Chapter 2 Descriptive Statistics

Chapter 2 Descriptive Statistics

Sample mean /
Sample variance /
Sample standard deviation /
Calculating 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 Event /
The Rule of Complements /
The Addition Rule /
Mutually Exclusive Events /
The Addition rule for two mutually exclusive events /
The Addition rule for N mutually exclusive events /
Conditional probability /

The General multiplication rule /
Independent Events /

The Multiplication rule for two independent events /
The Multiplication rule for N independent events /
Bayes’ 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 Variable /
The Poisson Distribution /
The Mean, Variance, and Standard Deviation of a Poisson Random Variable /
The Hypergeometric Distribution /
The Mean and Variance of a Hypergeometric Random Variable /

Chapter 5 Continuous Random Variables

Properties of a Continuous Probability Distribution /
The 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 np5 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 meanand 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 mean /
Central 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 np5 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 proportion /
Sampling 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 error /
Sample-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 σ12 = σ22 /
t test about μ1 – μ2 when σ12 ≠σ22 /
Hypothesis test about μd /
Sampling distribution of s12/s22 (independent random samples) / If , then has an F distribution with
df1 = n1 – 1 and df2 = n2 – 1.
Hypothesis test about the equality of σ12 and σ22 / For. For.

Chapter 9 Confidence Intervals

z-based confidence interval for a population mean μ with σ known /
t-based confidence interval for a population mean μ with σ unknown /
Sample size when estimating μ /
Large-sample confidence interval for a population proportion p /
Sample size when estimating p /
t-based confidence interval for μ1 – μ2 when σ12 = σ22 /
t-based confidence interval for μ1 – μ2 when σ12 ≠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) is /
The between-groups mean square (MSB) is /
The mean square error (MSE) is /
One-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 for /
Randomized 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 1 /
Estimation 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 - EPNS
Expected net gain of sampling / ENGS = EVSI – cost of sampling

Chapter 16 Time Series Forecasting

No trend /
Linear 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 method /
Simple exponential smoothing /
Double exponential smoothing /
Mean 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 /