# Α 2 z 005 2 575 by the above table as bp 14 14 bq 1

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α2=z0.005=2.575,by the above table.Asbp=14%=0.14,bq=1-bp=1-0.14=0.86.Finally,n=(zα2)2bpbqE2=2.5752×0.14×0.860.022=1995.82 round it up to 1996.A. Bourhim (Syracuse University)Review-Test4MAT121, Spring 2015
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FindingbpandEfrom a CIIf a CI is given, thenbp=upper limit+lower limit2andE=upper limit-lower limit2ExampleFor the confidence interval0.772<p<0.776find the point estimatebpand the margin errorE.bp=upper limit+lower limit2=0.776+0.7722=0.774.E=upper limit-lower limit2=0.776-0.7722=0.002.A. Bourhim (Syracuse University)Review-Test4MAT121, Spring 2015
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Confidence Interval forμRequirements1The sample is a simple random sample.2Either the population isnormally distributedor the sample sizen>30.A. Bourhim (Syracuse University)Review-Test4MAT121, Spring 201510 / 21
FormulaAconfidence interval (CI)for the population meanμbased on a sample ofsizenisx-E<μ<x+E,wherexis the sample mean.Eis the margin error that depends on whetherσis known or not.1Ifσis known, thenE=zα2σpn.Here,zα2is thez-score for which the area to the right isα2.2Ifσis not known, thenE=tα2spn.Here,sis the sample standard deviation andtα2is thet-score for which thearea to the right isα2anddf=n-1.Keep in mind thatα=1-The confidence level.Round the confidence interval limits forμto three significant digits.A. Bourhim (Syracuse University)Review-Test4MAT121, Spring 201511 / 21
ExampleBased on a simple random sample of 32 cars, the average braking distance is137ft. Construct 95% confidence interval for the average braking distance ofALL cars if standard deviation ofALL braking distances of all cars is 7ft.Requirements.The sample is a simple random sample (From the text).n=32>30.Thus the requirements are there. Sinceσ=7 ft is given andx=137 , let usfindzα2and thenE. From the table, the correspondingz-score for 95%confidence level iszα2=z0.025=1.96.Thus,E=zα2σpn=1.96×7p32=2.4ft.And so,The lower limit of the interval is:x-E=137-2.4=135.4.The upper limit of the interval is:x+E=137+2.4=139.4.Finally,134.6<μ<139.4.A. Bourhim (Syracuse University)Review-Test4MAT121, Spring 201512 / 21
ExampleThe bus company at Syracuse wishes to estimate the mean journey time forthe service from downtown to Liverpool. Based on a simple random sample of35 chosen timed journeys, it was found that the mean journey time was 32minutes with a standard deviation of 4.6 minutes. Construct a 98% confidence
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