) 229 225 222 219 19 438 352 313 290 274 263 254 248 242 238 231 226 221 218 216 20 435 349 310 287 271 260 251 2AS 239 235 228 222 218 215

21 432 347 307 284 268 2S7 249 242 237 232 225 220 216 212 2lC 22 430 344 305 282 266 255 246 240 234 230 223 217 213 210 207 23 428 303 280 264 253 2t-4 237 232 227 220 215 2 J1 208 205 24 426 40 301 278 262 51 242 236 230 225 218 213 209 205 203 25 424 299 276 260 249 228 224 216 211 204 20

26 422 337 298 274 259 247 39 232 227 222 25 209205 202 199 27 421 ) 296 273 257 246 237 231 225 220 21 208 204 200 197 28 420 334 295 271 256 245 236 229 219 212 206 202 99 ~ 96 29 418 333 293 270 255 243 235 228 218 210 205 201 197 194 30 417 332 292 269 253 242 233 227 221 216 209 204 199 196 193

35 4)2 327 287 264 249 237 229 216 211 204 199 194 191 188 40 408 323 284 261 245 234 225 218 212 208 200 195 190 L87 184 50 403 318 279 256 240 229 220 213 207 203 195 189 185 181 178

7 i ij60 400 315 276 2S3 237 225 217 204 199 192 J86 182 178 L7S 1

70 398 313 274 250 235 222 214 207 202 197 189 184 179 175 172

80 396 31 272 249 221 213 206 200 195 188 182 177 73 170 1190 395 310 L i 1 247 220 2 1 199 194 186 180 176 172 69

ro100 394 09 270 246 231 219 210 197 193 185 179 175 1 7 J 168 120 392 307 268 245229 218 209 202 196 191 183 178 173 169 166 150 390 306 266 243 227 216 207 200 194 189 182 l76 171 167 164

200 389 304 265 242 226 21 4 206 198 193 188 180 174 169 166 162 250 388 303 264 241 225 213 20- 198 192 187 179 173 168 65 61 300 387 303 263 240 224 213 204 197 191 L86 178 172 168 164 161 400 386 302 263 239 224 212 203 196 190 185 178 172 167 163 160 500 386 301 262 239 223 212 203 196 190 185 177 171 166 162 159

600 386 01 262 239 223 211 202 95 190 185 1 I 171 66 162 159 1 ~

750 385 01 262 238 223 211 202 95 189 1( i 170 66 162 158 1000 385 00 261 238 222 211 202 95 189 176 170 65 161 158

STATISTICAL TABLES

TABLE 41

Cumulative Standardized Normal Distribution

A(z) is the cf the standardized normal distribution from GO to z other the area under the c11i~e to the left It gives tbe

of a normal random variable not z deviatio1S above its

4~

n fj000 001 003 0C6 0G7 008 0[9~ vlt

00 05000 05040 05080 05120 05199 05279 05319 05359 05398 05438 05478 05517 05557 05596 05636 05675 05714 05753

02 05793 05832 05871 05910 05948 05987 06026 06064 e6103 06141 OJ 06179 Dmiddot6255 06331 06368 06406 06443 0648C 0651 04 06554 06591 06628 06664 0670~ 06772 06808 06844 06879 05 06915 06950 06985 07Cl9 O705~ C7088 0712~ 07157 07190 07224 OG 0 07291 0 07357 07389 07422 07454 07486 0517 0749

f0 O 07611 07642 07673 O Q7734 0776middot 07794 07823 07852 08 07881 07910 07939 07967 o1-CC 08C22 08051 08078 O81C)6 081 09 08 59 08186 08212 08238 08264 08289 08315 08340 08365 08389 10 08413 08438 08461 08485 08508 08554 08577 08middot599 08621 1 08643 08665 08686 08 7 08 O872S 08770 08790 08810 08830 12 08849 08869 08888 08907 08925 08944 08962 08980 08997 09015 l3 09032 09049 09082 09099 0911 09131 09147 09162 09177 14 09 09207 09236 09251 (19265 09279 09292 09306 9 15 09332 09345 09370 C9394 09406 09418 09429 09441 16 O9~ 52 09463 09~74 09484 (9505 G9515 09525 09535 09545

7 09554 09564 09573 09582 09599 09608 09616 09625 0963 0964 09649 09656 09664 0967 0908 09686 09693 09699 09706

19 0973 0979 09726 09732 (938 (94 09750 09756 09767 20 09772 09778 09788 0979 Q9803 09808 Q981 09817 21 09821 09826 09830 09834 O983S 09846 09850 09854 09857 22 09861 09864 09868 09871 0987- O9nS 09881 09884 09887 09890 23 09893 09896 09898 0990 J 89904 09906 09909 0991 i 09913 099 i 6 14 0991 S 09920 09922 09925 09927 09929 09931 09932 09934 09936 25 09938 09940 09941 09943 09945 09946 09948 09949 09951 09952

COQltq26 09953 09955 09956 09957 09960 09961 09962 09964 -

~ ~-r 09965 09966 09967 09968 C969 09970 09971 09972 09973 09974 28 09974 099 7 5 09976 0997 0997 09978 09979 09979 09980 0998 i 29 09981 09982 09982 09983 09984 09984 09985 09985 09986 09986 30 09987 09987 09987 09908 C9988 09989 09989 09989 09990 09990 31 J9990 09991 09991 0999i 09992 Q9992 09992 09992 O999~

32 09993 09993 09994 09994 0999 09994 09994 09995 09995 09995 i ~ 09995 09995 09995 09996 09996 G9996 (9996 09996 09996 09997 4 09997 09997 09997 09997 09997 09997 09997 09997 09998 35 09998 09998 09990 09998 09998 09998 09998 09998 26 09998 09998 09999

t Table tso t 75 t t[l t 55 lO tQ5 I 97E t 9S t 995 t999 t 9995

050 025 020 015 010 005 0025 001 0005 0001 oooos[ 100 050 040 030 020 010 005 002 001 0002 0001

08io

0741

076 -1061 CS7E 0941

~ 963 386 ~ 250 L~ to

37E

S8E 4632

O314 292G

127 4303 382 2J76

3~ S2 6965 454i

6366 8925 5641 46J4

3~ 83 22327 10215

63652 31599 12924 8610

I

-tablexs 7142007

3 STtTISl Ci TABLES

Tc3L A3

F Distribution Critical Values of F (5 significance level)

1 2 3 4 5 6 8 9 10 12 14 16 18 20

1 16145 199SD ~1571 22458 23016 ~339~l 77 23388 24054 24188 24391 24732 24801 2 lUI 190) 1916 192S1930 1933 1935 1938 1940 1~41 19t2 1943 1944 1945 3 1( 13 955 928 912 901 89~ 889 S81 879 87~ 8j 869 867 866 4 694 659 639 626 6 6 609 604600 596 91 587 581 582 580 5 66 579 54] 59 505 495 488 482 477 474 468 464 460 458 456

) ~o6 599 514 476 453 fi OS 421 ~ 15 410 406 400 396 392 390 387 f559 474 435 412 397 387 ) 37- 368 364 357 353 349 347 344

- bull lt8 531 446 407 384 369 35~ 35C 339 335 320 317 5 j_

9 512 386 363 348 3-9 38 303 296 294 10 496 371 348 333 Ii 02 298 291 286 280 277

11 484 398 359 336 32(1 309 301 290 285 279 274 270 267 265 12 475 349 326 311 300 29 285 280 264 260 257 254 13 467 318 303 292 283 i 271 267 260 255 251 248 246 14 46J 311 296 285 276 265 260 253 24R 24~ 241 239 15 451 3GS 329 306 259 254 248 242 238 235 23

16 U9 362 324 301 285 266 59 254 2A9 242 37 233 230 228 ~ cmiddot ~ ~o17 445 ) 296 281 2iO 261 25 249 245 238 i 229 226 22

- 4 18 AI 355 316 293 277 266 258 251 246 241 c) 229 225 222 219 19 438 352 313 290 274 263 254 248 242 238 231 226 221 218 216 20 435 349 310 287 271 260 251 2AS 239 235 228 222 218 215

21 432 347 307 284 268 2S7 249 242 237 232 225 220 216 212 2lC 22 430 344 305 282 266 255 246 240 234 230 223 217 213 210 207 23 428 303 280 264 253 2t-4 237 232 227 220 215 2 J1 208 205 24 426 40 301 278 262 51 242 236 230 225 218 213 209 205 203 25 424 299 276 260 249 228 224 216 211 204 20

26 422 337 298 274 259 247 39 232 227 222 25 209205 202 199 27 421 ) 296 273 257 246 237 231 225 220 21 208 204 200 197 28 420 334 295 271 256 245 236 229 219 212 206 202 99 ~ 96 29 418 333 293 270 255 243 235 228 218 210 205 201 197 194 30 417 332 292 269 253 242 233 227 221 216 209 204 199 196 193

35 4)2 327 287 264 249 237 229 216 211 204 199 194 191 188 40 408 323 284 261 245 234 225 218 212 208 200 195 190 L87 184 50 403 318 279 256 240 229 220 213 207 203 195 189 185 181 178

7 i ij60 400 315 276 2S3 237 225 217 204 199 192 J86 182 178 L7S 1

70 398 313 274 250 235 222 214 207 202 197 189 184 179 175 172

80 396 31 272 249 221 213 206 200 195 188 182 177 73 170 1190 395 310 L i 1 247 220 2 1 199 194 186 180 176 172 69

ro100 394 09 270 246 231 219 210 197 193 185 179 175 1 7 J 168 120 392 307 268 245229 218 209 202 196 191 183 178 173 169 166 150 390 306 266 243 227 216 207 200 194 189 182 l76 171 167 164

200 389 304 265 242 226 21 4 206 198 193 188 180 174 169 166 162 250 388 303 264 241 225 213 20- 198 192 187 179 173 168 65 61 300 387 303 263 240 224 213 204 197 191 L86 178 172 168 164 161 400 386 302 263 239 224 212 203 196 190 185 178 172 167 163 160 500 386 301 262 239 223 212 203 196 190 185 177 171 166 162 159

600 386 01 262 239 223 211 202 95 190 185 1 I 171 66 162 159 1 ~

750 385 01 262 238 223 211 202 95 189 1( i 170 66 162 158 1000 385 00 261 238 222 211 202 95 189 176 170 65 161 158

t Table tso t 75 t t[l t 55 lO tQ5 I 97E t 9S t 995 t999 t 9995

050 025 020 015 010 005 0025 001 0005 0001 oooos[ 100 050 040 030 020 010 005 002 001 0002 0001

08io

0741

076 -1061 CS7E 0941

~ 963 386 ~ 250 L~ to

37E

S8E 4632

O314 292G

127 4303 382 2J76

3~ S2 6965 454i

6366 8925 5641 46J4

3~ 83 22327 10215

63652 31599 12924 8610

I

-tablexs 7142007

3 STtTISl Ci TABLES

Tc3L A3

F Distribution Critical Values of F (5 significance level)

1 2 3 4 5 6 8 9 10 12 14 16 18 20

1 16145 199SD ~1571 22458 23016 ~339~l 77 23388 24054 24188 24391 24732 24801 2 lUI 190) 1916 192S1930 1933 1935 1938 1940 1~41 19t2 1943 1944 1945 3 1( 13 955 928 912 901 89~ 889 S81 879 87~ 8j 869 867 866 4 694 659 639 626 6 6 609 604600 596 91 587 581 582 580 5 66 579 54] 59 505 495 488 482 477 474 468 464 460 458 456

) ~o6 599 514 476 453 fi OS 421 ~ 15 410 406 400 396 392 390 387 f559 474 435 412 397 387 ) 37- 368 364 357 353 349 347 344

- bull lt8 531 446 407 384 369 35~ 35C 339 335 320 317 5 j_

9 512 386 363 348 3-9 38 303 296 294 10 496 371 348 333 Ii 02 298 291 286 280 277

11 484 398 359 336 32(1 309 301 290 285 279 274 270 267 265 12 475 349 326 311 300 29 285 280 264 260 257 254 13 467 318 303 292 283 i 271 267 260 255 251 248 246 14 46J 311 296 285 276 265 260 253 24R 24~ 241 239 15 451 3GS 329 306 259 254 248 242 238 235 23

16 U9 362 324 301 285 266 59 254 2A9 242 37 233 230 228 ~ cmiddot ~ ~o17 445 ) 296 281 2iO 261 25 249 245 238 i 229 226 22

- 4 18 AI 355 316 293 277 266 258 251 246 241 c) 229 225 222 219 19 438 352 313 290 274 263 254 248 242 238 231 226 221 218 216 20 435 349 310 287 271 260 251 2AS 239 235 228 222 218 215

21 432 347 307 284 268 2S7 249 242 237 232 225 220 216 212 2lC 22 430 344 305 282 266 255 246 240 234 230 223 217 213 210 207 23 428 303 280 264 253 2t-4 237 232 227 220 215 2 J1 208 205 24 426 40 301 278 262 51 242 236 230 225 218 213 209 205 203 25 424 299 276 260 249 228 224 216 211 204 20

26 422 337 298 274 259 247 39 232 227 222 25 209205 202 199 27 421 ) 296 273 257 246 237 231 225 220 21 208 204 200 197 28 420 334 295 271 256 245 236 229 219 212 206 202 99 ~ 96 29 418 333 293 270 255 243 235 228 218 210 205 201 197 194 30 417 332 292 269 253 242 233 227 221 216 209 204 199 196 193

35 4)2 327 287 264 249 237 229 216 211 204 199 194 191 188 40 408 323 284 261 245 234 225 218 212 208 200 195 190 L87 184 50 403 318 279 256 240 229 220 213 207 203 195 189 185 181 178

7 i ij60 400 315 276 2S3 237 225 217 204 199 192 J86 182 178 L7S 1

70 398 313 274 250 235 222 214 207 202 197 189 184 179 175 172

80 396 31 272 249 221 213 206 200 195 188 182 177 73 170 1190 395 310 L i 1 247 220 2 1 199 194 186 180 176 172 69

ro100 394 09 270 246 231 219 210 197 193 185 179 175 1 7 J 168 120 392 307 268 245229 218 209 202 196 191 183 178 173 169 166 150 390 306 266 243 227 216 207 200 194 189 182 l76 171 167 164

200 389 304 265 242 226 21 4 206 198 193 188 180 174 169 166 162 250 388 303 264 241 225 213 20- 198 192 187 179 173 168 65 61 300 387 303 263 240 224 213 204 197 191 L86 178 172 168 164 161 400 386 302 263 239 224 212 203 196 190 185 178 172 167 163 160 500 386 301 262 239 223 212 203 196 190 185 177 171 166 162 159

600 386 01 262 239 223 211 202 95 190 185 1 I 171 66 162 159 1 ~

750 385 01 262 238 223 211 202 95 189 1( i 170 66 162 158 1000 385 00 261 238 222 211 202 95 189 176 170 65 161 158

3 STtTISl Ci TABLES

Tc3L A3

F Distribution Critical Values of F (5 significance level)

1 2 3 4 5 6 8 9 10 12 14 16 18 20

1 16145 199SD ~1571 22458 23016 ~339~l 77 23388 24054 24188 24391 24732 24801 2 lUI 190) 1916 192S1930 1933 1935 1938 1940 1~41 19t2 1943 1944 1945 3 1( 13 955 928 912 901 89~ 889 S81 879 87~ 8j 869 867 866 4 694 659 639 626 6 6 609 604600 596 91 587 581 582 580 5 66 579 54] 59 505 495 488 482 477 474 468 464 460 458 456

) ~o6 599 514 476 453 fi OS 421 ~ 15 410 406 400 396 392 390 387 f559 474 435 412 397 387 ) 37- 368 364 357 353 349 347 344

- bull lt8 531 446 407 384 369 35~ 35C 339 335 320 317 5 j_

9 512 386 363 348 3-9 38 303 296 294 10 496 371 348 333 Ii 02 298 291 286 280 277

11 484 398 359 336 32(1 309 301 290 285 279 274 270 267 265 12 475 349 326 311 300 29 285 280 264 260 257 254 13 467 318 303 292 283 i 271 267 260 255 251 248 246 14 46J 311 296 285 276 265 260 253 24R 24~ 241 239 15 451 3GS 329 306 259 254 248 242 238 235 23

16 U9 362 324 301 285 266 59 254 2A9 242 37 233 230 228 ~ cmiddot ~ ~o17 445 ) 296 281 2iO 261 25 249 245 238 i 229 226 22

- 4 18 AI 355 316 293 277 266 258 251 246 241 c) 229 225 222 219 19 438 352 313 290 274 263 254 248 242 238 231 226 221 218 216 20 435 349 310 287 271 260 251 2AS 239 235 228 222 218 215

21 432 347 307 284 268 2S7 249 242 237 232 225 220 216 212 2lC 22 430 344 305 282 266 255 246 240 234 230 223 217 213 210 207 23 428 303 280 264 253 2t-4 237 232 227 220 215 2 J1 208 205 24 426 40 301 278 262 51 242 236 230 225 218 213 209 205 203 25 424 299 276 260 249 228 224 216 211 204 20

26 422 337 298 274 259 247 39 232 227 222 25 209205 202 199 27 421 ) 296 273 257 246 237 231 225 220 21 208 204 200 197 28 420 334 295 271 256 245 236 229 219 212 206 202 99 ~ 96 29 418 333 293 270 255 243 235 228 218 210 205 201 197 194 30 417 332 292 269 253 242 233 227 221 216 209 204 199 196 193

35 4)2 327 287 264 249 237 229 216 211 204 199 194 191 188 40 408 323 284 261 245 234 225 218 212 208 200 195 190 L87 184 50 403 318 279 256 240 229 220 213 207 203 195 189 185 181 178

7 i ij60 400 315 276 2S3 237 225 217 204 199 192 J86 182 178 L7S 1

70 398 313 274 250 235 222 214 207 202 197 189 184 179 175 172

80 396 31 272 249 221 213 206 200 195 188 182 177 73 170 1190 395 310 L i 1 247 220 2 1 199 194 186 180 176 172 69

ro100 394 09 270 246 231 219 210 197 193 185 179 175 1 7 J 168 120 392 307 268 245229 218 209 202 196 191 183 178 173 169 166 150 390 306 266 243 227 216 207 200 194 189 182 l76 171 167 164

200 389 304 265 242 226 21 4 206 198 193 188 180 174 169 166 162 250 388 303 264 241 225 213 20- 198 192 187 179 173 168 65 61 300 387 303 263 240 224 213 204 197 191 L86 178 172 168 164 161 400 386 302 263 239 224 212 203 196 190 185 178 172 167 163 160 500 386 301 262 239 223 212 203 196 190 185 177 171 166 162 159

600 386 01 262 239 223 211 202 95 190 185 1 I 171 66 162 159 1 ~

750 385 01 262 238 223 211 202 95 189 1( i 170 66 162 158 1000 385 00 261 238 222 211 202 95 189 176 170 65 161 158

4 STATISTICAL TABLES

TABLE A3 (continued)

F Distribution Critical Values of F (5 significance level)

30 35 40 50 60 75 100 150 200

1 249262501020692511425177 2526225304 25316253 2 1946 1946 1947 1947 1948 i948 1948 1949 1949 1949 3 863 862 860 859 858 857 856 855 S~4 854 4 577 575 573 572 570 569 568 566 565 565 5 452 45( 446 444 443 442 ~Al 429 439

G 383 38 379 377 375 374 3 70 369 7 340 338 336 334 332 330 329 327 326 325 8 3 1 308 306 304 302 3(] 299 29~ 296 295 9 289 286 284 283 280 279 274 273

10 270 268 266 264 262 26C 259 257 256

11 260 255 253 251 249 247 246 244 243 12 250 247 244 243 240 238 237 235 233 232 13 241 238 236 234 230 228 226 224 223 14 231 228 227 224 222 221 219 217 216 15 225 222 220 218 216 214 212 210 210

16 2~2~ 219 217 25 212 211 20S 20-- 205 204 17 218 25 212 210 208 206 204 202 200 199 18 214 211 208 206 204 202 200 198 196 193 19 2J 1 207 205 203 200 198 196 94 192 91 20 207 204 201 199 197 195 191 189 188

21 205 201 198 196 194 192 190 188 186 184 22 202 196 194 191 189 187 185 183 182 23 200 153 191 188 186 184 182 180 179 24 197 194 191 189 186 184 182 180 178 177 25middot 196 192 189 187 184 L32 180 178 176 175

26 194 190 187 185 un 180 U8 176 174 173 27 192 188 186 184 181 179 176 174 172 17 28 191 187 184 182 179 177 175 173 170 169 29 189 185 183 181 177 175 173 171 169 67 30 188 184 181 179 176 174 172 170 167 166

35 182 179 176 174 170 168 166 163 161 i60 40 172 169 166 164 161 159 156 155 50 173 169 166 163 160 158 152 LSO 148 60 169 165 162 159 156 153 151 148 145 144 70 1 162 159 157 153 150 48 145 142 lAO

80 164 160 157 l54 151 148 145 143 139 13f 90 159 155 153 149 146 144 141 138 136middot

100 157 154 152 148 145 142 139 136 134 120 160 155 152 1 146 143 lAO 13 7 133 132

150 158 1S4 150 144 141 138 134 131 129

200 156 152 148 146 141 13~ 135 132 250 155 LSD 147 144 ~40 137 3~ 131 300 154 150 lA6 139 136 133 130 126 123 400 153 149 45 142 138 135 132 128 124 122 500 153 148 145 142 13 8 135 Lo 1 128 123 121

600 152middot 148 144 141 13 134 13 27 123 120 750 152147 144 141 13 134 26 122 120

1000 152 147 143 141 bullL~

lt 130 26 L22 L19

5 STATISTiCAL TABLES

TABlE A3 tcontinued)

F Distribution Critical Values of F (1 significance level)

2 3 4 5 G 8 9 10 12 14 16 18 20 11

t 452J8 99950 54C33S 562458 576365 585899 60224 Ei42G7 6170JJ 6~9l53 610873

2 9850 9908 9917 9925 9930 993~ 996 99~7 9939 9943 9944 9944 9945 3 3412 3082 2946 2871 2824 27 2767 2749 2723 2705 2692 2683 2675 2669 4 2120 1800 1669 1598 IS52 J521 1498 1480 1466 455 1437 1425 1415 408 1402 5 1626 1327 1206 1139 1097 1067 1046 ]016 1005 989 977 968 961 955

G 375 1092 978 915 875 847 826 810 798 18 IL 60 52 745 740 225 955 845 785 i46 719 699 684 672 662 636 628 62i 616

8 126 865 759 701 663 67 618 603 591 581 567 556 548 541 536 9 CL~6 802 1599 642 606 58Q 561 547 535 526 511 50i 492 486 481

10 004 756 655 599 539 520 506 494 485 IJ 46Cl 452 446 441

11 965 721 622 567 532 507 489 474 463 ~54 440 429 421 415 410 12 933 693 595 541 506 482 464 450 439 430 416 405 397 391 386 13 907 6O 574 521 486 462 444 430 419 410 396 386 378 372 366 14 886 651 556 504 469 446 428 114 403 394 380 362 356 351 15 868 636 489 456 4]4 40C 389 38C -SS 349 342 337

16 853 623 5251 477 444 420 403 89 378 369 355 345 331 326 17 S40 bn 518 467 434 41( 393 79 368 359 346 5) 321 316

~M

18 829 6Gl 5C9 458 425 401 384 371 360 351 ) OL I 319 313 308 19 818 593 501 450 417 ~94 377 363 352 343 330 319 312 305 300 20 810 585 494 443 410 387 370 356 3A6 313 305 299 294

21 802 578 487 437 404 381 364 351 340 331 317 299 293 288 22 795 572 482 431 399 376 359 345 335 ~26 312 302 294 288 283

788 566 476 426 394 371 354 341 330 321 307 297 289 2amp3 278 2i- 561 472 422 390 367 350 336 326 317 303 293 285 279 274 25 777 557 468 418 385 363 332 322 3 299 289 281 27S 270

26 72 553 464 414 382 359 342 329 318 309 296 286 278 272 266 27 768 549 460 411 378 356 339 326 315 306 293 282 275 268 263 28 764 545 457 407 75 353 336 323 312 303 290 279 2 265 260 29 760 542 454 404 373 350 333 320 309 300 287 277 269 263 257 30 756 539 451 402 370 347 330 31 307 298 284 274 266 260 255

35 742 527 440 391 359 ) i 320 307 296 288 274 264 256 250 244 40 731 518 431 383 351 329 312 299 289 280 266 256 248 242 237 50 717 506 420 372 341 319 302 289 278 270 256 246 238 232 227 60 708 498 413 365 334 312 295 282 272 263 250 239 231 225 220 70 701 492 407 360 329 307 291 278 267 259 245 235 227 220 215

80 696 488 404 356 326 304 287 274 264 255 242 231 223 217 212 90 693 485 3 323 301 284 261 252 239 229 221 214 209

100 690 482 398 351 321 299 282 269 259 250 237 227 219 212 207 120 6S5 479 395 348 317 279 266 256 247 223 215 209 203 150 681 475 391 345 J14 276 263 253 214 231 220 212 206 200

200 676 4 88341 311 289 273 260 50 241 227 217 209 203 197 250 67t 469 86 34C 309 2S7 271 258 24S 226 215 207 201 195 300 672 468 85 338 30S 286 27G 247 224 214 206 199 194 400 670 466 83 337 306 285 268 256 245 237 223 205 198 192 500 669 465 82 336 305 284 268 244 236 222 204 97 192

GOO 668 46L 81 335 305 283 25L 244 235 221 2 1 203 196 191 750 667 463 81 334 304 266 253 243 234 221 2 1 202 196 190

1000 666 463 80 334 304 282 266 53 43 234 220 2 0 202 195 190

6 STATISTICAL TABLES

TABLE p 3 (continued)

F Distribution Critical Values of F(1 significance level)

30 35 40 50 60 15 100 150 200

1 6239f3 626065 275)7 628678 630252 62356 63S~i j 634468 63497

2 9946 9947 9947 9947 9948 9948 9945 9949 9949 9949 c( ~3middot 26SCJ 2645 2641 LO_ 2632 262~ 2624 262Q 2618

4 139 1384 13 1375 1369 1365 361 1358 1354 1352 5 945 938 933 929 924 920 91 93 909 908

G -730 723 718 714 709 06 702 699 695 693 7 606 599 594 591 586 582 579 575 52 570 8 526 520 515 512 507 03 50( 496 4 Q 491 9 471 465 460 457 452 448 445 L4 ~ 438 436

10 425 420 417 412 408 405 401 398 396 i

11 4(j) 394 389 386 381 378 374 367 366 11 376 370 365 362 357 354 50 347 343 341 13 357 S51 3AG 343 338 334 327 324 322

~ ~c14 34) 335 330 327 JLL 318 35 11 308 306 15 3 28 21 31 313 308 305 298 294 292

16 31 310 305 3C2 297 293 290 286 283 2S1 17 307 380 296 292 287 283 280 271 18 298 292 287 284 278 275 c 268 264 262 19 291 284 280 276 271 267 264 26emiddot 257 255 20 284 278 273 269 264 261 257 254 250 248

21 279 272 267 264 258 255 251 248 244 242 22 273 267 262 258 253 250 246 242 238 236 23 269 262 2S7 254 248 245 241 237 234 232 24 264 258 253 249 244 240 23 233 229 227 25 260 254 249 245 240 236 233 229 225 223

26 257 250 245 242 236 233 229 225 221 219 27middot 254 247 242 238 233 229 226 222 218 216 28 25 244 239 235 23G 226 219 215 213 29 248 241 236 233 223 220 216 212 210 30 245 239 234 230 225 221 217 23 209 207

35 235 228 223 219 214 210 206 202 198 196 40 227 220 215 211 206 202 198 94 190 187 50 217 210 205 20 195 19) 187 182 178 l76 60 210 203 198 194 188 184 1 175 170 168 70 205 198 193 189 183 178 174 170 165 162

80 201 194 189 185 179 175 70 L65 J6 158 90 199 192 186 182 176 167 162 157 155

100 197 189 184 180 174 169 165 160 155 152 middot120 193 186 181 176 170 161 56 151 148

150 190 183 77 73 166 162 157 152 146 1)3

200 187179 174 169 163 158 153 148 42 250 185 177 172 167 1 156 15 i 146 40 300184 176 70 166 59 155 150 144 38 35 400 182 175 169 164 58 153 48 142 36 32 500 L8i 74 168 163 157 147 141 34 31

600 180 73 167 163 56 151 46 140 34 30 150 180 172 166 162 S5 150 145 139 29

1000 179 122 166 161 54 150 144 138 32 28

7 STlTiSTICAL ABLES

TAB A3 (continued

F Distribution Critical Values of F (01 significance level)

3 4 5 G 8 9 10 12 14 16 18 20

1 45lt05 500005 ~40c05 576e05 5J6eJS S93e05 6J6eG 6 BeDS 6i-4eO~ G1ge05 621eQ5

2 998509990099917 99930 99933 99936 994099942 99943 99944 99944 99945 3 16703 14850 14ll1 13710 13458 13285 13L58 ]2986 12925 12832 12764 127 12674 1261+2 4 7414 6115 568 5344 5171 5053 4966 4900 4847 48C5 4741 4695 4632 4610 5 4718 3712 3320 3109 2975 28SO 2816 2~6S 2724 2692 2642 2606 2557 2539

in nG 3551 2700 2370 2192 2C80 -JV) lt46 869 841 199 76S 1712 1925 2l09 1877 1720 1621 1~52 502 lS63 433 408 1371 343 ]306 1293 8 2541 1849 1583 1439 48 1286 240 12cl) 154 1119 094 1060 1048 9 2286 J639 1390 1256 1171 ]1E 078 1D37 011 989 957 933 915 901 890

10 2104 191 1255 125 1048 993 952 ~20 896 875 845 822 805 791 7fiO

11 1969 138 1 1035 958 905 866 835 812 792 763 74 j 724 i11 701 12 1864 1297 1080 963 889 838 800 748 729 700 679 663 651 640 13 17S2 1231 1021 907 835 786 749 698 680 652 610 603 593 14 ~714 1178 973 862 792 744 708 658 640 613 593 578 56G 556 15 659 3 934 825 7S~ 709 6 626 608 5151 562 546 535 525

~16 1612 1097 9[1 94 727 680 Jl 598 581 55 535 509 499 17 1572 1066 873 68 102 656 622 ~9t 5 1lt 558 532 513 499 487 478 18 1538 1039 849 746 681 635 602 576 556 539 513 494 480 468 459 19 I5C8 1016 828 727 662 618 585 559 539 522 497 478 464 452 443 20 482 995 810 710 646 602 569 544 524 508 482 464 449 438 429

6 0

22 1438 780 681 619 576 5M 519 499 483 458 440 426 415 c~06

23 1420 670 608 565 533 5 09 489 473 448 420 416 405 396 24 1403 755 659 598 555 523 499 480 464 439 421 07 396 387 25 1388 922 45 649 589 546 51 491 471 456 43 413 399 388 379

21 1459 977 794 632 588 556 531 511 495 470 451 437 426 47

26 1374 912 736 641 580 538 507 464 448 424 406 392 381 372 27 1361 902 727 633 573 531 500 476 457 441 417 386 375 366 28 1350 893 719 625 566 524 469 450 435 411 380 369 360 29 1339 885 712 619 559 518 467 464 ~4S ~29 405 388 374 363 354 30 1329 877 705 612 553 512 482 58 439 424 401 382 369 358 349

35 1290 847 679 588 530 489 459 436 418 403 379 362 348 338 329 40 1261 825 659 570 53 473 444 421 402 387 364 347 334 323 34 50 1222 634 546 490 451 422 400 332 367 344 327 341 304 295 60 1197 617 531 476 437 409 386 369 354 332 315 302 291 283 70 1180 606 520 466 428 77 360 3~5 323 306 293 283 274

80 E67 54 597 Sl2 458 420 370 353 339 ]6 300 287 276 268 90 1157 747 591 506 453 41 365 348 334 1 295 282 271 263

100 1150 741 586 502 448 411 383 361 344 330 307 291 278 268 120 1138 732 578 495 442 404 377 355 338 324 302 285 272 262 150 1127 724 571 488 435 398 371 349 318 296 280 267 256248

200 1115 15 563 481 429 392 65 43 26 312 290 214 261 251 242 250 1109 710 559 477 425 388 61 40 23 309 287 271 US 248 239 300 1104 707 556 475 412 386 59 38 21 307 285 209 256 245 237 400 1099 703 553 471 419 383 56 35 18 304 282 266 253 243 234 500 1096 700 551 469 418 36 54 lt33 16 302 281 264 252 241 233

600 1094 699 549 468 416 380 53 32 l~ 301 280 251 240 750 1091 697 548 467 415 52 31 14 300 215 262 249 239 231

1000 1089 696 546 465 414 378 51 30 13 299 277 261 248 238 230

STATSTICAL TABLES 8

TABLE A3 (continued)

F Distribution Critical Values of F (01 significance level)

30 35 40 50 60 75 ]00 150 200

1 G2ampe05 6LStCOS 630cJ5

2 99946 99947 9994 9994 99948 99948 99949 3 12584 2545 12517 12496 12466 12447 12427 1 4 4570 4543 4523 4509 4488 4475 4461 5 250amp 248 2472 2460 2444 2433 2422

6 1685 1667 654 1644 1631 1621 1612 1593 1589 7 1269 1253 241 1233 122C 1212 1204 18 1182 8 1026 1011 000 992 980 973 965 957 949 945 9 869 855 S46 837 826 819 Sl 93

10 760 747 737 730 719 712 705 691 687

11 681 66S 659 652 642 635 628 621 611 6lG 12 622 60S 600 593 583 576 570 563 556 552 13 575 563 554 547 537 530 524 51 S1 () 507 14 538 525 517 51 G 500 494 487 48i 474 471 15 507 4SS 48~ 48C 4 -n 64 45 444 441

16 482 47C 46 44 4AS 439 432 42E 419 416 17 460 448 440 433 424 48 ~ 1 405 398 395 18 442 430 422 415 406 400 393 387 380 377 19 426 414 406 399 390 384 378 371 365 361 20 412 400 392 386 377 370 364 358 351 348

21 400 388 380 374 364 358 352 346 339 336 22 389 378 370 36~ 354 348 341 335 328 325

~ 1123 379 368 360 353 j I 319 316 24 371 359 351 345 336 323 310 307 25 363 352 343 337 328 322 315 309 303 299

26 356 344 330 321 315 308 302 295 292 27 349 338 330 323 314 308 302 296 289 286 28 343 332 318 309 302 296 290 283 280 29 338 327 318 312 3e3 297 291 284 278 274 30 333 322 313 307 298 286 279 273 269

35 313 302 293 287 2n 272 266 259 252 249 40 298 287 279 273 264 257 251 238 234 50 279 268 260 253 244 238 231 22 218 214 60 267 255 247 241 232 225 219 212 205 201 70 258 247 239 232 223 216 210 195 192

80 252 232 226 216 210 203 196 189 185 90 247 236 227 221 211 205 193 191 183 179

100 243 23 224 208 201 194 187 179 175 120 237 226 218 211 202 195 188 181 173 168 150 232 221 212 206 196 189 182 174 166 162

1 ~200 226 215 207 200 190 183 1 V 168 160 155 250 223 212 203 197 187 180 65 156 151 300 221 210 201 194 185178 170 162 153 148 400 218 207 198 182 175 167 159 150 AS 500 217 205 197 190 180 173 165 157 148 143

600 26 204 196 189 179 172 164 156 146 141 750 2 i5 203 195 188 178 171 163 155 145 140

1000 214 202 194 187 177 169 162 153 144 1

Chi-Square bution Taole