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Exam Number : A00-240

Exam Name : SAS Statistical Business Analysis SAS9: Regression and Model

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This test
is administered by SAS and Pearson VUE.

60 scored multiple-choice and short-answer questions.

(Must achieve score of 68 percent correct to pass)

In addition to the 60 scored items, there may be up to five unscored items.

Two hours to complete exam.

Use test
ID A00-240; required when registering with Pearson VUE.

ANOVA - 10%

Verify the assumptions of ANOVA

Analyze differences between population means using the GLM and TTEST procedures

Perform ANOVA post hoc test to evaluate treatment effect

Detect and analyze interactions between factors

Linear Regression - 20%

Fit a multiple linear regression model using the REG and GLM procedures

Analyze the output of the REG, PLM, and GLM procedures for multiple linear regression models

Use the REG or GLMSELECT procedure to perform model selection

Assess the validity of a given regression model through the use of diagnostic and residual analysis

Logistic Regression - 25%

Perform logistic regression with the LOGISTIC procedure

Optimize model performance through input selection

Interpret the output of the LOGISTIC procedure

Score new data sets using the LOGISTIC and PLM procedures

Prepare Inputs for Predictive Model Performance - 20%

Identify the potential challenges when preparing input data for a model

Use the DATA step to manipulate data with loops, arrays, conditional statements and functions

Improve the predictive power of categorical inputs

Screen variables for irrelevance and non-linear association using the CORR procedure

Screen variables for non-linearity using empirical logit plots

Measure Model Performance - 25%

Apply the principles of honest test
to model performance measurement

Assess classifier performance using the confusion matrix

Model selection and validation using training and validation data

Create and interpret graphs (ROC, lift, and gains charts) for model comparison and selection

Establish effective decision cut-off values for scoring

Verify the assumptions of ANOVA

Explain the central limit theorem and when it must be applied

Examine the distribution of continuous variables (histogram, box -whisker, Q-Q plots)

Describe the effect of skewness on the normal distribution

Define H0, H1, Type I/II error, statistical power, p-value

Describe the effect of trial
size on p-value and power

Interpret the results of hypothesis testing

Interpret histograms and normal probability charts

Draw conclusions about your data from histogram, box-whisker, and Q-Q plots

Identify the kinds of problems may be present in the data: (biased sample, outliers, extreme values)

For a given experiment, verify that the observations are independent

For a given experiment, verify the errors are normally distributed

Use the UNIVARIATE procedure to examine residuals

For a given experiment, verify all groups have equal response variance

Use the HOVTEST option of MEANS statement in PROC GLM to asses response variance

Analyze differences between population means using the GLM and TTEST procedures

Use the GLM Procedure to perform ANOVA

o CLASS statement

o MODEL statement

o MEANS statement

o OUTPUT statement

Evaluate the null hypothesis using the output of the GLM procedure

Interpret the statistical output of the GLM procedure (variance derived from MSE, Fvalue, p-value R**2, Levene's test)

Interpret the graphical output of the GLM procedure

Use the TTEST Procedure to compare means Perform ANOVA post hoc test to evaluate treatment effect

Use the LSMEANS statement in the GLM or PLM procedure to perform pairwise comparisons

Use PDIFF option of LSMEANS statement

Use ADJUST option of the LSMEANS statement (TUKEY and DUNNETT)

Interpret diffograms to evaluate pairwise comparisons

Interpret control plots to evaluate pairwise comparisons

Compare/Contrast use of pairwise T-Tests, Tukey and Dunnett comparison methods Detect and analyze interactions between factors

Use the GLM procedure to produce reports that will help determine the significance of the interaction between factors. MODEL statement

LSMEANS with SLICE=option (Also using PROC PLM)

ODS SELECT

Interpret the output of the GLM procedure to identify interaction between factors:

p-value

F Value

R Squared

TYPE I SS

TYPE III SS

Linear Regression - 20%

Fit a multiple linear regression model using the REG and GLM procedures

Use the REG procedure to fit a multiple linear regression model

Use the GLM procedure to fit a multiple linear regression model

Analyze the output of the REG, PLM, and GLM procedures for multiple linear regression models

Interpret REG or GLM procedure output for a multiple linear regression model:

convert models to algebraic expressions

Convert models to algebraic expressions

Identify missing degrees of freedom

Identify variance due to model/error, and total variance

Calculate a missing F value

Identify variable with largest impact to model

For output from two models, identify which model is better

Identify how much of the variation in the dependent variable is explained by the model

Conclusions that can be drawn from REG, GLM, or PLM output: (about H0, model quality, graphics)

Use the REG or GLMSELECT procedure to perform model selection

Use the SELECTION option of the model statement in the GLMSELECT procedure

Compare the differentmodel selection methods (STEPWISE, FORWARD, BACKWARD)

Enable ODS graphics to display graphs from the REG or GLMSELECT procedure

Identify best models by examining the graphical output (fit criterion from the REG or GLMSELECT procedure)

Assign names to models in the REG procedure (multiple model statements)

Assess the validity of a given regression model through the use of diagnostic and residual analysis

Explain the assumptions for linear regression

From a set of residuals plots, asses which assumption about the error terms has been violated

Use REG procedure MODEL statement options to identify influential observations (Student Residuals, Cook's D, DFFITS, DFBETAS)

Explain options for handling influential observations

Identify collinearity problems by examining REG procedure output

Use MODEL statement options to diagnose collinearity problems (VIF, COLLIN, COLLINOINT)

Logistic Regression - 25%

Perform logistic regression with the LOGISTIC procedure

Identify experiments that require analysis via logistic regression

Identify logistic regression assumptions

logistic regression concepts (log odds, logit transformation, sigmoidal relationship between p and X)

Use the LOGISTIC procedure to fit a binary logistic regression model (MODEL and CLASS statements)

Optimize model performance through input selection

Use the LOGISTIC procedure to fit a multiple logistic regression model

LOGISTIC procedure SELECTION=SCORE option

Perform Model Selection (STEPWISE, FORWARD, BACKWARD) within the LOGISTIC procedure

Interpret the output of the LOGISTIC procedure

Interpret the output from the LOGISTIC procedure for binary logistic regression models: Model Convergence section

Testing Global Null Hypothesis table

Type 3 Analysis of Effects table

Analysis of Maximum Likelihood Estimates table

Association of Predicted Probabilities and Observed Responses

Score new data sets using the LOGISTIC and PLM procedures

Use the SCORE statement in the PLM procedure to score new cases

Use the CODE statement in PROC LOGISTIC to score new data

Describe when you would use the SCORE statement vs the CODE statement in PROC LOGISTIC

Use the INMODEL/OUTMODEL options in PROC LOGISTIC

Explain how to score new data when you have developed a model from a biased sample

Prepare Inputs for Predictive Model

Performance - 20%

Identify the potential challenges when preparing input data for a model

Identify problems that missing values can cause in creating predictive models and scoring new data sets

Identify limitations of Complete Case Analysis

Explain problems caused by categorical variables with numerous levels

Discuss the problem of redundant variables

Discuss the problem of irrelevant and redundant variables

Discuss the non-linearities and the problems they create in predictive models

Discuss outliers and the problems they create in predictive models

Describe quasi-complete separation

Discuss the effect of interactions

Determine when it is necessary to oversample data

Use the DATA step to manipulate data with loops, arrays, conditional statements and functions

Use ARRAYs to create missing indicators

Use ARRAYS, LOOP, IF, and explicit OUTPUT statements

Improve the predictive power of categorical inputs

Reduce the number of levels of a categorical variable

Explain thresholding

Explain Greenacre's method

Cluster the levels of a categorical variable via Greenacre's method using the CLUSTER procedure

o METHOD=WARD option

o FREQ, VAR, ID statement

Use of ODS output to create an output data set

Convert categorical variables to continuous using smooth weight of evidence

Screen variables for irrelevance and non-linear association using the CORR procedure

Explain how Hoeffding's D and Spearman statistics can be used to find irrelevant variables and non-linear associations

Produce Spearman and Hoeffding's D statistic using the CORR procedure (VAR, WITH statement)

Interpret a scatter plot of Hoeffding's D and Spearman statistic to identify irrelevant variables and non-linear associations Screen variables for non-linearity using empirical logit plots

Use the RANK procedure to bin continuous input variables (GROUPS=, OUT= option; VAR, RANK statements)

Interpret RANK procedure output

Use the MEANS procedure to calculate the sum and means for the target cases and total events (NWAY option; CLASS, VAR, OUTPUT statements)

Create empirical logit plots with the SGPLOT procedure

Interpret empirical logit plots

Measure Model Performance - 25%

Apply the principles of honest test
to model performance measurement

Explain techniques to honestly assess classifier performance

Explain overfitting

Explain differences between validation and test data

Identify the impact of performing data preparation before data is split Assess classifier performance using the confusion matrix

Explain the confusion matrix

Define: Accuracy, Error Rate, Sensitivity, Specificity, PV+, PV-

Explain the effect of oversampling on the confusion matrix

Adjust the confusion matrix for oversampling

Model selection and validation using training and validation data

Divide data into training and validation data sets using the SURVEYSELECT procedure

Discuss the subset selection methods available in PROC LOGISTIC

Discuss methods to determine interactions (forward selection, with bar and @ notation)

Create interaction plot with the results from PROC LOGISTIC

Select the model with fit statistics (BIC, AIC, KS, Brier score)

Create and interpret graphs (ROC, lift, and gains charts) for model comparison and selection

Explain and interpret charts (ROC, Lift, Gains)

Create a ROC curve (OUTROC option of the SCORE statement in the LOGISTIC procedure)

Use the ROC and ROCCONTRAST statements to create an overlay plot of ROC curves for two or more models

Explain the concept of depth as it relates to the gains chart

Establish effective decision cut-off values for scoring

Illustrate a decision rule that maximizes the expected profit

Explain the profit matrix and how to use it to estimate the profit per scored customer

Calculate decision cutoffs using Bayes rule, given a profit matrix

Determine optimum cutoff values from profit plots

Given a profit matrix, and model results, determine the model with the highest average profit

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The leisure of this appendix gives textual content descriptions and SAS code examples that clarify some of the statistical ideas and terminology that you just may additionally come upon in the event you interpret the output of SAS procedures for fundamental information. For a more thorough dialogue, consult an introductory statistics textbook comparable to Mendenhall and Beaver (1994); Ott and Mendenhall; or Snedecor and Cochran (1989).

Populations and Parametersalways, there's a clearly described set of features during which you are interested. This set of aspects is known as the universe, and a group of values linked to these elements is called a population of values. The statistical term inhabitants has nothing to do with individuals per se. A statistical population is a set of values, not a set of americans. for instance, a universe is the entire college students at a selected school, and there could be two populations of interest: one in all peak values and one among weight values. Or, a universe is the set of all widgets manufactured by using a particular company, whereas the population of values may well be the length of time each and every widget is used before it fails.

A population of values will also be described in terms of its cumulative distribution feature, which offers the proportion of the inhabitants under or equal to each possible value. A discrete inhabitants can even be described by way of a likelihood feature, which gives the proportion of the inhabitants equal to every feasible value. A continual inhabitants can often be described through a density function, which is the by-product of the cumulative distribution feature. A density feature can be approximated via a histogram that offers the proportion of the inhabitants mendacity within every of a collection of intervals of values. A likelihood density feature is sort of a histogram with an unlimited variety of infinitely small intervals.

In technical literature, when the term distribution is used with out qualification, it frequently refers to the cumulative distribution feature. In casual writing, distribution sometimes potential the density feature instead. commonly the be aware distribution is used with no trouble to consult with an summary inhabitants of values in preference to some concrete population. as a consequence, the statistical literature refers to many kinds of summary distributions, reminiscent of typical distributions, exponential distributions, Cauchy distributions, and so forth. When a phrase similar to common distribution is used, it frequently does not count whether the cumulative distribution characteristic or the density function is supposed.

It can be expedient to explain a inhabitants in terms of a few measures that summarize wonderful elements of the distribution. One such measure, computed from the population values, is called a parameter. many different parameters will also be defined to measure distinct elements of a distribution.

probably the most regularly occurring parameter is the (arithmetic) imply. If the inhabitants incorporates a finite number of values, the population mean is computed because the sum of the entire values within the inhabitants divided by means of the variety of features in the inhabitants. For a limiteless population, the thought of the mean is identical but requires extra complex arithmetic.

E(x) denotes the mean of a inhabitants of values symbolized by means of x, comparable to peak, the place E stands for expected price. you could additionally trust anticipated values of derived features of the normal values. as an example, if x represents top, then is the expected cost of peak squared, this is, the imply value of the inhabitants received by way of squaring each price within the inhabitants of heights.

it is frequently unimaginable to measure all of the values in a population. a collection of measured values is referred to as a pattern. A mathematical characteristic of a trial of values is known as a statistic. A statistic is to a pattern as a parameter is to a population. it's general to denote information through Roman letters and parameters by means of Greek letters. as an example, the population imply is frequently written as , whereas the trial imply is written as . The field of records is basically involved with the look at of the behavior of pattern data.

Samples may also be selected in plenty of methods. Most SAS tactics assume that the information represent a simple random pattern, which capacity that the trial changed into selected in such a means that each one possible samples were equally likely to be chosen.

statistics from a trial may also be used to make inferences, or least expensive guesses, concerning the parameters of a population. for example, in case you take a random pattern of 30 college students from the high college, the mean height for these 30 students is an affordable wager, or estimate, of the suggest height of all of the students within the excessive college. other information, such because the average error, can provide assistance about how good an estimate is likely to be.

For any population parameter, a few statistics can estimate it. regularly, besides the fact that children, there's one specific statistic that is customarily used to estimate a given parameter. for example, the pattern imply is the common estimator of the population mean. in the case of the imply, the formulation for the parameter and the statistic are the equal. In different circumstances, the formulation for a parameter may be different from that of probably the most ordinary estimator. essentially the most familiar estimator isn't necessarily the most useful estimator in all applications.

Measures of vicinity include the mean, the median, and the mode. These measures describe the core of a distribution. in the definitions that follows, observe that if the total trial alterations by way of adding a set volume to every commentary, then these measures of place are shifted via the equal mounted volume.

The implyThe inhabitants suggest is constantly estimated via the pattern imply .

The MedianThe inhabitants median is the important cost, mendacity above and beneath half of the population values. The pattern median is the middle price when the records are arranged in ascending or descending order. For a good number of observations, the midpoint between both middle values is continually pronounced as the median.

The ModeThe mode is the cost at which the density of the inhabitants is at a optimum. Some densities have multiple native highest (top) and are said to be multimodal. The trial mode is the cost that occurs most commonly in the sample. by default, PROC UNIVARIATE reports the lowest such value if there's a tie for probably the most-regularly-happening trial value. PROC UNIVARIATE lists all viable modes for those who specify the MODES alternative in the PROC commentary. If the population is continual, then all pattern values happen as soon as, and the pattern mode has little use.

Percentiles, together with quantiles, quartiles, and the median, are effective for a detailed examine of a distribution. For a set of measurements organized in order of magnitude, the pth percentile is the cost that has p % of the measurements under it and (100-p) % above it. The median is the fiftieth percentile. because it may not be viable to divide your records in order that you get exactly the favored percentile, the UNIVARIATE procedure uses a greater precise definition.

The upper quartile of a distribution is the cost below which 75 % of the measurements fall (the 75th percentile). Twenty-five percent of the measurements fall under the lessen quartile price.In here illustration, SAS artificially generates the facts with a pseudorandom quantity characteristic. The UNIVARIATE procedure computes a lot of quantiles and measures of area, and outputs the values to a SAS facts set. an information step then uses the SYMPUT routine to assign the values of the records to macro variables. The macro %FORMGEN uses these macro variables to supply value labels for the layout technique. PROC CHART uses the resulting layout to display the values of the statistics on a histogram.

alternatives nodate pageno=1 linesize=64 pagesize=52; title 'illustration of Quantiles and Measures of location'; facts random; drop n; do n=1 to 1000; X=floor(exp(rannor(314159)*.eight+1.8)); output; conclusion; run; proc univariate information=random nextrobs=0; var x; output out=vicinity suggest=suggest mode=Mode median=Median q1=Q1 q3=Q3 p5=P5 p10=P10 p90=P90 p95=P95 max=Max; run; proc print statistics=vicinity noobs; run; information _null_; set vicinity; call symput('mean',round(suggest,1)); name symput('MODE',mode); name symput('MEDIAN',round(median,1)); call symput('Q1',circular(q1,1)); name symput('Q3',round(q3,1)); name symput('P5',circular(p5,1)); name symput('P10',circular(p10,1)); name symput('P90',circular(p90,1)); call symput('P95',circular(p95,1)); call symput('MAX',min(50,max)); run; %macro formgen; %do i=1 %to &max; %let value=&i; %if &i=&p5 %then %let value=&value P5; %if &i=&p10 %then %let cost=&price P10; %if &i=&q1 %then %let cost=&cost Q1; %if &i=&mode %then %let price=&price Mode; %if &i=&median %then %let cost=&value Median; %if &i=&mean %then %let price=&price mean; %if &i=&q3 %then %let price=&cost Q3; %if &i=&p90 %then %let cost=&price P90; %if &i=&p95 %then %let value=&cost P95; %if &i=&max %then %let value=>=&cost; &i="&value" %end; %mend; proc format print; cost stat %formgen; run; options pagesize=42 linesize=64; proc chart information=random; vbar x / midpoints=1 to &max by 1; structure x stat.; footnote 'P5 = fifth PERCENTILE'; footnote2 'P10 = tenth PERCENTILE'; footnote3 'P90 = ninetieth PERCENTILE'; footnote4 'P95 = 95th PERCENTILE'; footnote5 'Q1 = 1ST QUARTILE '; footnote6 'Q3 = third QUARTILE '; run;one more community of statistics is essential in discovering the distribution of a inhabitants. These data measure the variety, also referred to as the unfold, of values. within the definitions given in the sections that follow, word that if the total pattern is changed through the addition of a fixed quantity to every commentary, then the values of these facts are unchanged. If every observation in the trial is extended via a constant, however, the values of those data are as it should be rescaled.

The rangeThe pattern range is the difference between the biggest and smallest values within the sample. for a lot of populations, as a minimum in statistical theory, the latitude is countless, so the trial latitude might also not tell you tons in regards to the inhabitants. The pattern range tends to enhance because the pattern dimension increases. If all trial values are multiplied by means of a relentless, the trial latitude is accelerated by way of the identical steady.

The Interquartile rangeThe interquartile range is the change between the higher and lower quartiles. If all pattern values are multiplied by using a constant, the pattern interquartile range is improved by the same consistent.

The VarianceThe population variance, always denoted by way of , is the expected price of the squared change of the values from the inhabitants imply:

The pattern variance is denoted with the aid of . The change between a price and the mean is known as a deviation from the mean. as a result, the variance approximates the mean of the squared deviations.

When all of the values lie close to the suggest, the variance is small but by no means below zero. When values are greater scattered, the variance is larger. If all pattern values are elevated through a continuing, the trial variance is extended by way of the square of the consistent.

once in a while values other than are used within the denominator. The VARDEF= alternative controls what divisor the system makes use of.

The typical DeviationThe standard deviation is the square root of the variance, or root-imply-square deviation from the mean, in either a population or a sample. The ordinary symbols are for the population and s for a pattern. The usual deviation is expressed in the equal units because the observations, rather than in squared gadgets. If all pattern values are expanded by way of a relentless, the trial commonplace deviation is extended with the aid of the same regular.

Coefficient of editionThe coefficient of model is a unitless measure of relative variability. it is defined because the ratio of the standard deviation to the suggest expressed as a percentage. The coefficient of adaptation is significant best if the variable is measured on a ratio scale. If all pattern values are multiplied by way of a constant, the trial coefficient of edition continues to be unchanged.

SkewnessThe variance is a measure of the typical dimension of the deviations from the imply. considering the fact that the formula for the variance squares the deviations, both positive and negative deviations make a contribution to the variance in the identical manner. in lots of distributions, fine deviations may additionally are typically higher in magnitude than negative deviations, or vice versa. Skewness is a measure of the tendency of the deviations to be bigger in a single path than in the other. as an instance, the records within the last instance are skewed to the right.

population skewness is defined as

since the deviations are cubed in place of squared, the signals of the deviations are maintained. Cubing the deviations additionally emphasizes the effects of massive deviations. The formula comprises a divisor of to get rid of the impact of scale, so multiplying all values via a constant doesn't trade the skewness. Skewness can as a result be interpreted as a tendency for one tail of the inhabitants to be heavier than the other. Skewness will also be high quality or bad and is unbounded.

KurtosisThe heaviness of the tails of a distribution influences the behavior of many facts. hence it is advantageous to have a measure of tail heaviness. One such measure is kurtosis. The inhabitants kurtosis is always defined as

word: Some statisticians miss the subtraction of 3.

since the deviations are raised to the fourth vigor, fine and terrible deviations make the same contribution, while large deviations are strongly emphasized. as a result of the divisor , multiplying each cost by means of a relentless has no effect on kurtosis.

population kurtosis have to lie between and , inclusive. If represents population skewness and represents population kurtosis, then

Statistical literature occasionally studies that kurtosis measures the peakedness of a density. however, heavy tails have plenty extra affect on kurtosis than does the form of the distribution near the mean (Kaplansky 1945; Ali 1974; Johnson, et al. 1980).

pattern skewness and kurtosis are quite unreliable estimators of the corresponding parameters in small samples. they're more desirable estimators when your trial is terribly huge. despite the fact, massive values of skewness or kurtosis can also advantage attention even in small samples because such values point out that statistical strategies that are in response to normality assumptions could be inappropriate.

One primarily important family unit of theoretical distributions is the standard or Gaussian distribution. a traditional distribution is a smooth symmetric function regularly known as "bell-shaped." Its skewness and kurtosis are each zero. a traditional distribution can be completely certain by way of handiest two parameters: the mean and the regular deviation. about sixty eight percent of the values in a normal inhabitants are inside one standard deviation of the inhabitants imply; approximately 95 p.c of the values are inside two regular deviations of the imply; and about 99.7 p.c are inside three regular deviations. Use of the term regular to describe this selected kind of distribution does not indicate that other kinds of distributions are necessarily irregular or pathological.

Many statistical methods are designed under the belief that the population being sampled is perpetually disbursed. on the other hand, most precise-existence populations shouldn't have normal distributions. earlier than the usage of any statistical system in accordance with normality assumptions, make sure to consult the statistical literature to learn the way sensitive the components is to nonnormality and, if fundamental, check your pattern for facts of nonnormality.

In the following illustration, SAS generates a pattern from a standard distribution with a median of fifty and a standard deviation of 10. The UNIVARIATE procedure performs checks for area and normality. since the information are from a normal distribution, all p-values from the checks for normality are enhanced than 0.15. The CHART manner displays a histogram of the observations. The form of the histogram is a belllike, average density.

alternate options nodate pageno=1 linesize=sixty four pagesize=52; title 'ten thousand Obs trial from a standard Distribution'; title2 'with suggest=50 and common Deviation=10'; statistics normaldat; drop n; do n=1 to ten thousand; X=10*rannor(53124)+50; output; end; run; proc univariate records=normaldat nextrobs=0 typical mu0=50 loccount; var x; run; proc layout; graphic msd 20='20 3*Std' (noedit) 30='30 2*Std' (noedit) forty='40 1*Std' (noedit) 50='50 mean ' (noedit) 60='60 1*Std' (noedit) 70='70 2*Std' (noedit) 80='80 3*Std' (noedit) other=' '; run; alternatives linesize=sixty four pagesize=42; proc chart; vbar x / midpoints=20 to 80 by using 2; format x msd.; run;Sampling Distribution of the imply

if you again and again draw samples of size n from a inhabitants and compute the imply of every pattern, then the pattern potential themselves have a distribution. consider a new inhabitants which include the capability of all of the samples that might be drawn from the fashioned population. The distribution of this new population is referred to as a sampling distribution.

It can also be proven mathematically that if the normal inhabitants has mean and commonplace deviation , then the sampling distribution of the imply additionally has suggest , however its normal deviation is . The average deviation of the sampling distribution of the mean is called the common error of the mean. The usual error of the imply gives an illustration of the accuracy of a trial mean as an estimator of the population mean.

If the fashioned inhabitants has a standard distribution, then the sampling distribution of the suggest is additionally average. If the fashioned distribution is not typical but doesn't have excessively lengthy tails, then the sampling distribution of the mean can be approximated with the aid of a traditional distribution for huge trial sizes.

right here illustration incorporates three separate classes that show how the sampling distribution of the imply will also be approximated by means of a normal distribution because the trial measurement increases. the primary records step makes use of the RANEXP characteristic to create a pattern of a thousand observations from an exponential distribution.The theoretical inhabitants suggest is 1.00, whereas the pattern suggest is 1.01, to two decimal places. The population typical deviation is 1.00; the pattern standard deviation is 1.04.

here's an illustration of a nonnormal distribution. The inhabitants skewness is 2.00, which is close to the pattern skewness of 1.97. The inhabitants kurtosis is 6.00, but the pattern kurtosis is simply four.eighty.

alternate options nodate pageno=1 linesize=64 pagesize=forty two; title 'a thousand statement pattern'; title2 'from an Exponential Distribution'; statistics expodat; drop n; do n=1 to one thousand; X=ranexp(18746363); output; conclusion; run; proc format; cost axisfmt .05='0.05' .fifty five='0.fifty five' 1.05='1.05' 1.55='1.55' 2.05='2.05' 2.55='2.55' three.05='three.05' three.fifty five='three.fifty five' four.05='4.05' four.fifty five='four.fifty five' 5.05='5.05' 5.fifty five='5.fifty five' other=' '; run; proc chart data=expodat ; vbar x / axis=300 midpoints=0.05 to 5.fifty five through .1; structure x axisfmt.; run; options pagesize=sixty four; proc univariate statistics=expodat noextrobs=0 regular mu0=1; var x; run;The next facts step generates 1000 diverse samples from the identical exponential distribution. every trial includes ten observations. The potential manner computes the mean of each and every sample. in the records set that's created via PROC capability, each and every observation represents the mean of a trial of ten observations from an exponential distribution. as a consequence, the data set is a pattern from the sampling distribution of the imply for an exponential population.

PROC UNIVARIATE displays statistics for this trial of capability. observe that the suggest of the trial of potential is .99, virtually the equal because the mean of the normal inhabitants. Theoretically, the regular deviation of the sampling distribution is , whereas the general deviation of this trial from the sampling distribution is .30. The skewness (.55) and kurtosis (-.006) are nearer to zero in the pattern from the sampling distribution than within the common trial from the exponential distribution. here's so because the sampling distribution is closer to a traditional distribution than is the fashioned exponential distribution. The CHART method displays a histogram of the 1000-pattern potential. The form of the histogram is a good deal closer to a belllike, general density, but it is still particularly lopsided.

alternate options nodate pageno=1 linesize=64 pagesize=48; title 'a thousand pattern ability with 10 Obs per sample'; title2 'Drawn from an Exponential Distribution'; information samp10; drop n; do pattern=1 to 1000; do n=1 to 10; X=ranexp(433879); output; end; end; proc potential records=samp10 noprint; output out=mean10 imply=mean; var x; by using sample; run; proc format; cost axisfmt .05='0.05' .55='0.55' 1.05='1.05' 1.55='1.fifty five' 2.05='2.05' other=' '; run; proc chart facts=mean10; vbar mean/axis=300 midpoints=0.05 to 2.05 by using .1; layout mean axisfmt.; run; alternate options pagesize=sixty four; proc univariate facts=mean10 noextrobs=0 typical mu0=1; var suggest; run;In here facts step, the dimension of each and every trial from the exponential distribution is extended to 50. The commonplace deviation of the sampling distribution is smaller than in the outdated example since the size of each and every trial is greater. also, the sampling distribution is even closer to a traditional distribution, as can also be viewed from the histogram and the skewness.

alternate options nodate pageno=1 linesize=64 pagesize=forty eight; title 'a thousand trial ability with 50 Obs per sample'; title2 'Drawn from an Exponential Distribution'; statistics samp50; drop n; do sample=1 to one thousand; do n=1 to 50; X=ranexp(72437213); output; end; conclusion; proc skill information=samp50 noprint; output out=mean50 imply=suggest; var x; by using pattern; run; proc layout; cost axisfmt .05='0.05' .55='0.55' 1.05='1.05' 1.fifty five='1.55' 2.05='2.05' 2.55='2.55' different=' '; run; proc chart statistics=mean50; vbar mean / axis=300 midpoints=0.05 to 2.fifty five by using .1; format suggest axisfmt.; run; options pagesize=64; proc univariate facts=mean50 nextrobs=0 typical mu0=1; var suggest; run;The intention of the statistical strategies which have been discussed thus far is to estimate a population parameter with the aid of potential of a trial statistic. one more type of statistical strategies is used for testing hypotheses about inhabitants parameters or for measuring the amount of proof in opposition t a speculation.

consider the universe of scholars in a university. Let the variable X be the number of kilos through which a pupil's weight deviates from the most desirable weight for someone of the identical sex, top, and construct. You need to discover no matter if the population of scholars is, on the average, underweight or overweight. To this end, you've got taken a random pattern of X values from nine students, with results as given in the following statistics step:

title 'Deviations from usual Weight'; data x; enter X @@; datalines; -7 -2 1 3 6 10 15 21 30 ;that you may define a few hypotheses of hobby. One speculation is that, on the typical, the students are of exactly top-quality weight. If represents the population imply of the X values, you can write this speculation, known as the null hypothesis, as . The different two hypotheses, known as option hypotheses, are that the students are underweight on the usual, , and that the students are overweight on the standard, .

The null hypothesis is so called as a result of in many cases it corresponds to the assumption of "no effect" or "no change." despite the fact, this interpretation isn't appropriate for all testing issues. The null speculation is sort of a straw man that may also be toppled by using statistical proof. You come to a decision between the choice hypotheses in line with which means the straw man falls.

A naive method to strategy this issue can be to look on the pattern mean and choose among the many three hypotheses in response to the following rule:

The predicament with this strategy is that there may well be a high probability of creating an unsuitable resolution. If H0 is correct, you're essentially sure to make a incorrect determination because the chances of being exactly zero are nearly nil. If is a little lower than zero, so that H1 is true, there could be practically a 50 % opportunity that could be more suitable than zero in repeated sampling, so the chances of incorrectly deciding upon H2 would also be just about 50 percent. hence, you have got a high likelihood of making an error if is close zero. In such circumstances, there is not adequate proof to make a confident determination, so the most advantageous response could be to order judgment unless you can achieve extra facts.

The question is, how far from zero need to be that you should be able to make a confident decision? The reply may also be acquired by using because the sampling distribution of . If X has a roughly regular distribution, then has an about average sampling distribution. The imply of the sampling distribution of is . expect quickly that , the usual deviation of X, is accepted to be 12. Then the average error of for samples of nine observations is .

You know that about 95 percent of the values from a traditional distribution are inside two standard deviations of the mean, so about ninety five % of the viable samples of 9 X values have a pattern imply between and , or between -eight and eight. agree with the probabilities of making an error with the following resolution rule:

If H0 is true, then in about ninety five percent of the feasible samples might be between the vital values and eight, so that you will reserve judgment. In these circumstances the statistical evidence isn't potent enough to fell the straw man. within the different 5 percent of the samples you'll make an error; in 2.5 percent of the samples you'll incorrectly select H1, and in 2.5 percent you're going to incorrectly opt for H2.

The rate you pay for controlling the probabilities of making an error is the necessity of reserving judgment when there is not enough statistical facts to reject the null hypothesis.

magnitude and powerThe probability of rejecting the null hypothesis whether it is real is known as the type I error cost of the statistical examine and is usually denoted as . during this example, an value less than or stronger than eight is asserted to be statistically large on the 5 p.c level. that you would be able to alter the type I error rate in keeping with your wants by means of deciding on distinct vital values. for instance, essential values of -four and four would produce a value stage of about 32 %, whereas -12 and 12 would supply a sort I error expense of about 0.three p.c.

The decision rule is a two-tailed verify since the option hypotheses permit for population ability both smaller or larger than the price exact within the null hypothesis. in case you have been interested simplest in the chance of the students being obese on the regular, you might use a one-tailed check:

For this one-tailed examine, the type I error rate is 2.5 %, half that of both-tailed examine.

The probability of rejecting the null speculation if it is false is known as the energy of the statistical look at various and is usually denoted as . is known as the type II error rate, which is the chance of now not rejecting a false null hypothesis. The power depends on the real cost of the parameter. within the illustration, anticipate the inhabitants imply is 4. The power for detecting H2 is the chance of getting a trial suggest enhanced than 8. The important cost eight is one standard error better than the population suggest 4. The probability of getting a price at least one commonplace deviation more suitable than the imply from a normal distribution is about sixteen percent, so the vigor for detecting the alternative hypothesis H2 is ready sixteen %. If the population suggest have been 8, the vigour for H2 can be 50 %, whereas a population suggest of 12 would yield a power of about 84 %.

The smaller the type I error expense is, the less the opportunity of constructing an improper decision, but the higher the opportunity of getting to order judgment. In deciding upon a kind I error cost, be sure you agree with the resulting vigor for a variety of options of pastime.

student's t DistributionIn follow, you continually can't use any choice rule that uses a crucial cost in keeping with because you do not usually know the price of . you could, youngsters, use s as an estimate of . believe here statistic:

This t statistic is the change between the pattern suggest and the hypothesized imply divided by means of the estimated common error of the mean.

If the null hypothesis is correct and the population is always disbursed, then the t statistic has what's referred to as a pupil's t distribution with levels of freedom. This distribution appears very similar to a traditional distribution, however the tails of the student's t distribution are heavier. as the pattern size gets bigger, the trial normal deviation turns into a higher estimator of the population regular deviation, and the t distribution gets closer to a normal distribution.

which you can base a choice rule on the t statistic:

The value 2.3 was received from a desk of pupil's t distribution to provide a sort I error expense of 5 p.c for eight (that is, ) levels of freedom. Most usual records texts include a desk of pupil's t distribution. in case you shouldn't have a records textual content handy, that you could use the records step and the TINV function to print any values from the t distribution.

with the aid of default, PROC UNIVARIATE computes a t statistic for the null hypothesis that , together with connected records. Use the MU0= alternative within the PROC commentary to specify a further value for the null hypothesis.

This example makes use of the records on deviations from typical weight, which encompass 9 observations. First, PROC capacity computes the t statistic for the null hypothesis that . Then, the TINV feature in a data step computes the price of scholar's t distribution for a two-tailed examine on the 5 percent stage of value and 8 levels of freedom.

information devnorm; title 'Deviations from typical Weight'; input X @@; datalines; -7 -2 1 3 6 10 15 21 30 ; proc skill statistics=devnorm maxdec=three n imply std stderr t probt; run; title 'pupil''s t essential value'; data _null_; file print; t=tinv(.975,eight); put t 5.3; run;in the existing illustration, the price of the t statistic is 2.18, which is less than the vital t cost of 2.three (for a 5 percent importance stage and 8 degrees of freedom). thus, at a 5 percent importance degree you ought to reserve judgment. if you had elected to make use of a ten p.c significance degree, the essential cost of the t distribution would have been 1.86 and also you could have rejected the null hypothesis. The pattern dimension is so small, although, that the validity of your conclusion relies upon strongly on how shut the distribution of the population is to a normal distribution.

likelihood Valuesone more option to file the outcomes of a statistical test is to compute a likelihood value or p-price. A p-value gives the probability in repeated sampling of acquiring a statistic as a long way within the course(s) distinctive through the alternative hypothesis as is the value really followed. A two-tailed p-value for a t statistic is the chance of obtaining an absolute t value that is greater than the accompanied absolute t cost. A one-tailed p-price for a t statistic for the choice speculation is the chance of acquiring a t cost more advantageous than the accompanied t cost. as soon as the p-cost is computed, you can function a hypothesis test by evaluating the p-value with the desired value degree. If the p-price is lower than or equal to the class I error fee of the examine, the null hypothesis can also be rejected. the two-tailed p-cost, labeled Pr > |t| in the PROC capacity output, is .0606, so the null hypothesis may well be rejected at the 10 % importance degree however now not at the 5 percent degree.

A p-cost is a measure of the energy of the proof towards the null speculation. The smaller the p-price, the better the proof for rejecting the null hypothesis.

Copyright 1999 by using SAS Institute Inc., Cary, NC, united states. All rights reserved.

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