Chapter 2 Descriptive Statistics Sample mean Sample variance Sample standard deviation Calculating the Sample Variance (Computational formula for s 2 ) 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
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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)