Statistics for management and Economics, Seventh Edition



Statistics for Management and Economics, Tenth Edition

Formulas

Numerical Descriptive techniques

Population mean

[pic] = [pic]

Sample mean

[pic]

Range

Largest observation - Smallest observation

Population variance

[pic]= [pic]

Sample variance

[pic] = [pic]

Population standard deviation

[pic] = [pic]

Sample standard deviation

s = [pic]

Population covariance

[pic]

Sample covariance

[pic]

Population coefficient of correlation

[pic]

Sample coefficient of correlation

[pic]

Coefficient of determination

R2 = r2

Slope coefficient

[pic]

y-intercept

[pic]

Probability

Conditional probability

P(A|B) = P(A and B)/P(B)

Complement rule

P([pic]) = 1 – P(A)

Multiplication rule

P(A and B) = P(A|B)P(B)

Addition rule

P(A or B) = P(A) + P(B) - P(A and B)

Bayes’ Law Formula

[pic]

Random Variables and Discrete Probability Distributions

Expected value (mean)

E(X) = [pic]

Variance

V(x) =[pic]

Standard deviation

[pic]

Covariance

COV(X, Y) = σxy = [pic]

Coefficient of Correlation

[pic]

Laws of expected value

1. E(c) = c

2. E(X + c) = E(X) + c

3. E(cX) = cE(X)

Laws of variance

1.V(c) = 0

2. V(X + c) = V(X)

3. V(cX) = [pic]V(X)

Laws of expected value and variance of the sum of two variables

1. E(X + Y) = E(X) + E(Y)

2. V(X + Y) = V(X) + V(Y) + 2COV(X, Y)

Laws of expected value and variance for the sum of more than two variables

1. [pic]

2. [pic]if the variables are independent

Mean and variance of a portfolio of two stocks

E(Rp) = w1E(R1) + w2E(R2)

V(Rp) = [pic]V(R1) + [pic]V(R2) + 2[pic][pic]COV(R1, R2)

= [pic][pic] + [pic][pic] + 2[pic][pic][pic][pic][pic]

Mean and variance of a portfolio of k stocks

E(Rp) = [pic]

V(Rp) = [pic]

Binomial probability

P(X = x) = [pic][pic]

[pic]

[pic]

[pic]

Poisson probability

P(X = x) = [pic]

Continuous Probability Distributions

Standard normal random variable

[pic]

Exponential distribution

[pic]

[pic]

[pic]

[pic]

F distribution

[pic]= [pic]

Sampling Distributions

Expected value of the sample mean

[pic][pic]

Variance of the sample mean

[pic][pic]

Standard error of the sample mean

[pic]

Standardizing the sample mean

[pic]

Expected value of the sample proportion

[pic][pic]

Variance of the sample proportion

[pic]

Standard error of the sample proportion

[pic]

Standardizing the sample proportion

[pic]

Expected value of the difference between two means

[pic][pic]

Variance of the difference between two means

[pic]

Standard error of the difference between two means

[pic]

Standardizing the difference between two sample means

[pic]

Introduction to Estimation

Confidence interval estimator of [pic]

[pic]

Sample size to estimate [pic]

[pic]

Introduction to Hypothesis Testing

Test statistic for[pic]

[pic]

Inference about One Population

Test statistic for [pic]

[pic]

Confidence interval estimator of [pic]

[pic]

Test statistic for [pic]

[pic]

Confidence interval Estimator of [pic]

LCL = [pic]

UCL = [pic]

Test statistic for p

[pic]

Confidence interval estimator of p

[pic]

Sample size to estimate p

[pic]

Confidence interval estimator of the total of a large finite population

[pic]

Confidence interval estimator of the total number of successes in a large finite population

[pic]

Inference About Two Populations

Equal-variances t-test of [pic]

[pic] [pic]

Equal-variances interval estimator of [pic]

[pic] [pic]

Unequal-variances t-test of [pic]

[pic] [pic]

Unequal-variances interval estimator of [pic]

[pic] [pic]

t-Test of [pic]

[pic] [pic]

t-Estimator of [pic]

[pic] [pic]

F-test of [pic]

F = [pic] [pic]and [pic]

F-Estimator of [pic]

LCL = [pic]

UCL = [pic]

z-Test and estimator of [pic]

Case 1: [pic]

Case 2: [pic]

z-estimator of [pic]

[pic]

Analysis of Variance

One-way analysis of variance

SST = [pic]

SSE = [pic]

MST = [pic]

MSE = [pic]

F = [pic]

Two-way analysis of variance (randomized block design of experiment)

SS(Total) = [pic]

SST =[pic]

SSB =[pic]

SSE = [pic]

MST = [pic]

MSB = [pic]

MSE = [pic]

F = [pic]

F= [pic]

Two-factor experiment

SS(Total) = [pic]

SS(A) = [pic]

SS(B) =[pic]

SS(AB) =[pic]

SSE = [pic]

F = [pic]

F = [pic]

F = [pic]

Least Significant Difference Comparison Method

LSD = [pic]

Tukey’s multiple comparison method

[pic]

Chi-Squared Tests

Test statistic for all procedures

[pic]

Simple Linear Regression

Sample slope

[pic]

Sample y-intercept

[pic]

Sum of squares for error

SSE = [pic]

Standard error of estimate

[pic]

Test statistic for the slope

[pic]

Standard error of [pic]

[pic]

Coefficient of determination

[pic] [pic]

Prediction interval

[pic]

Confidence interval estimator of the expected value of y

[pic]

Sample coefficient of correlation

[pic]

Test statistic for testing [pic] = 0

[pic]

Multiple Regression

Standard Error of Estimate

[pic]

Test statistic for [pic]

[pic]

Coefficient of Determination

[pic] [pic]

Adjusted Coefficient of Determination

Adjusted [pic]

Mean Square for Error

MSE = SSE/k

Mean Square for Regression

MSR = SSR/(n-k-1)

F-statistic

F = MSR/MSE

Durbin-Watson statistic

[pic]

Nonparametric Statistical techniques

Wilcoxon rank sum test statistic

[pic]

E(T) = [pic]

[pic]

[pic]

Sign test statistic

x = number of positive differences

[pic]

Wilcoxon signed rank sum test statistic

[pic]

E(T) = [pic]

[pic]

[pic]

Kruskal-Wallis Test

[pic]

Friedman Test

[pic]

Spearman rank correlation coefficient

[pic]

Spearman Test statistic for n > 30

[pic]

Time Series Analysis and Forecasting

Exponential smoothing

[pic]

Statistical Process Control

Centerline and control limits for [pic]chart using S

Centerline = [pic]

Lower control limit = [pic]

Upper control limit = [pic]

Centerline and control limits for the p chart

Centerline = [pic]

Lower control limit = [pic]

Upper control limit = [pic]

Decision Analysis

Expected Value of perfect Information

EVPI = EPPI - EMV*

Expected Value of Sample Information

EVSI = EMV' - EMV*

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