Chapter 2 Descriptive Statistics

Sample mean

Sample variance

Sample standard deviationCalculating the Sample Variance (Computational formula for s2)

Empirical Rule For a normally distributed population, this rule tells us that 68.26 percent, 95.44 percent, and 99.73 percent of the population measurements are within one, two, and three standard deviations, respectively, of the population mean.

Chebyshev’s theorem A theorem that (for any population) allows us to find an interval that contains a specified percentage of the individual measurements in the population.

z score

Coefficient of variation

pth percentile For a set of measurements arranged in increasing order, a value such that p percent of the measurements fall at or below the value, and (100 − p) percent of the measurements fall at or above the value.

Weighted mean

Sample mean for grouped data

Sample variance for grouped data

Population mean for grouped data

Population variance for grouped data

Chapter 3 Probability

Computing the Probability of an EventThe Rule of ComplementsThe Addition Rule

Mutually Exclusive EventsThe Addition rule for two mutually exclusive eventsThe Addition rule for N mutually exclusive eventsConditional probability

The General multiplication rule

Independent Events

The Multiplication rule for two independent eventsThe Multiplication rule for N independent eventsBayes’ theorem

Chapter 4 Discrete Random Variables

Properties of a Discrete Probability Distribution P(x)

The Mean, or Expected Value, of a Discrete Random Variable

The Variance and standard deviation of a discrete random variable

The Binomial Distribution

The Mean, Variance, and Standard Deviation of a Binomial Random VariableThe Poisson Distribution

The Mean, Variance, and Standard Deviation of a Poisson Random VariableThe Hypergeometric Distribution

The Mean and Variance of a Hypergeometric Random Variable

Chapter 5 Continuous Random Variables

Properties of a Continuous Probability DistributionThe Uniform Distribution

The Normal Probability Distribution

z values

The Standard Normal Distribution

Normal approximation to the binomial distribution

Consider a binomial random variable x where n is the number of trials and p is the probability of success. If np 5 and n(1 – p) 5, then x is approximately normal with mean = np and standard deviation

.

To standardize, use or

.

The Exponential Distribution and

Mean and standard deviation of an exponential distribution and

Chapter 6 Sampling Distributions

Sampling distribution of the sample mean

If x has mean and standard deviation , then has mean and standard deviation . In addition, if x follows a normal distribution, then also follows a normal distribution.

Standard deviation of the sampling distribution of the sample meanCentral limit theorem If the sample size n is sufficiently large (at least 30),

then will follow an approximately normal distribution with mean and standard deviation .

Sampling distribution of the sample proportion

If np 5 and n(1 – p) 5, then is approximately normal with mean = p and standard deviation

.

Standard deviation of the sampling distribution of the sample proportion

Chapter 7 Hypothesis Testing

Hypothesis Testing Steps

1. State the null and alternative hypotheses. 2. Specify the level of significance. 3. Select the test statistic. 4. Find the critical value (or compute the p-value). 5. Compare the value of the test statistic to the critical value (or the p-value to the level of significance) and decide whether to reject H0.

Hypothesis test about a population mean (σ known)Large-sample hypothesis test about a population proportionSampling distribution of

(independent random samples)

has mean

and standard deviation

Hypothesis test about a difference in population mean (σ1 and σ2 known)Large-sample hypothesis test about a difference in population proportions where p1

= p2 Large-sample hypothesis test about a difference in population proportions where p1

p2 Calculating the probability of a Type II errorSample-size determination to achieve specified values of α and β

Chapter 8 Comparing Population Means and Variances Using t Tests and F Ratios

t test about μ

t test about μ1 – μ2 when σ1

2 = σ22

t test about μ1 – μ2 when σ1

2 ≠σ22

Hypothesis test about μd

Sampling distribution of s1

2/s22 (independent

random samples)If , then has an F distribution with

df1 = n1 – 1 and df2 = n2 – 1.Hypothesis test about the equality of σ1

2 and σ2

2

For . For .

Chapter 9 Confidence Intervalsz-based confidence interval for a population mean μ with σ knownt-based confidence interval for a population mean μ with σ unknownSample size when estimating μ

Large-sample confidence interval for a population proportion pSample size when estimating pt-based confidence interval for μ1 – μ2 when σ1

2 = σ22

t-based confidence interval for μ1 – μ2 when σ1

2 ≠σ22

Large-sample confidence interval for a difference in population proportions

Chapter 10 Experimental Design and Analysis of Variance

One-way ANOVA sums of squares

,

The sum of squares total (SST) isThe between-groups mean square (MSB) isThe mean square error (MSE) isOne-way ANOVA F test

Estimation in one-way ANOVA: Individual 100(1 – ) confidence interval for

Estimation in one-way ANOVA: Tukey simultaneous 100(1 – ) confidence interval for

Estimation in one-way ANOVA: Individual 100(1 – ) confidence interval forRandomized block sums of squares

, ,

Estimation in a randomized block experiment: Individual 100(1 – ) confidence interval for

Estimation in a randomized block experiment: Tukey simultaneous 100(1 – ) confidence

interval for

Two-way ANOVA sums of squares

,

, ,

SSE = SST – SS(1) – SS(2) – SS(int)Estimation in two-way ANOVA: Individual 100(1 – ) confidence interval for

Estimation in two-way ANOVA: Tukey simultaneous 100(1 – ) confidence interval for factor 1Estimation in two-way ANOVA: Tukey simultaneous 100(1 – ) confidence interval for factor 2

Estimation in two-way ANOVA: Individual 100(1 – ) confidence interval for

Chapter 11 Correlation Coefficient and Simple Linear Regression Analysis

Least squares point estimates of β0 and β1

and

The predicted value of yi

Point estimate of a mean value of y at x = x0

Point prediction of an individual value of y at x = x0

Chapter 12 Multiple Regression

Chapter 13 Nonparametric Methods

Sign test for a population median

If , then S = number of sample measurements less than M0. If , then S = number of sample measurements greater than M0.

Large-sample sign test

Wilcoxon rank sum test If D1 shifted to the right of D2, then reject H0 if or .

If D1 shifted to the left of D2, then reject H0 if or .

If D1 shifted to the right or left of D2, then reject H0 if or .

Wilcoxon rank sum test (large-sample approximation) , ,

Wilcoxon signed ranks test = sum of the ranks associated with the negative paired differences

= sum of the ranks associated with the positive paired differences If D1 shifted to the right of D2, then reject H0 if T = . If D1 shifted to the left of D2, then reject H0 if T = . If D1 shifted to the right or left of D2, then reject H0 if T = the smaller of and is .

Kruskal-Wallis H test, ,

Kruskal-Wallis H statistic

Spearman’s rank correlation coefficient

Spearman’s rank correlation test, where

Chapter 14 Chi-Square Tests

Goodness of fit test for multinomial probabilities

Test for homogeneity

Goodness of fit test for a normal distribution

Chi-Square test for independence

Chapter 15 Decision Theory

Maximin criterion Find the worst possible payoff for each alternative and then choose the alternative that yields the maximum worst possible payoff.

Maximax criterion Find the best possible payoff for each alternative and then choose the alternative that yields the maximum best possible payoff.

Expected monetary value criterion Choose the alternative with the largest expected payoff.

Expected value of perfect information EVPI = expected payoff under certainty – expected payoff under risk

Expected value of sample information EVSI = EPS - EPNSExpected net gain of sampling ENGS = EVSI – cost of sampling

Chapter 16 Time Series Forecasting

No trendLinear trend

Quadratic trend

Modelling constant seasonal variation by using dummy variables

For a time series with k seasons, define k – 1 dummy variables in a multiple regression model. (e.g. for quarterly data, define three dummy variables)

Multiplicative decomposition methodSimple exponential smoothing

Double exponential smoothingMean absolute deviation (MAD)

Mean squared deviation (MSD)

Percentage error (PE)

Mean absolute percentage error (MAPE)

A simple index

An aggregate price index

A Laspeyres index is

A Paasche index is