Source: From Donald B. Owen, Handbook of Statistical Tables.
Formulas and Tables by Mario F. TriolaCopyright 2014 Pearson Education, Inc.
Degrees of Freedom
n - 1 Confidence Interval or Hypothesis Test with a standard deviation or variance
k - 1 Goodness-of-Fit with k categories
(r - 1)(c - 1) Contingency Table with r rows and c columns
k - 1 Kruskal-Wallis test with k samples
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Ch. 6: Normal Distribution
z =x - ms
or x - x
s Standard score
mx = m Central limit theorem
sx =s2n
Central limit theorem (Standard error)
Ch. 7: Confidence Intervals (one population)
pn - E 6 p 6 pn + E Proportion
where E = za>2B pnqnn
x - E 6 m 6 x + E Mean
where E = ta>2s1n
(s unknown)
or E = za>2s1n
(s known)
(n - 1)s 2
x2R
6 s 2 6(n - 1)s 2
x2L
Variance
Ch. 7: Sample Size Determination
n =3za>2420.25
E2 Proportion
n =3za>242pnqn
E2 Proportion (pn and qn are known)
n = J za>2s
ER 2
Mean
Ch. 9: Confidence Intervals (two populations)
( pn1 - pn2) - E 6 ( p1 - p2) 6 ( pn1 - pn2) + E
where E = za>2B pn1qn1
n1+
pn2qn2
n2
(x1 - x2) - E 6 (m1 - m2) 6 (x1 - x2) + E (Indep.)
where E = ta>2B s 21
n1+
s 22
n2
Ch. 3: Descriptive Statistics
x =Σxn
Mean
x =Σ( f # x)
Σf Mean (frequency table)
s = AΣ(x - x)2
n - 1 Standard deviation
s = An (Σx2) - (Σx)2
n (n - 1) Standard deviation (shortcut)
s = An 3Σ( f # x2)4 - 3Σ( f # x)42
n (n - 1)
Formulas and Tables by Mario F. TriolaCopyright 2014 Pearson Education, Inc.
Standard deviation (frequency table)
variance = s2
Ch. 4: Probability
P (A or B) = P (A) + P (B) if A, B are mutually exclusiveP (A or B) = P (A) + P (B) - P (A and B) if A, B are not mutually exclusiveP (A and B) = P (A) # P (B) if A, B are independentP (A and B) = P (A) # P(B 0A) if A, B are dependentP (A) = 1 - P (A) Rule of complements
nPr =n!
(n - r)! Permutations (no elements alike)
n!n1! n2! c nk!
Permutations (n1 alike, c)
nCr =n!
(n - r)! r ! Combinations
Ch. 5: Probability Distributions
m = Σ 3x # P (x)4 Mean (prob. dist.)
s = 2Σ 3x2 # P (x)4 - m2 Standard deviation (prob. dist.)
P (x) =n!
(n - x)! x ! # px # q n- x Binomial probability
m = n # p Mean (binomial)s2 = n # p # q Variance (binomial)s = 1n # p # q Standard deviation (binomial)
P (x) =mx # e -m
x !
Poisson distributionwhere e = 2.71828
(s1 and s2 unknown and not assumed equal)
E = ta>2B s 2p
n1+
s 2p
n2 (df = n1 + n2 - 2)
s 2p =
(n1 - 1)s 21 + (n2 - 1)s 2
2
(n1 - 1) + (n2 - 1)
(s1 and s2 unknown but assumed equal)
E = za>2Bs21
n1+s2
2
n2
(s1, s2 known)
d - E 6 md 6 d + E (Matched pairs)
where E = ta>2sd1n
(df = n - 1)
(df = smaller ofn1 - 1, n2 - 1)
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Ch. 10: Linear Correlation/Regression
Correlation r =nΣxy - (Σx)(Σy)2n(Σx2) - (Σx)22n(Σy2) - (Σy)2
or r = a 1zx zy2n - 1
Slope: b1 =nΣxy - (Σx)(Σy)
n (Σx2) - (Σx)2
or b1 = r sysx
y-Intercept:
b0 = y - b1x or b0 =(Σy)(Σx2) - (Σx)(Σxy)
n (Σx2) - (Σx)2
yn = b 0 + b1x Estimated eq. of regression line
r 2 =explained variation
total variation
se = BΣ(y - yn)2
n - 2 or BΣy2 - b 0Σy - b1Σxy
n - 2
yn - E 6 y 6 yn + E Prediction interval
where E = ta>2se B1 +1n
+n(x0 - x)2
n(Σx2) - (Σx)2
where zx = z score for x zy = z score for y
Ch. 8: Test Statistics (one population)
z =pn - pB pq
n
Proportion—one population
t =x - m
s1n
Mean—one population (s unknown)
z =x - ms1n
Mean—one population (s known)
x2 =(n - 1)s 2
s2 Standard deviation or variance—
one population
Ch. 9: Test Statistics (two populations)
z =(pn1 - pn2) - (p1 - p2)B p q
n1+
p qn2
Two proportions
t =(x1 - x2) - (m1 - m2)B s 2
1
n1+
s 22
n2
Two means—independent; s1 and s2 unknown, and not assumed equal.
t =(x1 - x2) - (m1- m2)B s 2
p
n1+
s 2p
n2
(df = n1 + n2 - 2)
p =x1 + x2
n1 + n2
df = smaller of n1 - 1, n2 - 1
Formulas and Tables by Mario F. TriolaCopyright 2014 Pearson Education, Inc.
Ch. 11: Goodness-of-Fit and Contingency Tables
x2 = g (O - E)2
E Goodness-of-fit (df = k - 1)
x2 = g (O - E)2
E Contingency table [df = (r - 1)(c - 1)]
where E =(row total)(column total)
(grand total)
x2 =( 0 b - c 0 - 1)2
b + c McNemar’s test for matched pairs (df = 1)
s 2p =
(n1 - 1)s 21 + (n2 - 1)s 2
2
n1 + n2 - 2 Two means—independent; s1 and s2 unknown, but
assumed equal.
z =(x1 - x2) - (m1 - m2)Bs 2
1n1
+s 2
2n2
Two means—independent;
s1, s2 known.
t =d - md
sd1n
Two means—matched pairs (df = n - 1)
Ch. 12: One-Way Analysis of Variance
Procedure for testing H0: m1 = m2 = m3 = 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-value 7 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 2 is rejected. If H0 from Step 2 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.
F =s 2
1
s 22
Standard deviation or variance— two populations (where s 2
1 Ú s 22)
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Ch. 13: Nonparametric Tests
z =(x + 0.5) - (n>2)1n
2
Sign test for n 7 25
z =T - n (n + 1)>4Bn (n + 1)(2n + 1)
24
Formulas and Tables by Mario F. TriolaCopyright 2014 Pearson Education, Inc.
z =R - mR
sR=
R-n1(n1 + n2 + 1)
2Bn1n2(n1 + n2 + 1)12
Wilcoxon rank-sum (two independent samples)
H =12
N(N + 1) aR2
1
n1+
R22
n2+ . . . +
R2k
nkb - 3(N + 1)
Kruskal-Wallis (chi-square df = k - 1)
rs = 1 -6Σd 2
n(n2 - 1) Rank correlation
acritical values for n 7 30: { z1n - 1
b
z =G - mG
sG=
G - a 2n1n2
n1 + n2+ 1bB (2n1n2)(2n1n2 - n1 - n2)
(n1 + n2)2(n1 + n2 - 1)
Runs testfor n 7 20
Ch. 14: Control Charts
R chart: Plot sample ranges
UCL: D4R
Centerline: R
LCL: D3R
x chart: Plot sample means
UCL: x + A2R
Centerline: x
LCL: x - A2R
p chart: Plot sample proportions
UCL: p + 3 Bp qn
Centerline: p
LCL: p - 3 Bp qn
Table A-6 Critical Values of the Pearson Correlation Coefficient r
n a = .05 a = .01
4 .950 .990
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 D3 D4 A2
2 0.000 3.267 1.880
3 0.000 2.574 1.023
4 0.000 2.282 0.729
5 0.000 2.114 0.577
6 0.000 2.004 0.483
7 0.076 1.924 0.419
Wilcoxon signed ranks (matched pairs and n 7 30)
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Overview of Statistical Methods
Inferences about M: choosing between t and normal distributions
t distribution: s not known and normally distributed populationor s not known and n 7 30
Normal distribution: s known and normally distributed populationor s known and n 7 30
Nonparametric method or bootstrapping: Population not normally distributed and n … 30
P-value P-value istwice this area.
Test statistic Test statistic Test statistic Test statistic
Left
Left-tailed Right-tailed
Two-tailed
P-value istwice this area.
Right
P-value
Two-tailed
Start
P-valuePP ue ishis area.
Left
P-valPPtwice th
L
P-vaPPtwice t
Righ
P-valuePPalue isthis area.
ht
Whattype of test
?
Is the test statistic
to the right or left ofcenter
?
P-value = areato the left ofthe teststatistic
P-value = twicethe area to the left of the teststatistic
P-value = twicethe area to the right of the teststatistic
P-value = areato the right of the teststatistic
Finding P-Values
Conclude1. Statistical Significance Do the results have statistical significance? Do the results have practical significance?
Analyze1. Graph the Data2. Explore the Data Are there any outliers (numbers very far away from almost all of the other data)? What important statistics summarize the data (such as the mean and standard
deviation)? How are the data distributed? Are there missing data? Did many selected subjects refuse to respond?3. Apply Statistical Methods Use technology to obtain results.
Prepare1. Context What do the data mean? What is the goal of study? 2. Source of the Data Are the data from a source with a special interest so that there is pressure to obtain
results that are favorable to the source?3. Sampling Method Were the data collected in a way that is unbiased, or were the data collected in a way
that is biased (such as a procedure in which respondents volunteer to participate)?
Conclude1. Statistical Significance Do the results have statistical significance? Do the results have practical significance?
Analyze1. Graph the Data2. Explore the Data Are there any outliers (numbers very far away from almost all of the other data)? What important statistics summarize the data (such as the mean and standard
deviation)? How are the data distributed? Are there missing data? Did many selected subjects refuse to respond?3. Apply Statistical Methods Use technology to obtain results.
Prepare1. Context What do the data mean? What is the goal of study? 2. Source of the Data Are the data from a source with a special interest so that there is pressure to obtain
results that are favorable to the source?3. Sampling Method Were the data collected in a way that is unbiased, or were the data collected in a way
that is biased (such as a procedure in which respondents volunteer to participate)?
Conclude1. Statistical Significance Do the results have statistical significance? Do the results have practical significance?
Analyze1. Graph the Data2. Explore the Data Are there any outliers (numbers very far away from almost all of the other data)? What important statistics summarize the data (such as the mean and standard
deviation)? How are the data distributed? Are there missing data? Did many selected subjects refuse to respond?3. Apply Statistical Methods Use technology to obtain results.
Prepare1. Context What do the data mean? What is the goal of study? 2. Source of the Data Are the data from a source with a special interest so that there is pressure to obtain
results that are favorable to the source?3. Sampling Method Were the data collected in a way that is unbiased, or were the data collected in a way
that is biased (such as a procedure in which respondents volunteer to participate)?
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complement of event A
null hypothesis
alternative hypothesis
a alpha; probability of a type I error or thearea of the critical region
b beta; probability of a type II error
r sample linear correlation coefficient
r rho; population linear correlation coefficient
coefficient of determination
multiple coefficient of determination
Spearman’s rank correlation coefficient
point estimate of the slope of the regressionline
point estimate of the y-intercept of theregression line
predicted value of y
d difference between two matched values
mean of the differences d found frommatched sample data
standard deviation of the differences dfound from matched sample data
standard error of estimate
T rank sum; used in the Wilcoxon signed-ranks test
H Kruskal-Wallis test statistic
R sum of the ranks for a sample; used in theWilcoxon rank-sum test
expected mean rank; used in the Wilcoxonrank-sum test
expected standard deviation of ranks; usedin the Wilcoxon rank-sum test
G number of runs in runs test for randomness
expected mean number of runs; used in runstest for randomness
expected standard deviation for the numberof runs; used in runs test for randomness
mean of the population of all possible sam-ple means
standard deviation of the population of allpossible sample means
E margin of error of the estimate of a popula-tion parameter, or expected value
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