Confidence Interval Estimation for a Population ProportionPoint EstimatesInterval EstimatesSlide 4Approximate 95% Confidence IntervalsThe Target AnalogySlide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18The Confidence IntervalSlide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34ExampleStandard Confidence LevelsSlide 37Confidence LevelProbability of ErrorSlide 40Which Confidence Interval is Best?Slide 42Slide 43Slide 44TI-83 – Confidence IntervalsSlide 46Slide 47Slide 48Confidence Interval Confidence Interval Estimation for a Estimation for a Population Population ProportionProportionLecture 33Lecture 33Section 9.4Section 9.4Mon, Nov 6, 2006Mon, Nov 6, 2006Point EstimatesPoint EstimatesPoint estimatePoint estimate – A single value of the – A single value of the statistic used to estimate the statistic used to estimate the parameter.parameter.The problem with point estimates is The problem with point estimates is that we have no idea how close we that we have no idea how close we can expect them to be to the can expect them to be to the parameter.parameter.That is, we have no idea of how large That is, we have no idea of how large the error may be.the error may be.Interval EstimatesInterval EstimatesInterval estimateInterval estimate – an interval of – an interval of numbers that has a stated numbers that has a stated probability (often 95%) of containing probability (often 95%) of containing the parameter.the parameter.An interval estimate is more An interval estimate is more informative than a point estimate.informative than a point estimate.Interval EstimatesInterval EstimatesConfidence levelConfidence level – The probability – The probability that is associated with the interval.that is associated with the interval.If the confidence level is 95%, then If the confidence level is 95%, then the interval is called a the interval is called a 95% 95% confidence intervalconfidence interval..Approximate 95% Approximate 95% Confidence IntervalsConfidence IntervalsHow do we find a 95% confidence How do we find a 95% confidence interval for interval for pp??Begin with the sample size Begin with the sample size nn and the and the sampling distribution of sampling distribution of pp^^..We know that the sampling We know that the sampling distribution is normal with mean distribution is normal with mean pp and standard deviationand standard deviation nppp1ˆThe Target AnalogyThe Target AnalogySuppose a shooter hits within 4 Suppose a shooter hits within 4 rings (4 inches) of the bull’s eye 95% rings (4 inches) of the bull’s eye 95% of the time.of the time.Then each individual shot has a 95% Then each individual shot has a 95% chance of hitting within 4 inches.chance of hitting within 4 inches.The Target AnalogyThe Target AnalogyThe Target AnalogyThe Target AnalogyThe Target AnalogyThe Target AnalogyThe Target AnalogyThe Target AnalogyThe Target AnalogyThe Target AnalogyThe Target AnalogyThe Target AnalogyThe Target AnalogyThe Target AnalogyNow suppose we are shown where Now suppose we are shown where the shot hit, but we are not shown the shot hit, but we are not shown where the bull’s eye is.where the bull’s eye is.What is the probability that the What is the probability that the bull’s eye is within 4 inches of that bull’s eye is within 4 inches of that shot?shot?The Target AnalogyThe Target AnalogyThe Target AnalogyThe Target AnalogyThe Target AnalogyThe Target AnalogyWhere is the bull’s eye?The Target AnalogyThe Target Analogy4 inchesThe Target AnalogyThe Target Analogy4 inches95% chance that thebull’s eye is within this circle.The Confidence IntervalThe Confidence IntervalIn a similar way, 95% of the sample In a similar way, 95% of the sample proportions proportions pp^^ should lie within 1.96 should lie within 1.96 standard deviations (standard deviations (pp^^) of the ) of the parameter parameter pp..The Confidence IntervalThe Confidence IntervalpThe Confidence IntervalThe Confidence Intervalp1.96 p^The Confidence IntervalThe Confidence Intervalp1.96 p^The Confidence IntervalThe Confidence Intervalp1.96 p^The Confidence IntervalThe Confidence Intervalp1.96 p^The Confidence IntervalThe Confidence Intervalp1.96 p^The Confidence IntervalThe Confidence Intervalp1.96 p^The Confidence IntervalThe Confidence IntervalTherefore, if we compute a single Therefore, if we compute a single pp^^, , then we expect that there is a 95% then we expect that there is a 95% chance that it lies within a distance chance that it lies within a distance 1.961.96pp^^ of of pp..The Confidence IntervalThe Confidence IntervalThe Confidence IntervalThe Confidence IntervalThe Confidence IntervalThe Confidence IntervalWhere is p?p^The Confidence IntervalThe Confidence Interval1.96 p^p^The Confidence IntervalThe Confidence Interval1.96 p^95% chance that p is within this intervalp^Approximate 95% Approximate 95% Confidence IntervalsConfidence IntervalsThus, the confidence interval isThus, the confidence interval isThe trouble is, to know The trouble is, to know pp^^, we must , we must know know pp. (See the formula for . (See the formula for pp^^.).)The best we can do is to use The best we can do is to use pp^^ in in place of place of pp to estimate to estimate pp^^..ppˆ96.1ˆApproximate 95% Approximate 95% Confidence IntervalsConfidence IntervalsThat is,That is,This is called the This is called the standard errorstandard error of of pp^^ and is denoted SE( and is denoted SE(pp^^).).Now the 95% confidence interval Now the 95% confidence interval isis npppˆ1ˆˆ ppˆSE96.1ˆExampleExampleExample 9.6, p. 585 – Study: Chronic Example 9.6, p. 585 – Study: Chronic Fatigue Common.Fatigue Common.Rework the problem supposing that Rework the problem supposing that 350 out of 3066 people reported that 350 out of 3066 people reported that they suffer from chronic fatigue they suffer from chronic fatigue syndrome.syndrome.How should we interpret the How should we interpret the confidence interval?confidence interval?Standard Confidence Standard Confidence LevelsLevelsThe standard confidence levels are The standard confidence levels are 90%, 95%, 99%, and 99.9%. (See p. 90%, 95%, 99%, and 99.9%. (See
View Full Document
Unlocking...