HWK

Dadato

tables__formulas.pdf

Home >Mathematics homework help >HWK

Chapter 2 Organizing and Summarizing Data

• Class midpoint: The sum of consecutive lower class limits divided by 2.

• Relative frequency = frequency

sum of all frequencies

Chapter 3 Numerically Summarizing Data

• Population Mean:

• Sample Mean:

•

• Population Variance:

• Sample Variance:

• Population Standard Deviation:

• Sample Standard Deviation:

• Empirical Rule: If the shape of the distribution is bell- shaped, then

• Approximately 68% of the data lie within 1 standard deviation of the mean

• Approximately 95% of the data lie within 2 standard deviations of the mean

• Approximately 99.7% of the data lie within 3 stan- dard deviations of the mean

• Population Mean from Grouped Data:

• Sample Mean from Grouped Data: x = gxifi gfi

m = gxifi gfi

s = 2s2 s = 2s2

s2 = g1xi - x22

n - 1 = gx2i -

1gxi22 n

n - 1

s2 = g1xi - m22

N = gx2i -

1 gxi22 N

Range = Largest Data Value - Smallest Data Value

x = gxi n

m = gxi N

• Weighted Mean:

• Population Variance from Grouped Data:

• Sample Variance from Grouped Data:

• Population z-score:

• Sample z-score:

• Interquartile Range:

• Lower and Upper Fences:

• Five-Number Summary

Minimum, Q1 , M, Q3 , Maximum

Lower fence = Q1 - 1.51IQR2 Upper fence = Q3 + 1.51IQR2

IQR = Q3 - Q1

z = x - x

z = x - m s

s2 = g1xi - m22fi Agfi B - 1 =

gx2i fi - 1gxifi22 gfi

gfi - 1

s2 = g1xi - m22fi

gfi = gx2i fi -

1gxifi22 gfi

gfi

xw = gwixi gwi

CHAPTER 4 Describing the Relation between Two Variables

• observed

• for the least-squares regression model

• The coefficient of determination, measures the proportion of total variation in the response variable that is explained by the least-squares regression line.

R2,

yN = b1x + b0 R2 = r2

y - predicted y = y - yNResidual =

• Correlation Coefficient:

• The equation of the least-squares regression line is

where is the predicted value,

is the slope, and is the intercept.b0 = y - b1x

b1 = r # sy

sx yNyN = b1x + b0,

r = a axi - xsx b a

yi - y sy b

n - 1

CHAPTER 5 Probability

• Empirical Probability

• Classical Probability

P1E2 = number of ways that E can occur number of possible outcomes

= N1E2 N1S2

P1E2 L frequency of E number of trials of experiment

• Addition Rule for Disjoint Events

• Addition Rule for n Disjoint Events

• General Addition Rule

P1E or F2 = P1E2 + P1F2 - P1E and F2

P1E or F or G or Á 2 = P1E2 + P1F2 + P1G2 + Á

P1E or F2 = P1E2 + P1F2

Z05_SULL8028_03_SE_BARREL.QXD 8/26/08 4:06 PM Page 1

CHAPTER 6 Discrete Probability Distributions

• Mean and Standard Deviation of a Binomial Random Variable

• Poisson Probability Distribution Function

• Mean and Standard Deviation of a Poisson Random Variable

mX = lt sX = 2lt P1x2 = 1lt2

x! e-lt x = 0, 1, 2, Á sX = 4np11 - p2mX = np

• Mean (Expected Value) of a Discrete Random Variable

• Variance of a Discrete Random Variable

• Binomial Probability Distribution Function

P1x2 = nCxpx11 - p2n- x

s2X = g1x - m22 # P1x2 = gx2P1x2 - m2X

mX = gx # P1x2

CHAPTER 7 The Normal Distribution

• Standardizing a Normal Random Variable

z = x - m s

• Finding the Score: x = m + zs

CHAPTER 8 Sampling Distributions

• Mean and Standard Deviation of the Sampling Distribu- tion of

• Sample Proportion: pN = x

mx = m and sx = s

2n x

• Mean and Standard Deviation of the Sampling Distribution of

mpN = p and spN = C p11 - p2

CHAPTER 9 Estimating the Value of a Parameter Using Confidence Intervals

Confidence Intervals • A confidence interval about with

known is .

• A confidence interval about with

unknown is . Note: is computed using

degrees of freedom.

• A confidence interval about p is

• A confidence interval about is

. 1n - 12s2 xa/2

2 6 s 2 6 1n - 12s2 x1 -a/2

s211 - a2 # 100% pn ; za/2 # C

pN11 - pn2 n

11 - a2 # 100% n - 1

ta/2x ; ta/2 # s1n sm11 - a2 # 100%

x ; za/2 # s1n sm11 - a2 # 100%

Sample Size • To estimate the population mean with a margin of error E

at a level of confidence:

rounded up to the next integer.

• To estimate the population proportion with a margin of error E at a level of confidence:

rounded up to the next integer,

where is a prior estimate of the population proportion,

or rounded up to the next integer when

no prior estimate of p is available.

n = 0.25 a za/2 E b2

n = pN11 - pN2a za/2 E b2

11 - a2 # 100%

n = a za/2 # s E b211 - a2 # 100%

• Complement Rule

• Multiplication Rule for Independent Events

• Multiplication Rule for n Independent Events

• Conditional Probability Rule

• General Multiplication Rule

P1E and F2 = P1E2 # P1F ƒ E2

P1F ƒ E2 = P1E and F2 P1E2 =

N1E and F2 N1E2

P1E and F and G Á 2 = P1E2 # P1F2 # P1G2 # Á

P1E and F2 = P1E2 # P1F2

P1Ec2 = 1 - P1E2 • Factorial

• Permutation of n objects taken r at a time:

• Combination of n objects taken r at a time:

• Permutations with Repetition:

n! n1! # n2! # Á # nk!

nCr = n!

r!1n - r2!

nPr = n!

1n - r2! n! = n # 1n - 12 # 1n - 22 # Á # 3 # 2 # 1

Z05_SULL8028_03_SE_BARREL.QXD 8/26/08 4:06 PM Page 2

CHAPTER 10 Testing Claims Regarding a Parameter

•

• x20 = 1n - 12s2 s0

z0 = pN - p0

C p011 - p02

CHAPTER 11 Inferences on Two Samples

• Test Statistic for Matched-Pairs data

where is the mean and is the standard deviation of the differenced data.

• Confidence Interval for Matched-Pairs data:

Note: is found using degrees of freedom.

• Test Statistic Comparing Two Means (Independent Sampling):

• Confidence Interval for the Difference of Two Means (Independent Samples):

1x1 - x22 ; ta/2C s1

n1 +

s2 2

t0 = 1x1 - x22 - 1m1 - m22

C s1

n1 +

s2 2

n - 1ta/2

d ; ta/2 # sd1n

sdd

t0 = d - md sdn1n

Note: is found using the smaller of or degrees of freedom.

• Test Statistic Comparing Two Population Proportions

where

• Confidence Interval for the Difference of Two Proportions

• Test Statistic for Comparing Two Population Standard Deviations

• Finding a Critical F for the Left Tail

F1 -a,n1 - 1,n2 - 1 = 1

Fa,n2 - 1,n1 - 1

F0 = s1

s2 2

1pN1 - pN22 ; za/2C pN111 - pN12

n1 +

pN211 - pN22 n2

pN = x1 + x2 n1 + n2

.z0 = pN1 - pN2 - (p1 - p2)

4pN11 - pN2B 1 n1

+ 1 n2

n2 - 1n1 - 1ta/2

Test Statistics

• single mean, known

• single mean, unknownst0 = x - m0 sn1n

s z0 =

x - m0 sn1n

CHAPTER 12 Inference on Categorical Data

• Chi-Square Test Statistic

All and no more than 20% less than 5.Ei Ú 1

i = 1, 2, Á , k

x20 = a 1observed - expected22

expected = a

1Oi - Ei22 Ei

• Expected Counts (when testing for goodness of fit)

• Expected Frequencies (when testing for independence or homogeneity of proportions)

Expected frequency = 1row total21column total2

table total

Ei = mi = npi for i = 1, 2, Á , k

CHAPTER 13 Comparing Three or More Means

• Test Statistic for One-Way ANOVA

where

MSE = 1n1 - 12s12 + 1n2 - 12s22 + Á + 1nk - 12sk2

n - k

MST = n11x1 - x22 + n21x2 - x22 + Á + nk1xk - x22

k - 1

F = Mean square due to treatment

Mean square due to error =

MST MSE

• Test Statistic for Tukey’s Test after One-Way ANOVA

q = 1x2 - x12 - 1m2 - m12

A s2

2 # a 1

n1 +

1 n2 b

= x2 - x1

A s2

2 # a 1

n1 + 1

n2 b

Z05_SULL8028_03_SE_BARREL.QXD 8/26/08 4:06 PM Page 3

CHAPTER 14 Inference on the Least-squares Regression Model and Multiple Regression

• Confidence Interval about the Mean Response of

where is the given value of the explanatory variable and is the critical value with degrees of freedom.

• Prediction Interval about an Individual Response,

where is the given value of the explanatory variable and is the critical value with degrees of freedom.

n - 2ta/2 x…

yN ; ta/2 # seC1 + 1 n

+ 1x… - x22 g1xi - x22

n - 2ta/2 x…

yN ; ta/2 # seC 1 n

+ 1x… - x22 g1xi - x22

y, yN• Standard Error of the Estimate

• Standard error of

• Teststatistic fortheSlopeoftheLeast-SquaresRegressionLine

• Confidence Interval for the Slope of the Regression Line

where is computed with degrees of freedom.n - 2ta/2

b1 ; ta/2 # se4g1xi - x22

t0 = b1 - b1

sen4g1xi - x22 =

b1 - b1 sb1

sb1 = se

4g1xi - x22 b1

se = C g1yi - yNi22

n - 2 = C

g residuals2

n - 2

Two-Tailed Left-Tailed Right-Tailed

CHAPTER 15 Nonparametric Statistics

• Test Statistic for a Runs Test for Randomness Small-Sample Case If and the test statistic in the runs test for randomness is r, the number of runs.

Large-Sample Case If or the test statistic is

where

• Test Statistic for a One-Sample Sign Test

mr = 2n1n2

n + 1 and sr = B

2n1n212n1n2 - n2 n2 1n - 12

z0 = r - mr sr

n2 7 20,n1 7 20

n2 … 20,n1 … 20

Large-Sample Case The test statistic, , is

where n is the number of minus and plus signs and k is obtained as described in the small sample case.

z0 = 1k + 0.52 - n

1n 2

z0(n > 25)

The test statistic, k, will The test statistic, The test statistic, be the smaller of the k, will be the k, will be the number of minus signs number of number of or plus signs. plus signs. minus signs.

H1 : M 7 MoH1 : M 6 MoH1 : M Z Mo

Ho : M = MoHo : M = MoHo: M = Mo

Small-Sample Case (n ◊ 25)

Large-Sample Case

where T is the test statistic from the small-sample case.

• Test Statistic for the Mann–Whitney Test

Small-Sample Case and

If S is the sum of the ranks corresponding to the sample from population X, then the test statistic, T, is given by

Note: The value of S is always obtained by summing the ranks of the sample data that correspond to in the hypothesis.

Large-Sample Case or

• Test Statistic for Spearman’s Rank Correlation Test

where the difference in the ranks of the two observations in the ordered pair.

• Test Statistic for the Kruskal–Wallis Test

where is the sum of the ranks in the ith sample.Ri

= 12

N1N + 12 B R1

n1 +

R2 2

n2 + Á +

Rk 2

nk R - 31N + 12

H = 12

N1N + 12a 1 ni BRi - ni1N + 122 R

ith di =

rs = 1 - 6gdi2

n1n2 - 12

z0 = T -

n1n2 2

B n1n21n1 + n2 + 12

(n2>20)(n1>20)

T = S - n11n1 + 12

n2 ◊ 20)(n1 ◊ 20

z0 = T -

n1n + 12 4

C n1n + 12 12n + 12

(n>30)

Two-Tailed Left-Tailed Right-Tailed

Test Statistic: T is the Test Statistic: Test Statistic: smaller of or T = ƒ T- ƒT = T+T-T+

Ho : MD 7 0H1 : MD 6 0H1 : MD Z 0 Ho : MD = 0Ho : MD = 0Ho : MD = 0

• Test Statistic for the Wilcoxon Matched-Pairs Signed-Ranks Test

Small-Sample Case (n ◊ 30)

Z05_SULL8028_03_SE_BARREL.QXD 8/26/08 4:06 PM Page 4

Table I

Row Number 01–05 06–10 11–15 16–20 21–25 26–30 31–35 36–40 41–45 46–50

01 89392 23212 74483 36590 25956 36544 68518 40805 09980 00467 02 61458 17639 96252 95649 73727 33912 72896 66218 52341 97141 03 11452 74197 81962 48443 90360 26480 73231 37740 26628 44690 04 27575 04429 31308 02241 01698 19191 18948 78871 36030 23980

05 36829 59109 88976 46845 28329 47460 88944 08264 00843 84592 06 81902 93458 42161 26099 09419 89073 82849 09160 61845 40906 07 59761 55212 33360 68751 86737 79743 85262 31887 37879 17525 08 46827 25906 64708 20307 78423 15910 86548 08763 47050 18513 09 24040 66449 32353 83668 13874 86741 81312 54185 78824 00718 10 98144 96372 50277 15571 82261 66628 31457 00377 63423 55141

11 14228 17930 30118 00438 49666 65189 62869 31304 17117 71489 12 55366 51057 90065 14791 62426 02957 85518 28822 30588 32798 13 96101 30646 35526 90389 73634 79304 96635 06626 94683 16696 14 38152 55474 30153 26525 83647 31988 82182 98377 33802 80471 15 85007 18416 24661 95581 45868 15662 28906 36392 07617 50248

16 85544 15890 80011 18160 33468 84106 40603 01315 74664 20553 17 10446 20699 98370 17684 16932 80449 92654 02084 19985 59321 18 67237 45509 17638 65115 29757 80705 82686 48565 72612 61760 19 23026 89817 05403 82209 30573 47501 00135 33955 50250 72592 20 67411 58542 18678 46491 13219 84084 27783 34508 55158 78742

Column Number

Random Numbers

Table II

Critical Values for Correlation Coefficient n

3 0.997 4 0.950 5 0.878 6 0.811 7 0.754 8 0.707 9 0.666

10 0.632 11 0.602 12 0.576 13 0.553 14 0.532 15 0.514 16 0.497

17 0.482 18 0.468 19 0.456 20 0.444 21 0.433 22 0.423 23 0.413

24 0.404 25 0.396 26 0.388 27 0.381 28 0.374 29 0.367 30 0.361

Z05_SULL8028_03_SE_BARREL.QXD 8/26/08 4:06 PM Page 5

�3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002 �3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003 �3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 �3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007 �3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010

�2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 �2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 �2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 �2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036 �2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048

�2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064 �2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084 �2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110 �2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143 �2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183

�1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233 �1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294 �1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 �1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 �1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559

�1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 �1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 �1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985 �1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170 �1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379

�0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 �0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867 �0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148 �0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 �0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776

�0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121 �0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483 �0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859 �0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 �0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641

Standard Normal Distribution

Table V

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 Area

0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879

0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389

1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319

1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767

2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936

2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986

3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998

Confidence Interval Critical Values, Level of Confidence Critical Value,

0.90 or 90% 1.645 0.95 or 95% 1.96 0.98 or 98% 2.33 0.99 or 99% 2.575

zA/2 zA/2 Hypothesis Testing Critical Values

Level of Significance, Left Tailed Right Tailed Two-Tailed

0.10 �1.28 1.28 �1.645 0.05 �1.645 1.645 �1.96 0.01 �2.33 2.33 �2.575

Z05_SULL8028_03_SE_BARREL.QXD 8/26/08 4:06 PM Page 6

1 1.000 1.376 1.963 3.078 6.314 12.706 15.894 31.821 2 0.816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 3 0.765 0.978 1.250 1.638 2.353 3.182 3.482 4.541 5.841 4 0.741 0.941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5 0.727 0.920 1.156 1.476 2.015 2.571 2.757 3.365 4.032 6 0.718 0.906 1.134 1.440 1.943 2.447 2.612 3.143 3.707 7 0.711 0.896 1.119 1.415 1.895 2.365 2.517 2.998 3.499 8 0.706 0.889 1.108 1.397 1.860 2.306 2.449 2.896 3.355 9 0.703 0.883 1.100 1.383 1.833 2.262 2.398 2.821 3.250 10 0.700 0.879 1.093 1.372 1.812 2.228 2.359 2.764 3.169 11 0.697 0.876 1.088 1.363 1.796 2.201 2.328 2.718 3.106 12 0.695 0.873 1.083 1.356 1.782 2.179 2.303 2.681 3.055 13 0.694 0.870 1.079 1.350 1.771 2.160 2.282 2.650 3.012 14 0.692 0.868 1.076 1.345 1.761 2.145 2.264 2.624 2.977 15 0.691 0.866 1.074 1.341 1.753 2.131 2.249 2.602 2.947 16 0.690 0.865 1.071 1.337 1.746 2.120 2.235 2.583 2.921 17 0.689 0.863 1.069 1.333 1.740 2.110 2.224 2.567 2.898 18 0.688 0.862 1.067 1.330 1.734 2.101 2.214 2.552 2.878 19 0.688 0.861 1.066 1.328 1.729 2.093 2.205 2.539 2.861 20 0.687 0.860 1.064 1.325 1.725 2.086 2.197 2.528 2.845 21 0.686 0.859 1.063 1.323 1.721 2.080 2.189 2.518 2.831 22 0.686 0.858 1.061 1.321 1.717 2.074 2.183 2.508 2.819 23 0.685 0.858 1.060 1.319 1.714 2.069 2.177 2.500 2.807 24 0.685 0.857 1.059 1.318 1.711 2.064 2.172 2.492 2.797 25 0.684 0.856 1.058 1.316 1.708 2.060 2.167 2.485 2.787 26 0.684 0.856 1.058 1.315 1.706 2.056 2.162 2.479 2.779 27 0.684 0.855 1.057 1.314 1.703 2.052 2.158 2.473 2.771 28 0.683 0.855 1.056 1.313 1.701 2.048 2.154 2.467 2.763 29 0.683 0.854 1.055 1.311 1.699 2.045 2.150 2.462 2.756 30 0.683 0.854 1.055 1.310 1.697 2.042 2.147 2.457 2.750 31 0.682 0.853 1.054 1.309 1.696 2.040 2.144 2.453 2.744 32 0.682 0.853 1.054 1.309 1.694 2.037 2.141 2.449 2.738 33 0.682 0.853 1.053 1.308 1.692 2.035 2.138 2.445 2.733 34 0.682 0.852 1.052 1.307 1.691 2.032 2.136 2.441 2.728 35 0.682 0.852 1.052 1.306 1.690 2.030 2.133 2.438 2.724 36 0.681 0.852 1.052 1.306 1.688 2.028 2.131 2.434 2.719 37 0.681 0.851 1.051 1.305 1.687 2.026 2.129 2.431 2.715 38 0.681 0.851 1.051 1.304 1.686 2.024 2.127 2.429 2.712 39 0.681 0.851 1.050 1.304 1.685 2.023 2.125 2.426 2.708 40 0.681 0.851 1.050 1.303 1.684 2.021 2.123 2.423 2.704 50 0.679 0.849 1.047 1.299 1.676 2.009 2.109 2.403 2.678 60 0.679 0.848 1.045 1.296 1.671 2.000 2.099 2.390 2.660 70 0.678 0.847 1.044 1.294 1.667 1.994 2.093 2.381 2.648 80 0.678 0.846 1.043 1.292 1.664 1.990 2.088 2.374 2.639 90 0.677 0.846 1.042 1.291 1.662 1.987 2.084 2.368 2.632 100 0.677 0.845 1.042 1.290 1.660 1.984 2.081 2.364 2.626 1000 0.675 0.842 1.037 1.282 1.646 1.962 2.056 2.330 2.581 z 0.674 0.842 1.036 1.282 1.645 1.960 2.054 2.326 2.576 63.657 127.321 318.309 636.619 9.925 14.089 22.327 31.599 7.453 10.215 12.924 5.598 7.173 8.610 4.773 5.893 6.869 4.317 5.208 5.959 4.029 4.785 5.408 3.833 4.501 5.041 3.690 4.297 4.781 3.581 4.144 4.587 3.497 4.025 4.437 3.428 3.930 4.318 3.372 3.852 4.221 3.326 3.787 4.140 3.286 3.733 4.073 3.252 3.686 4.015 3.222 3.646 3.965 3.197 3.610 3.922 3.174 3.579 3.883 3.153 3.552 3.850 3.135 3.527 3.819 3.119 3.505 3.792 3.104 3.485 3.768 3.091 3.467 3.745 3.078 3.450 3.725 3.067 3.435 3.707 3.057 3.421 3.690 3.047 3.408 3.674 3.038 3.396 3.659 3.030 3.385 3.646 3.022 3.375 3.633 3.015 3.365 3.622 3.008 3.356 3.611 3.002 3.348 3.601 2.996 3.340 3.591 2.990 3.333 3.582 2.985 3.326 3.574 2.980 3.319 3.566 2.976 3.313 3.558 2.971 3.307 3.551 2.937 3.261 3.496 2.915 3.232 3.460 2.899 3.211 3.435 2.887 3.195 3.416 2.878 3.183 3.402 2.871 3.174 3.390 2.813 3.098 3.300 2.807 3.090 3.291

t-Distribution

Table VI

df 0.25 0.20 0.15 0.10 0.05 0.025 0.02 0.01 0.005 0.0025 0.001 0.0005

Area in Right Tail

Area in right tail

Z05_SULL8028_03_SE_BARREL.QXD 8/26/08 4:06 PM Page 7

1 — — 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879 2 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597 3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838 4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860 5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.833 15.086 16.750

6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548 7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278 8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955 9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589 10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188

11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757 12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300 13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819 14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319 15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801

16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267 17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718 18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156 19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582 20 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997

21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401 22 8.643 9.542 10.982 12.338 14.041 30.813 33.924 36.781 40.289 42.796 23 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181 24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559 25 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928

26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290 27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.645 28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993 29 13.121 14.256 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52.336 30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672

40 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766 50 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.490 60 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952 70 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215 80 51.172 53.540 57.153 60.391 64.278 96.578 101.879 106.629 112.329 116.321

90 59.196 61.754 65.647 69.126 73.291 107.565 113.145 118.136 124.116 128.299 100 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169

Chi-Square (X2) Distribution

Table VII

0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 0.005

Area to the Right of Critical Value Degrees of Freedom

Two tails

The area to the right of this value is .

The area to the right of this value is 1� .

2X1�a 2

2Xa 2 a

2 a

Left tail Area � 1�a

The area to the right of this value is 1�a.

2X1�a

Right tail

The area to the right of this value is a.

Z05_SULL8028_03_SE_BARREL.QXD 8/26/08 4:06 PM Page 8

Formulas
Tables

Table I
Table II
Table V
Table VI
Table VII