Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data
Chapter 2: Organizing and Summarizing Data
Relative frequency = frequencysum of all frequenciesClass midpoint: The sum of consecutive lower class limits divided by 2.
Chapter 3: Numerically Summarizing Data
Population Mean: μ= ΣxiNSample Mean: x = ΣxinRange = Largest Data Value – Smallest Data Value
Population Standard Deviation: σ = Σ(xi- μ)2NSample Standard Deviation: s = Σ(xi- x)2n-1Population Variance: σ2Sample Variance: s2
Empirical Rule: If the shape of the distribution is bell-shaped, then
Approximately 68% of the data will lie within 1 standard deviation of the mean
Approximately 95% of the data will lie within 2 standard deviations of the mean
Approximately 99.7% of the data will lie within 3 standard deviations of the mean
Population z-score: z = x- μσSample z-score: z = x- xsInterquartile Range: IQR = Q3 – Q1Lower Fences = Q1-1.5(IQR)
Upper Fence = Q3+1.5(IQR)Five-Number Summary
Minimum, Q1, M, Q3, MaximumChapter 4: Describing the Relation between Two Variables
Correlation Coefficient: -1 ≤ r ≤ 1
The equation of the least-squares regression line is y = b1x +b0,
where y is the predicted value, b1 is the slope, and b0 is the y-intercept.
Residual = observed y – predicted y
= y – yThe coefficient of determination, R2, measures the proportion of total variation in the response variable that is explained by the least-squares regression line.
Chapter 5: Probability
Empirical Probability
P(E) = frequency of Enumber of trials of experimentClassical Probability
P(E) = number of ways that E can occurnumber of possible outcomes = N(E)N(S)Addition Rule for Disjoint Events
P(E or F) = P(E) + P(F)
General Addition Rule
P(E or F) = P(E) + P(F) – P(E and F)
Compliment Rule
P(EC) = 1– P(E)
Multiplication Rule for Independent Events
P(E and F) = P(E)⋅P(F)
Conditional Probability Rule
P (F|E) = P(E and F)P(E)= N(E and F)N(E)General Multiplication Rule
P (E and F) = P(E) ⋅ P(F|E)
Chapter 6: Discrete Probability Distributions
Mean (Expected Value) of a Discrete Random Variable
μX = Σx ⋅ P(x)
Standard Deviation of a Discrete Random Variable
σX = Σ (x- μ)2 ⋅P(x)
Binomial Probability Distribution Function
13970002032000P(x) = n C x px(1-p)n-xMean and Standard Deviation of a Binomial Random Variable
μX=npσX = np(1-p)
Chapter 7: The Normal Distribution
Standardizing a Normal Random Variable
z = x – μσ
Finding the Score: x = μ+zσChapter 8: Sampling Distributions
Mean and Standard Deviation of the Sampling Distribution of xμx = μ andσx = σnSample Proportion: p = xnMean and Standard Deviation of the Sampling Distribution of pμp = p and σp = p(1-p)nChapter 9: Estimating the Value of a Parameter
Confidence Intervals
A (1 – α)⋅100% confidence interval about p is p ± zα/2⋅p(1 – p)nA (1 – α) ⋅100% confidence interval about μ is x ± tα/2⋅snNote: tα/2 is computed using n – 1 degrees of freedom.
Sample Size
To estimate the population proportion with a margin of error E at a (1 – α)⋅ 100% level of confidence:
n = p (1 – p) zα/2Ε2 rounded up to the next integer, where p is a prior estimate of the population proportion, or n = 0.25 zα/2Ε2rounded up to the next integer when no prior estimate of p is available.
To estimate the population mean with a margin of error E at a (1 – α) ⋅ 100% level of confidence:
n = zα/2⋅ sΕ2 rounded up to the next integer.
Chapter 10: Hypothesis Tests Regarding a Parameter
Test Statistics
z0 = p- p0p0 (1- p0)nt0= x – μ0 snChapter 11: Inferences on Two Samples
Test Statistic Comparing Two Population Proportions (Independent Samples)
z0 = p1- p2-(p1- p2)σ(p1-p2) where p = x1+ x2n1+ n2Confidence Interval for the Difference of Two Proportions (Independent Samples)
(p1- p2) ± zα/2 ⋅ σp1- p2Test Statistic for Matched-Pairs Data
t0= d- μdsdnwhere d is the mean and sd is the standard deviation of the differenced data.
Confidence Interval for Matched-Pairs Data
d ± tα2 ⋅ sdnNote: tα2 is found using n-1 degrees of freedom.
Test Statistic Comparing Two Means (Independent Sampling)
t0 = x1-x2-(μ1-μ2)s12n1+ s22n2Confidence Interval for the Difference of Two Means (Independent Samples)
(x1 – x2) ± tα2s12n1+ s22n2Note: tα2 is found using the smaller of n1-1 or n2-1 degrees of freedom.
Chapter 12: Additional Inferential Procedures
Chi-Square Procedures
Expected Counts (when testing for goodness of fit)
Εi = μi = npi for i = 1, 2, …, k
Expected frequencies (when testing for independence or homogeneity of proportions)
Expected frequency = (row total)(column total)table totalChi-Square Test Statistic
02 = (observed-expected)2expected = (Oi – Ei)2 Ei i = 1, 2, …, k
All Ei ≥ 1 and no more than 20% less than 5.
Tables:
Table II
Critical Values (CV) for Correlation Coefficient
n CV
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
Table VI
Critical Values for Normal Probability Plots
Sample Size, n Critical Value
5 0.880
6 0.888
7 0.898
8 0.906
9 0.912
10 0.918
11 0.923
12 0.928
13 0.932
14 0.935
15 0.939
16 0.941
17 0.944
18 0.946
19 0.949
20 0.951
21 0.952
22 0.954
23 0.956
24 0.957
25 0.959
30 0.960
