For Elementary Statistics, Tenth Edition, by Mario F ...
Formulas and Tables for Elementary Statistics, Tenth Edition, by Mario F. Triola
Copyright 2006 Pearson Education, Inc.
Ch. 3: Descriptive Statistics
x Sx Mean n
Sf . x
x
Mean (frequency table)
Sf
S(x 2 x)2
s
Standard deviation
? n21
n(Sx2) 2 (Sx)2 Standard deviation
s ? n(n 2 1)
(shortcut)
n3S(f . x2) 4 2 3S(f . x) 42 Standard deviation
s?
n(n 2 1)
(frequency table)
variance s2
Ch. 4: Probability
P(A or B) 5 P(A) 1 P(B) if A, B are mutually exclusive
P(A or B) 5 P(A) 1 P(B) 2 P(A and B)
if A, B are not mutually exclusive P(A and B) 5 P(A) . P(B) if A, B are independent P(A and B) 5 P(A) . P(B 0A) if A, B are dependent
P(A) 5 1 2 P(A) Rule of complements
n! nPr 5 (n 2 r)! Permutations (no elements alike)
n! n1! n2! . . . nk!
Permutations (n1 alike, ...)
n! nCr 5 (n 2 r)! r!
Combinations
Ch. 5: Probability Distributions
x . P(x) Mean (prob. dist.)
[x2 . P(x)] 2 Standard deviation (prob. dist.)
P(x)
n!
. px . qnx Binomial probability
(n x)! x!
n.p
Mean (binomial)
2 n . p . q
Variance (binomial)
n . p . q x . e
P(x) x!
Standard deviation (binomial)
Poisson Distribution where e 2.71828
Ch. 6: Normal Distribution
z
x
s
x
or
x
Standard score
x Central limit theorem
x
n
Central limit theorem (Standard error)
Ch. 7: Confidence Intervals (one population)
p^ E p p^ E Proportion p^ q^
where E 5 za>2? n
x 2 E , m , x 1 E Mean
s where E 5 za>2 !n (s known )
or
E
5
ta>2
s !n
(s unknown)
(n 2 1)s2
(n 2 1)s2
xR2
, s2 ,
x
2 L
Variance
Ch. 7: Sample Size Determination
3za>242 . 0.25
n5
Proportion
E2
3za>242p^ q^
n 5 E2
Proportion (p^ and q^ are known)
za>2s 2 n 5 B R Mean
E
Ch. 9: Confidence Intervals (two populations)
(p^ 1 2 p^ 2) 2 E , (p1 2 p2) , (p^ 1 2 p^ 2) 1 E
where
E
5
za>2?
p^ 1q^ 1 n1
1
p^ 2q^ 2 n2
(x1 2 x2) 2 E , (m1 2 m2) , (x1 2 x2) 1 E (Indep.)
< where
E
5
ta>2?
s21 n1
1
s22 n2
(df smaller of n1 1, n2 1)
(s1 and s2 unknown and not assumed equal)
< E
5
ta>2?
sp2 n1
1
sp2 n2
(df 5 n1 1 n2 2 2)
sp2
5
(n1 2 1)s21 (n1 2 1)
1 1
(n2 (n2
2 2
1 ) s22 1)
(s1 and s2 unknown but assumed equal)
< E
5
za>2?
s
2 1
n1
1
s
2 2
n2
(s1, s2 known)
d 2 E , md , d 1 E (Matched Pairs)
where
E
5
ta>2
sd !n
(df n 1)
Copyright 2007 Pearson Education, publishing as Pearson Addison-Wesley.
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Formulas and Tables for Elementary Statistics, Tenth Edition, by Mario F. Triola
Copyright 2006 Pearson Education, Inc.
Ch. 8: Test Statistics (one population)
p^ 2 p
z5
Proportion--one population
pq
?n
x 2 m Mean--one population
z5 s> !n
( known)
x 2 m Mean--one population
t5 s> !n
( unknown)
(n 2 1)s2
x2 5
s2
Standard deviation or variance-- one population
Ch. 9: Test Statistics (two populations)
z 5 (p^ 1 2 p^ 2) 2 (p1 2 p2) pq pq
? n1 1 n2
Two proportions
t
5
(x1
2
x2) 2 (m1 2
s21 ? n1
1
s22 n2
m2)
df smaller of n1 1, n2 1
Two means--independent; s1 and s2 unknown, and
not assumed equal.
< t
5
(x1
2 x2) 2 (m1 2 sp2 1 sp2
? n1 n2
m2)
(df n1 n2 2)
sp2
5
(n1
2 1)s21 1 (n2 2 n1 1 n2 2 2
1 ) s22
Two means--independent; s1 and s2 unknown, but assumed equal.
z
5
(x1
2 x2) 2 (m1 2
s 12 ? n1
1
s
2 2
n2
m2)
Two means--independent; 1, 2 known.
t 5 d 2 md sd> !n
Two means--matched pairs (df n 1)
F
5
s21 s22
Standard deviation or variance--
two
populations
(where
s
2 1
s
22)
Ch. 11: Multinomial and Contingency Tables
(O 2 E)2 x2 5 g
E
Multinomial (df k 1)
(O 2 E)2 x2 5 g
E
Contingency table [df (r 1)(c 1)]
(row total) (column total)
where E 5
(grand total)
( 0 b 2 c 0 2 1)2 McNemar's test
x2 5
b1c
for matched pairs
(df 1)
Ch. 10: Linear Correlation/Regression
nSxy 2 (Sx) (Sy) Correlation r 5
"n(Sx2) 2 (Sx)2"n(Sy2) 2 (Sy)2
nSxy 2 (Sx) (Sy) b1 5 n(Sx2) 2 (Sx)2
(Sy) (Sx2) 2 (Sx) (Sxy)
b0 5 y 2 b1x or b0 5
n(Sx2) 2 (Sx)2
y^ 5 b0 1 b1x Estimated eq. of regression line
explained variation r2 5
total variation
se
5
S(y ?n
2 2
y^ )2 2
or
Sy2 ?
2
b0Sy n2
2 2
b1Sxy
y^ E y y^ E Prediction interval
where E t2se
1
1 n
n(x0 n(x2)
x)2 (x)2
Ch. 12: One-Way Analysis of a Variance
Procedure for testing H0: m1 5 m2 5 m3 5 c
1. Use software or calculator to obtain results. 2. Identify the P-value. 3. Form conclusion:
If P-value a, reject the null hypothesis of equal means.
If P a, fail to reject the null hypothesis of equal means.
Ch. 12: Two-Way Analysis of Variance
Procedure:
1. Use software or a calculator to obtain results. 2. Test H0: There is no interaction between the row factor
and column factor. 3. Stop if H0 from Step 1 is rejected.
If H0 from Step 1 is not rejected (so there does not appear to be an interaction effect), proceed with these two tests:
Test for effects from the row factor. Test for effects from the column factor.
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Copyright 2007 Pearson Education, publishing as Pearson Addison-Wesley.
Formulas and Tables for Elementary Statistics, Tenth Edition, by Mario F. Triola
Copyright 2006 Pearson Education, Inc.
Ch. 13: Nonparametric Tests
(x 1 0.5) 2 (n>2)
z5
Sign test for n 25
!n>2
z5
T 2 n(n 1 1)>4 Wilcoxon signed ranks n(n 1 1) (2n 1 1) (matched pairs and n 30)
?
24
z
5
R2 mR
5
R2
n1(n1 1 n2 1 1) 2
sR
n1n2(n1 1 n2 1 1)
?
12
Wilcoxon rank-sum (two independent samples)
H5
12
a R21 1 R22 1 . . . 1 R2k b 2 3(N 1 1)
N(N 1 1) n1 n2
nk
Kruskal-Wallis (chi-square df k 1)
6Sd2 rs 5 1 2 n(n2 2 1) Rank correlation acritical value for n . 30: 6 z b
!n 2 1
z 5 G 2 mG 5 sG
G 2 a 2n1n2 1 1b n1 1 n2
(2n1n2) (2n1n2 2 n1 2 n2)
? (n1 1 n2)2(n1 1 n2 2 1)
Runs test for n 20
Ch. 14: Control Charts
R chart: Plot sample ranges UCL: D4R Centerline: R LCL: D3R
x chart: Plot sample means UCL: x 1 A2R Centerline: x LCL: x 2 A2R
p chart: Plot sample proportions pq
UCL: p 1 3 ? n Centerline: p
pq LCL: p 2 3 ? n
TABLE A-6 Critical Values of the Pearson Correlation Coefficient r
n
a .05
a .01
4
.950
.999
5
.878
.959
6
.811
.917
7
.754
.875
8
.707
.834
9
.666
.798
10
.632
.765
11
.602
.735
12
.576
.708
13
.553
.684
14
.532
.661
15
.514
.641
16
.497
.623
17
.482
.606
18
.468
.590
19
.456
.575
20
.444
.561
25
.396
.505
30
.361
.463
35
.335
.430
40
.312
.402
45
.294
.378
50
.279
.361
60
.254
.330
70
.236
.305
80
.220
.286
90
.207
.269
100
.196
.256
NOTE: To test H0: r 0 against H1: r 0, reject H0 if the absolute value of r is greater than the critical value in the table.
Control Chart Constants
Subgroup Size n
2 3 4 5 6 7
A2
1.880 1.023 0.729 0.577 0.483 0.419
D3
0.000 0.000 0.000 0.000 0.000 0.076
D4
3.267 2.574 2.282 2.114 2.004 1.924
Copyright 2007 Pearson Education, publishing as Pearson Addison-Wesley.
FINDING P-VALUES
Start
Left -tailed Left
What type of test
?
Two-tailed
Right-tailed
Is the test statistic to the right or left of
center ?
Right
P-value area to the left of the
test statistic
P-value twice the area to the left of the test statistic
P - value
P-value is twice this area.
P-value twice the area to the right of the test statistic
P-value area to the right of the
test statistic
P-value is twice this area.
P-value
Test statistic
Test statistic
Test statistic
Test statistic
Start HYPOTHESIS TEST: WORDING OF FINAL CONCLUSION
Wording of final conclusion
Does the original claim contain
the condition of equality ?
Yes
(Original claim contains equality)
Do you reject
H0?
Yes (Reject H0 )
"There is sufficient evidence to warrant rejection of the claim
that . . . (original claim)."
No (Original claim does not contain equality and becomes H1)
No (Fail to reject H0 )
"There is not sufficient evidence to warrant rejection of the claim
that . . . (original claim)."
Do you reject
H0?
Yes
"The sample data
support the claim
(Reject H0 ) that . . . (original claim)."
No (Fail to reject H0 )
"There is not sufficient sample evidence to support the claim
that . . . (original claim)."
(This is the only case in which the original claim
is rejected.)
(This is the only case in which the original claim
is supported.)
Inferences about M: choosing between t and normal distributions
t distribution:
s not known and normally distributed population
or s not known and n 30
Normal distribution: or
s known and normally distributed population s known and n 30
Nonparametric method or bootstrapping: Population not normally distrubted and n 30
Copyright 2007 Pearson Education, publishing as Pearson Addison-Wesley.
NEGATIVE z Scores
z
0
TABLE A-2 Standard Normal (z) Distribution: Cumulative Area from the LEFT
z
3.50 and lower 3.4 3.3 3.2 3.1 3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
.00
.0001 .0003 .0005 .0007 .0010 .0013 .0019 .0026 .0035 .0047 .0062 .0082 .0107 .0139 .0179 .0228 .0287 .0359 .0446 .0548 .0668 .0808 .0968 .1151 .1357 .1587 .1841 .2119 .2420 .2743 .3085 .3446 .3821 .4207 .4602 .5000
.01
.0003 .0005 .0007 .0009 .0013 .0018 .0025 .0034 .0045 .0060 .0080 .0104 .0136 .0174 .0222 .0281 .0351 .0436 .0537 .0655 .0793 .0951 .1131 .1335 .1562 .1814 .2090 .2389 .2709 .3050 .3409 .3783 .4168 .4562 .4960
.02
.0003 .0005 .0006 .0009 .0013 .0018 .0024 .0033 .0044 .0059 .0078 .0102 .0132 .0170 .0217 .0274 .0344 .0427 .0526 .0643 .0778 .0934 .1112 .1314 .1539 .1788 .2061 .2358 .2676 .3015 .3372 .3745 .4129 .4522 .4920
.03
.0003 .0004 .0006 .0009 .0012 .0017 .0023 .0032 .0043 .0057 .0075 .0099 .0129 .0166 .0212 .0268 .0336 .0418 .0516 .0630 .0764 .0918 .1093 .1292 .1515 .1762 .2033 .2327 .2643 .2981 .3336 .3707 .4090 .4483 .4880
.04
.05
.0003 .0003 .0004 .0004 .0006 .0006 .0008 .0008 .0012 .0011 .0016 .0016 .0023 .0022 .0031 .0030 .0041 .0040 .0055 .0054 .0073 .0071 .0096 .0094 .0125 .0122 .0162 .0158 .0207 .0202 .0262 .0256 .0329 .0322 .0409 .0401 .0505 * .0495 .0618 .0606 .0749 .0735 .0901 .0885 .1075 .1056 .1271 .1251 .1492 .1469 .1736 .1711 .2005 .1977 .2296 .2266 .2611 .2578 .2946 .2912 .3300 .3264 .3669 .3632 .4052 .4013 .4443 .4404 .4840 .4801
.06
.0003 .0004 .0006 .0008 .0011 .0015 .0021 .0029 .0039 .0052 .0069 .0091 .0119 .0154 .0197 .0250 .0314 .0392 .0485 .0594 .0721 .0869 .1038 .1230 .1446 .1685 .1949 .2236 .2546 .2877 .3228 .3594 .3974 .4364 .4761
.07
.08
.0003 .0003 .0004 .0004 .0005 .0005 .0008 .0007 .0011 .0010 .0015 .0014 .0021 .0020 .0028 .0027 .0038 .0037 .0051 * .0049 .0068 .0066 .0089 .0087 .0116 .0113 .0150 .0146 .0192 .0188 .0244 .0239 .0307 .0301 .0384 .0375 .0475 .0465 .0582 .0571 .0708 .0694 .0853 .0838 .1020 .1003 .1210 .1190 .1423 .1401 .1660 .1635 .1922 .1894 .2206 .2177 .2514 .2483 .2843 .2810 .3192 .3156 .3557 .3520 .3936 .3897 .4325 .4286 .4721 .4681
NOTE: For values of z below 3.49, use 0.0001 for the area.
*Use these common values that result from interpolation:
z score Area
1.645 0.0500 2.575 0.0050
.09
.0002 .0003 .0005 .0007 .0010 .0014 .0019 .0026 .0036 .0048 .0064 .0084 .0110 .0143 .0183 .0233 .0294 .0367 .0455 .0559 .0681 .0823 .0985 .1170 .1379 .1611 .1867 .2148 .2451 .2776 .3121 .3483 .3859 .4247 .4641
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