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