Presentation on theme: "Statistics for Business and Economics"— Presentation transcript:
1Statistics for Business and Economics Estimation of Population Parameters: Confidence Intervals Chapter 8
2Learning Objectives 1. State What Is Estimated 2. Distinguish Point & Interval Estimates3. Explain Interval Estimates4. Compute Confidence Interval Estimates for Population Mean & Proportion5. Compute Sample SizeAs a result of this class, you will be able to ...
3Thinking ChallengeSuppose you’re interested in the average amount of money that students in this class (the population) have on them. How would you find out?
5I am 95% confident that mX is between 40 & 60. Estimation ProcessPopulationRandom SampleI am 95% confident that mX is between 40 & 60.Mean `X = 50Mean, mX, is unknown
6Unknown Population Parameters Are Estimated Estimate Populationwith SampleParameter...StatisticMeanm`xxProportionpps2Variance2ssxDifferencesm- m`x-`x1212
7Estimation Methods Estimation Point Interval Estimation Estimation ConfidenceBoot-PredictionIntervalstrappingInterval
8Point Estimation 1. Provides Single Value Based on Observations from 1 Sample2. Gives No Information about How Close Value Is to the Unknown Population Parameter3. Sample Mean`X = 3 Is Point Estimate of Unknown Population Mean
9Estimation Methods Estimation Point Interval Estimation Estimation ConfidenceBoot-PredictionIntervalstrappingInterval
10Interval Estimation 1. Provides Range of Values Based on Observations from 1 Sample2. Gives Information about Closeness to Unknown Population ParameterStated in terms of ProbabilityKnowing Exact Closeness Requires Knowing Unknown Population Parameter3. e.g., Unknown Population Mean Lies Between 50 & 70 with 95% Confidence
11Key Elements of Interval Estimation A Probability That the Population Parameter Falls Somewhere Within the Interval.Sample Statistic (Point Estimate)Confidence IntervalConfidence Limit (Lower)Confidence Limit (Upper)
13Many Samples Have Same Confidence Interval `X = mx ± Zs`xsx_`Xmx-2.58s`x mx-1.65s`xmxmx+1.65s`x mx+2.58s`xmx-1.96s`xmx+1.96s`x90% Samples95% Samples99% Samples
14Level of Confidence1. Probability that the Unknown Population Parameter Falls Within Interval2. Denoted (1 - a) %a Is Probability That Parameter Is Not Within Interval3. Typical Values Are 99%, 95%, 90%
15Intervals & Level of Confidence _Sampling Distribution of Meansxa/2a/21 -a_Notice that the interval width is determined by 1-a in the sampling distribution.Xm=m`xxIntervals Extend from `X - Zs`X to `X + Zs`X(1 - a) % of Intervals Contain mX .a % Do Not.Large Number of Intervals
18Confidence Interval Estimate Mean (sX Known) 1. AssumptionsPopulation Standard Deviation Is KnownPopulation Is Normally DistributedIf Not Normal, Can Be Approximated by Normal Distribution (n ³ 30)2. Confidence Interval EstimatessXXX-Z×mX+Z×a/2Xa/2nn
19Estimation Example Mean (sX Known) The mean of a random sample of n = 25 is`X = 50. Set up a 95% confidence interval estimate for mX if sX = 10.ssXXX-Z×mX+Z×a/2Xa/2nn101050-1.96×m50+1.96×X252546.08m53.92X
23Confidence Interval Estimate Mean (sX Unknown) 1. AssumptionsPopulation Standard Deviation Is UnknownPopulation Must Be Normally Distributed2. Use Student’s t Distribution3. Confidence Interval EstimateSSX-t×mX+t×a/2,n-1Xa/2,n-1nn
24Student’s t Distribution Standard NormalBell-ShapedSymmetric‘Fatter’ Tailst (df = 13)t (df = 5)Zt
25Student’s t Table .05 2 t a / 2 .05 2.920 t Values Upper Tail Area df Assume: n = 3 df = n - 1 = 2 a = .10 a/2 =.05a / 2Upper Tail Areadf.25.10.05Confidence intervals use a/2, so divide a!11.0003.0786.31420.8171.8862.920.0530.7651.6382.353t2.920t Values
26Degrees of Freedom (df) 1. Number of Observations that Are Free to Vary After Sample Statistic Has Been Calculated2. ExampleSum of 3 Numbers Is 6 X = 1 (or Any Number) X = 2 (or Any Number) X = 3 (Cannot Vary) Sum = 6degrees of freedom = n -1 = 3 -1 = 2
27Estimation Example Mean (sX Unknown) A random sample of n = 25 has`X = 50 & S = 8. Set up a 95% confidence interval estimate for mX.SSX-t×mX+t×a/2,n-1Xa/2,n-1nn8850-2.0639×m50+2.0639×X252546.69m53.30X
28Thinking ChallengeYou’re a time study analyst in manufacturing. You’ve recorded the following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1.What is the 90% confidence interval estimate of the population mean task time?Allow students about 20 minutes to solve.
31Estimation for Finite Populations 1. AssumptionsSample Is Large Relative to Populationn / N > .052. Use Finite Population Correction Factor3. Confidence Interval (Mean, sX Unknown)SN-nSN-nX-t××mX+t××a/2,n-1Xa/2,n-1nN-1nN-1
34Estimation Example Proportion A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for p.p×(1-p)p×(1-p)ssssp-Z×pp+Z×sa/2sa/2nn.08×(1-.08).08×(1-.08).08-1.96×p.08+1.96×400400.053p.107
35Thinking ChallengeYou’re a production manager for a newspaper. You want to find the % defective. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion defective?
37Estimation Methods Estimation Point Interval Estimation Estimation ConfidenceBoot-PredictionIntervalstrappingInterval
38Bootstrapping Method 1. Used If Population Is Not Normal 2. Requires Computer3. StepsTake Initial SampleSample Repeatedly from Initial SampleCompute Sample StatisticForm Resampling DistributionLimits Are Values That Cut Off Smallest & Largest a/2 %
39Estimation Methods Estimation Point Interval Estimation Estimation ConfidenceBoot-PredictionIntervalstrappingInterval
40Prediction Interval 1. Used to Estimate Future Individual X Value 2. Not Used to Estimate Unknown Population Parameter3. Prediction Interval Estimate
41I don’t want to sample too much or too little! Finding Sample SizesI don’t want to sample too much or too little!X-mErrorx(1)Z==ssxxsx(2)Error=Zs=Zxn22Zsx(3)n=2Error
42( ) ( ) ( ) Sample Size Example What sample size is needed to be 90% confident of being correct within ± 5? A pilot study suggested that the standard deviation is 45.()()2222Zs1.64545xn==()=219.2@22022Error5
43Thinking ChallengeYou work in Human Resources at Merrill Lynch. You plan to survey employees to find their average medical expenses. You want to be 95% confident that the sample mean is within ± $50. A pilot study showed that sX was about $400. What sample size do you use?
44( ) ( ) ( ) Sample Size Solution* Z s n = Error 1 . 96 400 = 50 = 245 xn=2Error()()221.96400=()250=245.86@246
45Conclusion 1. Stated What Is Estimated 2. Distinguished Point & Interval Estimates3. Explained Interval Estimates4. Computed Confidence Interval Estimates for Population Mean & Proportion5. Computed Sample Size
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