DeterminingSampleSizDeterminingSampleSize eWhat does Statistics Mean?Descriptive StatisticsNumber of PeopleTrends in EmploymentDataInferential StatisticsMake an inference about a population from a samplePopulation Parameter Versus Sample StatisticsCopyright 2000 by Harcourt, Inc. All rights reserved.Population ParameterVariables in a populationMeasured characteristics of a populationGreek lower-case letters as notationSample StatisticsVariables in a sampleMeasures computed from dataEnglish letters for notationMaking Data UsableFrequency DistributionsProportionsCentral TendencyMeanMedianModeMeasures of DispersionFrequency Distribution of Deposits Frequency (number ofpeople making depositsAmount in each range)less than $3,000 499$3,000 - $4,999 530$5,000 - $9,999 562$10,000 - $14,999 718$15,000 or more 8113,120Amount Percentless than $3,000 16$3,000 - $4,999 17$5,000 - $9,999 18$10,000 - $14,999 23$15,000 or more 26100Percentage Distribution of Amounts of DepositsAmount Probabilityless than $3,000 .16$3,000 - $4,999 .17$5,000 - $9,999 .18$10,000 - $14,999 .23$15,000 or more .261.00Probability Distribution of Amounts of DepositsMeasures of Central TendencyMean - Arithmetic Average, population; , sampleMedian - Midpoint of the DistributionMode - the Value that occurs most oftenPopulation MeanCopyright 2000 by Harcourt, Inc. All rights reserved.Sample MeanCopyright 2000 by Harcourt, Inc. All rights reserved.Number of Sales Calls Per Day by SalespersonsNumber ofSalesperson Sales callsMike 4Patty 3Billie 2Bob 5John 3Frank 3Chuck 1Samantha 526Sales for Products A and B, Both Average 200Product AProduct B196150198160199176199181200192200200200201201202201213201224202240202261Measures of DispersionThe RangeStandard DeviationMeasures of Dispersion or SpreadRangeMean absolute deviationVarianceStandard deviationThe Range as a Measure of SpreadThe range is the distance between the smallest and the largest value in the set.Range = largest value smallest value Deviation ScoresThe differences between each observation value and the mean:Low Dispersion Verses High Dispersion150 160 170 180 190 20021054321Low DispersionValue on VariableFrequency150 160 170 180 190 20021054321FrequencyHigh dispersionValue on VariableAverage DeviationCopyright 2000 by Harcourt, Inc. All rights reserved.Mean Squared DeviationCopyright 2000 by Harcourt, Inc. All rights reserved.The VarianceCopyright 2000 by Harcourt, Inc. All rights reserved.VarianceCopyright 2000 by Harcourt, Inc. All rights reserved.The variance is given in squared unitsThe standard deviation is the square root of variance:Sample Standard DeviationCopyright 2000 by Harcourt, Inc. All rights reserved.The Normal DistributionNormal CurveBell ShapedAlmost all of its values are within plus or minus 3 standard deviationsI.Q. is an exampleNormal DistributionMEAN2.14%13.59%34.13% 34.13%13.59%Normal Distribution2.14%Normal Curve: IQ Example 8511510014570Standardized Normal DistributionSymetrical about its meanMean identifies highest pointInfinite number of cases - a continuous distributionArea under curve has a probability density = 1.0Mean of zero, standard deviation of 1Standard Normal CurveThe curve is bell-shaped or symmetricalAbout 68% of the observations will fall within 1 standard deviation of the meanAbout 95% of the observations will fall within approximately 2 (1.96) standard deviations of the meanAlmost all of the observations will fall within 3 standard deviations of the meanA Standardized Normal Curve01-1-22zThe Standardized Normal is the Distribution of Z z+zStandardized ScoresCopyright 2000 by Harcourt, Inc. All rights reserved.Standardized ValuesUsed to compare an individual value to the population mean in units of the standard deviationLinear Transformation of Any Normal Variable into a Standardized Normal Variable-2 -1 0 1 2Sometimes thescale is stretchedSometimes thescale is shrunkmmssXCopyright 2000 by Harcourt, Inc. All rights reserved.Population DistributionSample DistributionSampling Distribution Population Distributionmxs-sSample Distribution_CXSSampling DistributionStandard Error of the MeanStandard deviation of the sampling distributionCENTRAL LIMIT THEORMCopyright 2000 by Harcourt, Inc. All rights reserved.Standard Error of the MeanCopyright 2000 by Harcourt, Inc. All rights reserved.Parameter EstimatesPoint EstimatesConfidence interval estimatesConfidence IntervalCopyright 2000 by Harcourt, Inc. All rights reserved.Copyright 2000 by Harcourt, Inc. All rights reserved.Copyright 2000 by Harcourt, Inc. All rights reserved.Copyright 2000 by Harcourt, Inc. All rights reserved.Estimating the Standard Error of the MeanCopyright 2000 by Harcourt, Inc. All rights reserved.Copyright 2000 by Harcourt, Inc. All rights reserved.Random Sampling Error and Sample Size are RelatedSample SizeVariance (Standard Deviation)Magnitude of ErrorConfidence LevelSample Size FormulaCopyright 2000 by Harcourt, Inc. All rights reserved.Sample Size FormulaCopyright 2000 by Harcourt, Inc. All rights reserved.Sample Size Formula - exampleSuppose a survey researcher, studying expenditures on lipstick, wishes to have a 95 percent confident level (Z) and a range of error (E) of less than $2.00. The estimate of the standard deviation is $29.00.Copyright 2000 by Harcourt, Inc. All rights reserved.Sample Size Formula - exampleCopyright 2000 by Harcourt, Inc. All rights reserved.Sample Size Formula - exampleSuppose, in the same example as the one before, the range of error (E) is acceptable at $4.00, sample size is reduced.Copyright 2000 by Harcourt, Inc. All rights reserved.Sample Size Formula - exampleCopyright 2000 by Harcourt, Inc. All rights reserved.99% ConfidenceCalculating Sample Size1389=265.372=253.742=2)29)(57. 2(n2=347=6325.182=453.742=4)29)(57. 2(n2=Copyright 2000 by Harcourt, Inc. All rights reserved.Standard Error of the ProportionCopyright 2000 by Harcourt, Inc. All rights reserved.Confidence Interval for a ProportionCopyright 2000 by Harcourt, Inc. All rights reserved.Sample Size for a ProportionCopyright 2000 by Harcourt, Inc. All rights reserved.