Formula Sheet for Statistics

Faculty of Computer Science and Information System

Formula Sheet for CPIT 603 (Quantitative Analysis)PROBABILITY

Probability of any event: 0 P (event) 1Permutation: subsets of r elements from n different elements

Permutation of similar objects: n1 is of one type, n2 is of second type, among n = n1+n2++nr elements

Combinations: subsets of size r from a set of n elements

P(AB) = P(A or B) = P(A) + P(B) P(A and B)P(ABC) = P(A) + P(B) + P(C) P(AB) P(AC) - P(BC) + P(ABC)P(B) = P(BA) + P(BA) = P(B|A)P(A)+P(B|A)P(A)For Mutually exclusive events AB=: P(AB) = P(A or B) = P(A) + P(B)Independent Events:P(AB) = P(A and B) = P(A)P(B)P(A|B) = P(A)P(B|A)=P(B)Dependent Events:P(A and B) = P(A) * P(B given A)P(AB)=P(A|B)P(B)=P(BA)=P(B|A)P(A)P(A and B and C) = P(A) * P(B | A) * P(C given A and B)

P(AB) = P(AB) = P(A | B) P(B) = P(B | A) P(A)

Bayes Theorem

A, B= any two eventsA= complement of A

Markovs InequalityIf X is a non-negative random variable with mean , then for any constant a > 0

Chebyshevs InequalityIf X is a random variable with a finite mean and variance 2, then for any constant a > 0

DECISION ANALYSIS

Criterion of RealismWeighted average =(best in row) + (1 )(worst in row)

For Minimization:Weighted average =(best in row) + (1 )(worst in row)Expected Monetary Value

Xi =payoff for the alternative in state of nature iP(Xi) =probability of achieving payoff Xi (i.e., probability of state of nature i) =summation symbol

EMV (alternative i) = (payoff of first state of nature) x (probability of first state of nature) + (payoff of second state of nature) x (probability of second state of nature) + + (payoff of last state of nature) x (probability of last state of nature)

Expected Value with Perfect InformationEVwPI = (best payoff in state of nature i)(probability of state of nature i)EVwPI = (best payoff for first state of nature) x (probability of first state of nature) + (best payoff for second state of nature) x (probability of second state of nature) + + (best payoff for last state of nature) x (probability of last state of nature)

Expected Value of Perfect InformationEVPI = EVwPI Best EMV

Expected Value of Sample Information EVSI = (EV with SI + cost) (EV without SI)

Utility of other outcome = (p)(utility of best outcome, which is 1) + (1 p)(utility of the worst outcome, which is 0)

REGRESSION MODELS

Error = (Actual value) (Predicted value)

Correlation Coefficient =

degrees of freedom for the numerator = df1 = kdegrees of freedom for the denominator = df2 = n k 1

Y = 0 + 1X1 + 2X2 + + kXk + Y =dependent variable (response variable)Xi =ith independent variable (predictor or explanatory variable)0 =intercept (value of Y when all Xi = 0)i =coefficient of the ith independent variablek =number of independent variables =random error

=predicted value of Yb0 =sample intercept (an estimate of 0)bi =sample coefficient of the i th variable (an estimate of i)

Where

Null hypothesis:

Test Statistic: : Reject null hypothesis if

Null hypothesis:


Multiple Regression

Hypothesis of ANOVA TestNull hypothesis:

Null hypothesis:


Single-Factor Experiments

Randomized Block Experiment

Two Factors experiment

Three-Factor Fixed Effects ModelSource of VariationSum of SquaresDegrees of FreedomMean SquareExpected Mean SquaresF0

ASSAa 1 MSA

BSSBb 1MSB

CSSCc 1 MSC

ABSSAB(a 1)(b 1)MSAB

ACSSAC(a 1)(c 1)MSAC

BCSSBC(b 1)(c 1)MSBC

ABCSSABC(a 1)(b 1)(c 1)MSABC

ErrorSSEabc(n 1)MSE

TotalSSTabcn 1

2k Factorial Designs(l) Represents the treatment combination with both factors at the low level.

2k Factorial Designs for k 3 factors

FORECASTING

INVENTORY CONTROL MODELS

Economic Order QuantityAnnual ordering cost = Annual holding cost

Total cost (TC) = Order cost + Holding cost

Cost of storing one unit of inventory for one year = Ch = IC, where C is the unit price or cost of an inventory item and I is Annual inventory holding charge as a percentage of unit price or cost

ROP without Safety Stock:Reorder Point (ROP) = Demand per day x Lead time for a new order in days d LInventory position = Inventory on hand + Inventory on orderEOQ without instantaneous receipt assumptionMaximum inventory level (Total produced during the production run) (Total used during the production run) (Daily production rate)(Number of days production) (Daily demand)(Number of days production) (pt) (dt)

Total produced Q pt

D = the annual demand in unitsQ number of pieces per order, or production run

Production Run Model: EOQ without instantaneous receipt assumptionAnnual holding cost Annual setup cost

Quantity Discount Model

If EOQ < Minimum for discount, adjust the quantity to Q = Minimum for discountTotal cost Material cost + Ordering cost + Holding cost

Holding cost per unit is based on cost, so Ch = ICWhere I = holding cost as a percentage of the unit cost (C)

Safety StockROP = Average demand during lead time + Safety StockService level = 1 Probability of a stockoutProbability of a stockout = 1 Service levelSafety Stock with Normal DistributionROP = (Average demand during lead time) + ZsdLTZ= number of standard deviations for a given service leveldLT= standard deviation of demand during the lead timeSafety stock = ZdLT

Demand is variable but lead time is constant

Demand is constant but lead time is variable

Both demand and lead time are variable

Total Annual Holding Cost with Safety StockTotal Annual Holding Cost = Holding cost of regular inventory + Holding cost of safety stock

The expected marginal profit = P(MP)The expected marginal loss = (1 P)(ML)The optimal decision ruleStock the additional unit if P(MP) (1 P)MLP(MP) ML P(ML)P(MP) + P(ML) MLP(MP + ML) ML

PROJECT MANAGEMENT

Expected Activity Time

Earliest finish time =Earliest start time + Expected activity timeEF =ES + tEarliest start =Largest of the earliest finish times of immediate predecessorsES =Largest EF of immediate predecessors

Latest start time = Latest finish time Expected activity timeLS =LF tLatest finish time =Smallest of latest start times for following activitiesLF =Smallest LS of following activities

Slack = LS ES, or Slack = LF EFProject Variance = sum of variances of activities on the critical path

Value of work completed = (Percentage of work complete) x (Total activity budget)Activity difference =Actual cost Value of work completed

WAITING LINES AND QUEUING THEORY MODELS

Single-Channel Model, Poisson Arrivals, Exponential Service Times (M/M/1)= mean number of arrivals per time period (arrival rate)= mean number of customers or units served per time period (service rate)The average number of customers or units in the system, L

The average time a customer spends in the system, W

The average number of customers in the queue, Lq

The average time a customer spends waiting in the queue, Wq

The utilization factor for the system, (rho), the probability the service facility is being used

The percent idle time, P0, or the probability no one is in the system

The probability that the number of customers in the system is greater than k, Pn>k

Multichannel Model, Poisson Arrivals, Exponential Service Times (M/M/m or M/M/s)m = number of channels open= average arrival rate = average service rate at each channelThe probability that there are zero customers in the system

The average number of customers or units in the system

The average time a unit spends in the waiting line or being served, in the system

The average number of customers or units in line waiting for service

The average number of customers or units in line waiting for service

The average number of customers or units in line waiting for service (Utilization rate)

Finite Population Model (M/M/1 with Finite Source)= mean arrival rate = mean service rateN = size of the populationProbability that the system is empty

Average length of the queue

Average number of customers (units) in the system

Average waiting time in the queue

Average time in the system

Probability of n units in the system

Total service cost = (Number of channels) x (Cost per channel)Total service cost = mCsm = number of channelsCs = service cost (labor cost) of each channel

Total waiting cost = (Total time spent waiting by all arrivals) x (Cost of waiting)= (Number of arrivals) x (Average wait per arrival)Cw= (W)Cw

Total waiting cost (based on time in queue) = (Wq)Cw

Total cost = Total service cost + Total waiting costTotal cost = mCs + WCw

Total cost (based on time in queue) = mCs + WqCw

Constant Service Time Model (M/D/1)Average length of the queue

Average waiting time in the queue

Average number of customers in the system

Average time in the system

Littles Flow EquationsL = W(or W = L/)Lq = Wq(or Wq = Lq/)

Average time in system = average time in queue + average time receiving serviceW = Wq + 1/

N(t) = Number of customers in queuing system at time t (t >= 0)Pn(t) = Probability of exactly n customers in queuing system at time t, given number at time 0.s = number of servers (parallel service channels) in queuing systemn= mean arrival rate (expected number of arrivals per unit time) of new customers when n customers are in system= expected interarrival time n= mean service rate for overall system (expected number of customers completing service per unit time) when n customers are in system. Represents combined rate at which all busy servers (those serving customers) achieve service completions= expected service timeUtilization factor = = /(s)Steady-state condition

Impact of Exponential distribution on Queuing Model

T1, T2, be independent service-time random variables having an exponential distribution with parameter, and letSn+1 = T1 + T2 + + Tn+1, for n = 0, 1, 2, Sn+1 represents the conditional waiting time given n customers already in the system. Sn+1 is known to have an Erlang distribution.

M/M/s, with s > 1

Finite Queue Variation of M/M/s (M/M/s/K model)Finite Queue: Number of customers in the system cannot exceed a specified number, K.Queue capacity is K-s

M/M/1/K

M/M/s/K

Finite Calling Population variation of M/M/sM/M/1 with finite population

M/M/s with finite population and s > 1

M/G/1 Model

M/D/s Model

M/Ek/s Model

T1, T2, , Tk are k independent random variables with an identical exponential distribution whose mean is 1/(k).T = T1 + T2 + + Tk, has an Erlang distribution with parameters and k. Exponential and degenerate (constant) are special cases of Erlang distribution with k=1 and k=respectively.M/Ek/1 Model

Nonpreemptive Priorities Model

Jackson NetworksWith m service facilities where facility i (i=1, 2, .,., m)1. Infinite Queue2. Customers arriving from outside the system according to a Poisson input process with parameters ai3. si servers with an exponential service-time distribution with parameter A Customer leaving facility i is routed next to facility j (j= 1, 2, , m) with probability pij or departs the system with probability

Jackson network behaves as if it were an independent M/M/s queueing system with arrival rate

Preemptive Priorities Model

MARKOV ANALYSIS

(i)=vector of state probabilities for period i= (1, 2, 3, , n)wheren= number of states1, 2, , n=probability of being in state 1, state 2, , state nPij =conditional probability of being in state j in the future given the current state of i

For any period n we can compute the state probabilities for period n + 1 (n + 1) = (n)PEquilibrium condition = P

Fundamental MatrixF = (I B)1Inverse of Matrix

r = ad bcM represent the amount of money that is in each of the nonabsorbing statesM = (M1, M2, M3, , Mn)n= number of nonabsorbing statesM1= amount in the first state or categoryM2= amount in the second state or categoryMn= amount in the nth state or category

Partition of Matrix for absorbing states

I = identity matrixO = a matrix with all 0sComputing lambda and the consistency index

Consistency Ratio

Stochastic process {Xt} is said to have the Markovian property if P{Xt+1 = j | X0 = k0, X1 = k1, , Xt-1 = kt-1, Xt = i} = P{Xt+1=j|Xt=i}, for t=0,1, and every sequence i, j, k0, k1, , kt-1.Stochastic process {Xt} (t = 0, 1, ) is a Markov chain if it has the Markovian property.

Pij = P{Xt+1 = j | Xt = i}n-step transition probabilities:

n-step transition matrix:

STATISTICAL QUALITY CONTROL

= mean of the sample meansz = number of normal standard deviations (2 for 95.5% confidence, 3 for 99.7%)

= standard deviation of the sampling distribution of the sample means =

= average of the samplesA2 = Mean factor

= mean of the sample means

UCLR= upper control chart limit for the rangeLCLR= lower control chart limit for the rangeD4 and D3 = Upper range and lower rangep-charts

= mean proportion or fraction defective in the sample

z = number of standard deviations

= standard deviation of the sampling distribution

is estimated by Estimated standard deviation of a binomial distribution where n is the size of each sample

c-charts

The mean is and the standard deviation is equal to

To compute the control limits we use (3 is used for 99.7% and 2 is used for 95.5%)

Range of the sample = Xmax - Xmin

Control Chart Modelk is the distance of control limits from the center line, expressed in Standard Deviation units. Common choice is k = 3.

OTHERS

Computing lambda and the consistency index

Consistency Ratio

The input to one stage is also the output from another stagesn1 = Output from stage n The transformation functiontn = Transformation function at stage n General formula to move from one stage to another using the transformation functionsn1 = tn (sn, dn) The total return at any stagefn = Total return at stage n Transformation Functions

Return Equations

Probability of breaking even

P(loss)= P(demand < break-even) P(profit) = P(demand > break-even)

whereK = loss per unit when sales are below the break-even pointX = sales in units

Using the unit normal loss integral, EOL can be computed usingEOL = KN(D)EOL =expected opportunity lossK = loss per unit when sales are below the break-even point = standard deviation of the distributionN(D) =value for the unit normal loss integral for a given value of D

Determinant Value = (a)(d) (c)(b)

Determinant Value = aei + bfg + cdh gec hfa idb

Equation for a lineY = a + bXwhere b is the slope of the lineGiven any two points (X1, Y1) and (X2, Y2)

For the Nonlinear functionY = X2 4X + 6 Find the slope using two points and this equation

Total cost (Total ordering cost) + (Total holding cost) + (Total purchase cost)

Q = order quantityD = annual demandCo = ordering cost per orderCh = holding cost per unit per yearC = purchase (material) cost per unitEconomic Order Quantity

32NameWhen to UseApproximations / ConditionsProbability Mass function, Mean and Variance

E(X) is Expected Value = Mean; Xi = random variables possible values; P(Xi) = Probability of each of the random variables possible values

Cumulative F(x) = ; xi x

Uniform (Discrete)Equal probabilityFinite number of possible values

For a series of n values, f(x) = 1/ n, a bFor a range that starts from a and ends with b (a, a+1, a+2, , b) and a b

Binomial / Bernoulli (Discrete)Bernoulli Trials: Each trial is independent Probability of success in a trial is constant Only two possible outcomesUnknown: Number of successesKnown: Number of trials Number of trials that result in a successIf n is large (np > 5, n(1-p) > 5), approximate binomial to normal. P(X x) = P(X x+0.5)P(x X) = P(x-0.5 X)If n is large & p is small, approximate to Poisson as = np

Binomial expansion Probability of r success in n trials x = 0,1,,n, 0 p 1, n = 1,2,

Expected value (mean) E(X) = = npVariance = V(X) = = np(1 p)

Geometric (Discrete)Bernoulli trial; Memoryless Number of trails until first successx = 1,2,,n, 0 p 1Expected value (mean) = E(X) = = 1/p Variance = V(X) = = (1 p)/p2

Negative Geometric (Discrete)Unknown: Number of trialsKnown: Number of success Number of trials required to obtain r successesx = r,r+1,r+2,, 0 p 1E(X) = = r/p= Variance = r(1-p)/p2

Hypergeometric (Discrete)Trials are not independentWithout replacement Number of success in the sampleV(X) = V(X) of binomial * ((N-n)/(N-1)) where ((N-n)/(N-1)) is called finite population correction factorn 5, n(1-p) > 5 and (n/N) < 0.1 x =max(0,n-N+k) to min(K,n), K N, n NK objects classed as successes; N K objects classified as failures; Sample size of n objectsE(X) = = npwhere p = K/N

Poisson (Discrete)Poisson Process: Probability of more than one event in a subinterval is zero. Probability of one event in a subinterval is constant & proportional to length of subinterval Event in each subinterval is independent Number of events in the intervalArrival rate does not change over time; Arrival pattern does not follow regular pattern; Arrival of disjoint time intervals are independent.Approximated to normal if > 5

x = 0,1,2,, 0 < P(X) = probability of exactly X arrivals or occurrencesExpected value = Variance =

Taylor series:

NameWhen to UseApproximations / ConditionsProbability Density function, Mean and Variance

P(x1 X x2) = P(x1 X x2) = P(x1 X x2) = P(x1 X x2)

Cumulative F(x) = P(Xx) = ; for -< x 5, n(1-p) > 5), binomial is approximated to normal.

Adding or subtracting 0.5 is called continuity correction.Normal is approximated to Poisson if > 5

- < x < - < < , 0 < E(X) = V(X) =Standard normal means mean = = 0 and variance = = 1

Cumulative distribution of a standard normal variable - < < + = 68% -2 < < +2 = 95%-3 < < +3 = 99.7%

Exponential (Continuous)Memoryless

distance between successive events of Poisson process with mean > 0 length until first count in a Poisson process

for 0 x for x < 0

The probability that an exponentially distributed time (X) required to serve a customer is less than or equal to time t is given by the formula,

Erlang (Continuous)r shape scaleTime between events are independent length until r counts in a Poisson process or exponential distribution

For mean and variance: Exponential multiplied by r gives Erlang.

P(X>0.1) = 1 F(0.1)If r = 1, Erlang random variable is an exponential random variable

Gamma (Continuous)For r is an integer (r=1,2,), gamma is ErlangErlang random variable is time until the rth event in a Poisson process and time between events are independentFor = , r = , 1, 3/2, 2, gamma is chi-squareGamma Function

E(X) and V(X) = E(X) and V(X) of exponential distribution multiplied by r

Weibull (Continuous)Includes memory property- Scale; - Shape Time until failure of many different physical systems=1, Weibull is identical to exponential=2, Weibull is identical to Raleigh

x > 0,

Lognormal (Continuous)Includes memory propertyX = exp(W); W is normally distributed with mean and variance ln(X) = W; X is lognormalEasier to understand than WeibullWeibull can be approximated to lognormal with and

Beta (Continuous)Flexible but bounded over a finite range

Power Law (Continuous)Called as heavy-tailed distribution.f(x) decreases rapidly with x but not as rapid as exponential distribution.A random variable described by its minimum value xmin and a scale parameter > 1 is is said to obey the power law distribution if its density function is given by

Normalize the function for a given set of parameters to ensure that

Central Limit TheoremIf X1, X2, , Xn is a random sample of size n taken from a population (either finite or infinite) with mean and finite variance 2, and if is the sample mean, the limiting form of the distribution of

as n , is the standard normal distribution.

NameProbability Density function, Mean and Variance

Two or more Discrete Random Variables

Joint Probability Mass Fn: for all points (x1,x2,,xp) in the range of X1,X2,,Xp

Joint Probability Mass For subset: for all points in the range of X1,X2,,Xp for which X1=x1, X2=x2,, Xk=xk

Marginal Probability Mass Function:

Mean:Variance:

Multinomial Probability DistributionThe random experiment that generates the probability distribution consists of a series of independent trials. However, the results from each trial can be categorized into one of k classes.

Two or more Continuous Random Variables

Marginal Probability Density Function:

Independence:

Joint Probability Density Fn:

Joint Probability Mass For subset: for all points in the range of X1,X2,,Xp for which X1=x1, X2=x2,, Xk=xk

Marginal Probability Density Function:where the integral is over all points of X1,X2,,Xp for which Xi=xi

Mean:Variance:

Covariance is a measure of linear relationship between the random variables. If the relationship between the random variables is nonlinear, the covariance might not be sensitive to the relationship.Two random variables with nonzero correlation are said to be correlated. Similar to covariance, the correlation is a measure of the linear relationship between random variables.

Covariance:Correlation:

If X and Y are independent random variables,

Bivariate Normal

for - < x < and - < y < , with parameters x > 0, y > 0, - < X < , - < Y < and -1 < < 1. Marginal Distribution: If X and Y have a bivariate normal distribution with joint probability density fXY(x, y;X,Y,X,Y, ), the marginal probability distribution of X and Y are normal with means x and y and standard deviation x and y, respectively.Conditional Distribution: If X and Y have a bivariate normal distribution with joint probability density fXY(x, y;X,Y,X,Y, ), the conditional probability distribution of Y given X = x is normal with mean

and varianceCorrelation: If X and Y have a bivariate normal distribution with joint probability density function fXY(x, y;X,Y,X,Y, ), the correlation between X and Y is If X and Y have a bivariate normal distribution with = 0, X and Y are independent

Linear Functions of random variablesGiven random variables X1, X2, , Xp and constants c1,c2, , cp, Y = c1X1 + c2X2+ + cpXp is a linear combination of X1, X2, , XpMean E(Y) = c1E(X1) + c2E(X2) + + cpE(Xp)

Variance:

If X1, X2, , Xp are independent, variance:

Mean and variance on average:

General Functions of random variablesDiscrete: y = h(x) and x = u(y):

Continuous: y = h(x) and x = u(y):

NameConfidence IntervalSample SizeOne-Sided Confidence Bounds

Type I Error: Rejecting the null Hypothesis H0 when it is true; Type II Error: Failing to reject null hypothesis H0 when it is false.Probability of Type I Error = = P(type I error) = Significance level = -error = -level = size of the test.Probability of Type II Error = = (type II error)Power = Probability of rejecting the null hypothesis H0 when the alternative hypothesis is true = 1 - = Probability of correctly rejecting a false null hypothesis.P-value = Smallest level of significance that would lead to the rejection of the null hypothesis H0.

Normal Distribution with mean unknown and variance known

100(1-)% upper confidence bound for

100(1-)% lower confidence bound for

Large scale sample size n: Using central limit theorem, has approximately a normal distribution with mean and variance /n.

Null hypothesis: H0:= 0

Test Statistic: Alternative hypothesisp-valueRejection Criteria

H1: 0Probability above |z0| & below -|z0|, P = 2[1 - (|z0|)]

H1: 0Probability above z0P = 1 - (z0)

H1: 0Probability below z0P = (z0)

Probability of Type II Error for a two-sided test

Sample size for a two-sided test Sample size for a one-sided test Large set: for n > 40, replace sample standard deviation s for .Parameter for Operating Characteristic Curves:

Normal Distribution with mean unknown and variance unknownt distribution (Similar to normal in symmetry and unimodal. But t distribution is heavier tails than normal) with n-1 degrees of freedom. k = n-1

Mean = 0Variance=k/(k-2) for k>2

Finding s: Can be obtained only using trial and error as s is unknown until the data is collected.100(1-)% upper confidence bound for




H1: 0Probability above |t0| & below -|t0|

H1: 0Probability above t0

H1: 0Probability below t0

Parameter for Operating Characteristic Curves:

Probability of Type II Error for a two-sided test

When the true value of mean = + , the distribution for T0 is called the noncentral t distribution with n-1 degrees of freedom and noncentrality parameter . If = 0, the noncentral t distribution reduces to the usual central t distribution. denotes the noncentral t random variable.

Normal Distribution;CI on variance and Standard deviationdistribution with n-1 degrees of freedom. k = n-1

Mean = kVariance = 2k

upper and lower 100/2 percentage points of chi-square distribution with n-1 degrees of freedom.100(1-)% upper confidence bound for


Null hypothesis:

Test Statistic: Alternative hypothesisRejection Criteria

H1:

H1:

H1:

Parameter for Operating Characteristic Curves:

Large scale CI for a population proportionNormal approximation for a binomial proportion: If n is large, the distribution of

is approximately standard normal. is the proportional population. Mean = p. Variance = p(1-p)/n

p can be computed as from a preliminary sample or use the maximum value of p, which is 0.5.



Null hypothesis:

Test Statistic:

Alternative hypothesisp-valueRejection Criteria

H1:p p0Probability above |z0| & below -|z0|, P = 2[1 - (|z0|)]

H1: p p0Probability above z0P = 1 - (z0)

H1: p p0Probability below z0P = (z0)

Probability of Type II Error for a two-sided test Sample size for a two-sided test Sample size for a one-sided test

Inference of the difference in means of two normal distributions, variance known



Probability of Type II Error for a two-sided test Sample size for a two-sided test, with n1n2, Sample size for a two-sided test, with n1n2, Sample size for a one-sided test, with n1=n2, Parameter for Operating Characteristic Curves:



H1: 0Probability above |z0| & below -|z0|,P = 2[1 - (|z0|)]

H1: 0Probability above z0P = 1 - (z0)

H1: 0Probability below z0P = (z0)

Inference of the difference in means of two normal distributions, variance unknown

Pooled estimator of 2, denoted by , is:

Null hypothesis: H0:= 0H1: 0

Test Statistic: has t distribution with n1+n2-2 degrees of freedom; Called as pooled t-testAlternative hypothesisp-valueRejection Criteria

H1: 0Probability above |t0| & below -|t0|

H1: 0Probability above t0

H1: 0Probability below t0

If variances are not assumed equal

If H0:= 0 is true, the statistic with t degrees of freedom given by

Goodness of Fit Test StatisticWhere Oi is the observed frequency and Ei is the expected frequency in the ith class.Approximated to Chi-square distribution with k-p-1 degrees of freedom. p represents the number of parameters. If test statistic is large, we reject null hypothesis. P-value is .

Expected Frequency of each cell

For large n, the statistic P-value is .

Prediction Interval

Prediction interval for Xn+1 will always be longer than the CI for because there is more variability associated with the prediction error than with the error of estimation.

Tolerance Interval

Where k is a tolerance interval factor for the given confidence.

Sign TestNumber of positive differences = r+If P-value is less than some preselected level , we will reject H0.

Normal approximation for sign test statistic:

Null hypothesis:

P-value: One-sided hypothesis:

P-value: Two-sided hypothesis:

If r+ < n/2, P-value:

If r+ > n/2, P-value:

Wilcoxon Signed-Rank Test

Null hypothesis: Sort based on differences; Give the ranks the signs of their corresponding differences.Sum of Positive Ranks: W+; Absolute value of Sum of Negative Ranks: W-. W = min(W+,W-)

Reject Null hypothesis, if observed value of statistic

For one-sided tests, Reject Null hypothesis, if observed value of statistic

For one-sided tests, Reject Null hypothesis, if observed value of statistic

Normal approximation for Wilcoxon signed-rank test statistic: Arrange all n1 and n2 observations in ascending order of magnitude and assign ranks to them. If two or more observations are tied(identical), use the mean of the ranks that would have assigned if the observations differed. W1 is sum of ranks in smaller sample. W2 is sum of ranks in other sample. Reject null hypothesis, if w1 or w2 is less than or equal to tabulated critical value w.

For one-sided hypothesis: reject H0 if w1 wFor reject H0 if w2 wNormal approximation when n1 and n2 > 8,

Paired t-test

Null hypothesis: H0:D= 0

Test Statistic: where is the sample average of n differences D1, D2, , Dn, and SD is the sample standard deviation of these differences.Alternative hypothesisp-valueRejection Criteria

H1:D 0Probability above |t0| & below -|t0|

H1:D 0Probability above t0

H1:D 0Probability below t0

Inference on the variances of two normal distributions (F Distribution)Let W and Y be independent chi-square random variables with u and v degrees of freedom respectively.Ratio has the probability density function

Mean Variance Lower tail percentage point, F Distribution n1-1 numerator degrees of freedom and n2-1 denominator degrees of freedomNull hypothesis: Test Statistic: Alternative hypothesisRejection Criteria

H1:

H1:

H1:

P-value is the area (probability) under the F distribution with n1-1 and n2-1 degrees of freedom that lies beyond the computed value of the test statistic f0.

Inference on the population proportionsNull hypothesis:

Test Statistic: Alternative hypothesisp-valueRejection Criteria for Fixed-Level tests

H1:p1 p2Probability above |z0| & below -|z0|,P = 2[1 - (|z0|)]

H1: p1 p2Probability above z0P = 1 - (z0)

H1: p1 p2Probability below z0P = (z0)

Probability of Type II Error for a two-sided test Where and Sample size for a two-sided test where q1 = 1 p1 and q2 = 1 p2

Inferences:1. Population is normal: Sign test or t-test.a. t-test has the smallest value of for a significance level , thus t-test is superior to other tests.2. Population is symmetric but not normal (but with finite mean):a. t-test will have the smaller (or a higher power) than sign test.b. Wilcoxon Signed-rank test is comparable to t-test.3. Distribution with heavier tails:a. Wilcoxon Signed-rank test is better than t-test as t-test depends on the sample mean, which is unstable in heavy-tailed distributions.4. Distribution is not close to normal:a. Wilcoxon Signed-rank test is preferred.5. Paired observations:a. Both sign test and Wilcoxon Signed-rank test can be applied. In sign test, median of the differences is equal to zero in null hypothesis. In Wilcoxon Signed-rank test, mean of the differences is equal to zero in null hypothesis.6.

Formula Sheet for Statistics

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