Chapter 3 Descriptive Statistics

Population Mean

Sample Mean

Interquartile RangeQ3 – Q1 Sum of Deviations from the Arithmetic Mean is Always Zero

Mean Absolute Deviation

Population Variance

Population Standard Deviation

Empirical Rule*Distance from the Mean Values within the Distance

68%

95% 99.7%*Based on the assumption that the data are approximately normally distributed.

Chebyshev’s Theorem

Within k standard deviations of the mean, , lie at least

proportion of the values.Assumption: k > 1

Sample Variance

Sample Standard Deviation

Computational Formulas for Population Variance and Standard Deviation

Computational Formulas for Sample Variance and Standard Deviation

z Score

Coefficient of Variation

Mean of Grouped Data

wherei = the number of classesf = class frequencyM = class midpointN = total frequencies (total number of data values)

Medium of Grouped Data

wherel = lower endpoint of the class containing the medianw = width of the class containing the medianf = frequency of the class containing the medianF = cumulative frequency of classes preceding the

class containing the medianN = total frequencies (total number of data values)

Formulas for Population Variance and Standard Deviation of Grouped Data Original Formula Computational Version

wheref = frequencyM = class midpointN = ∑ , or total of the frequencies of the population

= grouped mean for the population

Formulas for Sample Variance and Standard Deviation of Grouped Data Original Formula Computational Version

wheref = frequencyM = class midpointN = ∑ , or total of the frequencies of the population

= grouped mean for the sample

Coefficient of Skewness

where

= coefficient of skewness

= median

Chapter 4 Probability

Classical Method of Assigning Probabilities

whereN = total possible number of outcomes of an

experiment

= the number of outcomes in which the event occurs out of N outcomes

Range of Possible Probabilities

Probability by Relative Frequency of Occurrence

Mutually Exclusive Events X and Y

Independent Events X and Y

Probability of the Complement of A

The mn Counting RuleFor an operation that can be done m ways and a second operation that can be done n ways, the two operations can then occur, in order, in mn ways. This rule can be extended to cases with three or more operations.

General Law of Addition

where X, Y, are events and is the intersection of X and Y.

Special Law of AdditionIf X, Y are mutually exclusive,

General Law of Multiplication

Special Law of MultiplicationIf X, Y are independent,

Law of Conditional Probability

Independent Events X, Y

If X and Y are independent events, the following must be true:

Bayes’ Rule

Chapter 5 Discrete Distributions

Mean or Expected Value of a Discrete Distribution

whereE(x) = long-run average x = an outcomeP(x) = probability of that outcome

Variance of a Discrete Distribution

wherex = an outcome

P(x) = probability of a given outcome= mean

Standard Deviation of a Discrete Distribution

Assumptions of the Binomial Distribution- The experiment involves n identical trials.- Each trial has only two possible outcomes denoted as

success or failure.- Each trial is independent of the previous trials.- The terms p and q remain constant throughout the

experiment, where the term p is the probability of getting a success on any one trial and the term q = 1 – p is the probability of getting a failure on any one trial.

Binomial Formula

wheren = the number of trials (or the number being

sampled)x = the number of successes desiredp = the probability of getting a success in one trial

= 1 – p = the probability of getting a failure in one trial

Mean and Standard Deviation of a Binomial Distribution

Poisson Formula

wherex = 0, 1, 2, 3, … = long-run average

e = 2.718281…Hypergeometric Formula

where

N = size of the populationn = sample sizeA = number of successes in the populationx = number of successes in the sample; sampling is done without replacement

Chapter 6 Continuous Distributions

Probability Density Function of a Uniform Distribution

Mean and Standard Deviation of a Uniform Distribution

Probabilities in a Uniform Distribution

where

z formula

Exponential Probability Density Function

where

x

and e = 2.271828…

Probabilities of the Right Tail of the Exponential Distribution

where

Chapter 7 Sampling and Sampling Distributions

Determining the Value of k

wheren = sample sizeN = population sizek = size of interval for selection

Central Limit TheoremIf samples of size n are drawn randomly from a population that has a mean of and a standard

deviation of , the sample means, , are approximately

normally distributed for sufficiently large samples (n

30*) regardless of the shape of the population distribution. If the population is normally distributed, the sample means are normally distributed for any sample size.From mathematical expectation, it can be shown that the mean of the sample means is the population mean:

and the standard deviation of the sample means (called the standard error of the mean) is the standard deviation of the population divided by the square root of the sample size:

z Formula for Sample Means

z Formula for Sample Means of a Finite Population

Sample Proportion

wherex = number of items in a sample that have the

characteristicn = number of items in the sample

z Formula for Sample Proportions for n and

n

where

= sample proportionn = sample sizep = population proportionq = 1 – p

Chapter 8 Statistical Inference: Estimation for Single Populations

100(1 – a)% Confidence Interval to Estimate : Known (8.1)

where = the area under the normal curve outside the

confidence interval area = the area in one end (tail) of the distribution

outside the confidence interval

Confidence Interval to Estimate Using the Finite Correction Factor (8.2)

Confidence Interval to Estimate : Population Standard Deviation Unknown and the Population Normally Distributed (8.3)

Confidence Interval to Estimate p (8.4)

where

= sample proportion

= sample sizep = population proportionn = sample size

Formula for Single Variance (8.5)

Confidence Interval to Estimate the Population Variance (8.6)

Sample Size When Estimating (8.7)

Sample Size When Estimating p (8.8)

wherep = population proportionq= 1 – p E = error of estimationn = sample size

Chapter 9 Statistical Inference: Hypothesis Testing for Single Populations

z Test for a Single Mean (9.1)

Formula to Test Hypotheses about with a Finite Population (9.2)

t Test for (9.3)

z Test of a Population Proportion (9.4)

where

= sample proportionp = population proportionq = 1 – p

Formula for Testing Hypotheses about a Population Variance (9.5)

Chapter 10 Statistical Inference: About Two Populations

z Formula for the Difference in Two Sample Means(Independent Samples and Population VariancesKnown) (10.1)

where = mean of population 1 = mean of population 2 = size of sample 1 = size of sample 2Confidence Interval to Estimate − (10.2)

t Formula to Test the Difference in Means Assuming and are Equal (10.3)

where

t Formula to Test the Difference in Means (10.4)

Confidence Interval to Estimate μ1 − μ2 Assumingthe Population Variances are Unknown and Equal(10.5)

t Formula to Test the Difference in two Dependent Populations (10.6)

wheren = number of pairsd = sample difference in pairsD = mean population differencesd = standard deviation of sample differenced = mean sample difference

Formulas for d and sd (10.7 and 10.8)

Confidence Interval Formula to Estimate the Difference in Related Populations, D (10.9)

z Formula for the Difference in Two PopulationProportions (10.10)

where

= proportion from sample 1

= proportion from sample 2

= size of sample 1

= size of sample 2

= proportion from population 1

= proportion from population 2

=

=

z Formula to Test the Difference in PopulationProportions (10.11)

where

and

Confidence Interval to Estimate p1 - p2 (10.12)

F Test for Two Population Variances (10.13)

Formula for Determining the Critical Value for the Lower-Tail F (10.14)

Chapter 12 Correlation and Simple Regression Analysis

(12.1) Pearson product-moment correlation coefficient

Equation of the simple regression line

Sum of squares

(12.2) Slope of the regression line

(12.3) Alternative formula for slope

(12.4) y intercept of the regression line

Sum of squares of error

Standard error of the estimate

(12.5) Coefficient of determination

Computational formula for r2

t test of slope

(12.6) Confidence interval to estimate E(yx) for a given value of x

(12.7) Prediction interval to estimate y for a given value of x