D EFINITION M ARGIN OF ERROR \uf0a2 The margin of error denoted by e is the upper bound on the absolute difference between the estimator and the

# D efinition m argin of error  the margin of error

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DEFINITION: MARGIN OF ERRORThe margin oferror, denoted by e, is the upper bound on the absolute difference between the estimator and the parameter called the error of estimation37
NOTES ONMARGIN OFERRORClearly, the margin of error is . Note that we can also view the margin of error as one-half of the length of the interval. 382zen.
NOTES ONMARGIN OFERRORFor this reason, some define the margin of error as: estimate margin of error, in the confidence interval estimate . Some researchers report the margin of error without mentioning the size of the associated risk, . In such cases, it is understood that =0.05.39
CONFIDENCEINTERVAL FOR THEPROPORTIONIf the population proportion is not expected to be too close to 0 or 1 and the sample size n is large, then an approximate 100(1-α)% confidence interval estimator for the population proportion, p, is given by:where z/2is the z-value leaving an area of /2 to the right.40
EXAMPLE:In a random sample of 250 persons who took the civil service exam, 148 passed the exam. Find an approximate 99% confidence interval for the population proportion of persons who passed the exam.41
42Confidence Interval Estimate for the ProportionUSING PHStatDataSample Size250Number of Successes148Confidence Level95%Intermediate CalculationsSample Proportion0.592Z Value-1.95996398Standard Error of the Proportion0.031082857Interval Half Width0.06092128Confidence IntervalInterval Lower Limit0.53107872Interval Upper Limit0.65292128
CONFIDENCE INTERVAL FORTWOPOPULATIONSDifference of Means and Difference of ProportionsIn the same manner that we use a 100(1-α)% confidence interval estimator for the population mean of a single population to incorporate the assessment of the “goodness” of our estimate, the 100(1-α)% C.I. for the difference of two population means or proportions may also be constructed for the same purpose. 43
If we have two populations with means μ1and μ2and standard deviations σ1and σ2, respectively, a point estimator of the difference between μ1and μ2is the statistic Types of sampling:Selecting two independent samplesPaired sampling44X1-X2
C.I. FOR THE DIFFERENCE OF MEANS(TWOINDEPENDENT SAMPLES)Case 1: Population variances of the two populations are knownCase 2: Population variances are unknown but are assumed to be equal45(X1-X2)-zα/2σ12n1+σ22n2, (X1-X2)+zα/2σ12n1+σ22n2
Case 3: Population variances are once again unknown, but this time we cannot assume that these variances are equal46(X1-X2)-tα/2(v)S12n1+S22n2, (X1-X2)+tα/2(v)S12n1+S22n2v=(S12/n1+S22/n2)2(S12/n1)2n1-1+(S22/n2)2n2-1