VIEWS: 3 PAGES: 26 POSTED ON: 8/15/2012 Public Domain
Confidence Intervals Estimation Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 1 Lesson Objective Learn how to construct a confidence interval estimate for many situations. L.O.P. Understand the meaning of being “95%” confident by using a simulation. Learn how confidence intervals are used in making decisions about population parameters. Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 2 Statistical Inference Generalizing from a sample to a population, by using a statistic to estimate a parameter. Goal: To make a decision. Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 3 Statistical Inference Estimation of parameter: 1. Point estimators 2. Confidence intervals Testing parameter values using: 1. Confidence intervals 2. p-values 3. Critical regions. Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 4 Confidence Interval point estimate ± margin of error Choose the appropriate statistic and its corresponding m.o.e. based on the problem that is to be solved. Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 5 Estimation of Parameters A (1-a)100% confidence interval estimate of a parameter is point estimate m.o.e. Population Point Estimator Margin of Error Parameter at (1-a)100% confidence Mean, m x m.o.e. = Z α σ if s is known: 2 n Mean, m x m.o.e. = t( α s if s is unknown: 2 , n-1) n Proportion, p: ^ X / n, p p m.o.e. = Z α ˆ ˆ p(1-p) n 2 Diff. of two 2 s12 s2 means, m1 - m2 : x1 x2 m.o.e. = Zα + (for large sample sizes only) 2 n1 n2 Diff. of two ˆ ˆ ˆ ˆ p1 (1-p1 ) p2 (1-p2 ) p1 p2 ˆ ˆ m.o.e. = Zα + proportions, p1 - p2 : 2 n1 n2 Slope of regression b m.o.e. = t( α , n-2) s line, b : 2 Equ.2 where s MSE Mean from a y a bx * ˆ 1 (x * -x)2 regression m.o.e. =t( α s + when X = x*: 2 , n-2) n Equ.2 Estimation of Parameters A (1-a)100% confidence interval estimate of a parameter is point estimate m.o.e. Population Margin of Error Parameter Point Estimator at (1-a)100% confidence Mean, m if s is known: x m.o.e. Za s 2 n Mean, m x m.o.e. t( a , n 1) s if s is unknown: 2 n Proportion, p: ^ X / n, p p m.o.e. Za p(1 p ) n ˆ ˆ 2 When is the population of all possible X values Normal? Anytime the original pop. is Normal, (“exactly” for any n). Anytime the original pop. is not Normal, but n is BIG; (n > 30). Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 8 Margin of Error for 95% confidence: = 1.96 l s n To get a smaller Margin of Error: Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 9 Confidence Intervals point estimate ± margin of error Estimate the true mean net weight of 16 oz. bags of Golden Flake Potato Chips with a 95% confidence interval. Data: s = .24 oz. (True population standard deviation.) Sample size = 9. Sample mean = 15.90 oz. Must assume ori. pop. is Normal Distribution of individual bags is ______ . Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 10 s = .24 oz. For 95% confidence when s is known: n = 9. X = 15.90 oz. m.o.e. = 95% confidence interval for m: 15.90 Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 11 Statement in the L.O.P. “I am 95% confident that the true mean net weight of Golden Flake 16 oz. bags of potato chips falls in the interval 15.5472 to 16.2528 oz.” A statement in L.O.P. must contain four parts: 1. amount of confidence. 2. the parameter being estimated in L.O.P. 3. the population to which we generalize in L.O.P. 4. the calculated interval. Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 12 Meaning of being 95% Confident If we took many, many, samples from the same population, under the same conditions, and we constructed a 95% CI from each, then we would expect that 95% of all these many, many different confidence intervals would contain the true mean, and 5% would not. Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 13 Reality: We will take only ONE sample. l 15.7 15.9 16.1 X-axis - m.o.e. X + m.o.e. Is the true population mean in this interval? I cannot tell with certainty; but I am 95% confident it does. Making a decision using a CI. Question of interest: Is there evidence that the hypothesized mean is not true, at the “a” level of significance? If the “hypothesized value” is inside the CI, this value may be a plausible value. Make a vague conclusion. If the “hypothesized value” is not in the CI, this value IS NOT a plausible value. Reject it! Make a strong conclusion. Take appropriate action! Confidence level = 1-a = .95 Level of significance = a = .05 Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 16 0.5 The “true” population Population of mean is hypothesized all possible to be 13.0. p2 0.4 X-bar values, assuming . . . . 0.3 Conclusion: The hypothesis is 0.2 wrong. The “true” My ONE mean not 13.0! sample mean. 0.1 Middle 95% 0.0 l l 5 8 11 14 17 20 X-axis 23 l 5.6 7.9 10.2 X 13.0 does NOT fall in My ONE my confidence interval; Confidence Interval. it is not a plausible mean. Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 17 A more likely location 0.5 The data of the population. The “true” population convince mean is hypothesized to be 13.0. p1 me the 0.4 true 0.3 Conclusion: mean is The hypothesis is smaller wrong. The “true” 0.2 than 13.0, mean not 13.0! around 0.1 7.9! 0.0 l l 5 8 11 14 17 20 X-axis 23 l 5.6 7.9 10.2 X 13.0 does NOT fall in my confidence interval; it is not a plausible mean. Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 18 Net weight of potato chip bags should be 16.00 oz. FDA inspector takes a sample. If 95% CI is, say, (15.81 to 15.95), X = 15.88 then 16.00 is NOT in the interval. Therefore, reject 16.00 as a plausible value. Take action against the company. If 95% CI is, say, (15.71 to 16.05), X = 15.88 then 16.00 IS in the interval. Therefore, __________________________ ___________________________________ Net weight of potato chip bags should be 16.00 oz. FDA inspector takes a sample. If 95% CI is, say, (16.05 to 16.15), X = 16.10 then 16.00 is NOT in the interval. Therefore, __________________________. But, the FDA does not care that the company is giving away potato chips. The FDA would obviously _____________ against the company Meaning of being 95% Confident If we took many, many, samples from the same population, under the same conditions, and we constructed a 95% CI from each, then we would expect that 95% of all these many, many different confidence intervals would contain the true mean, and 5% would not. Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 21 Interpretation of “Margin of Error” A sample mean X calculated from a simple random sample has a 95% chance of being “within the range of the true population mean, m, plus and minus the margin of error.” True - m.o.e. True True mean mean mean + m.o.e. A sample mean is likely to fall in this interval, but it may not. Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 22 A common misconception m m.o.e. X m.o.e. This region DOES contain This region 95% of all possible X-bars. DOES NOT. .95 m Xm -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 -4.0 3.0 -3.0 4.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 X-axis X-axis A random x-bar Department of ISM, University of Alabama, 1995-2002 M32 Margin of Error 23 Concept questions. Our 95% confidence interval is 15.7 to 16.1. X = 15.9 Yes or No or ? Is our confidence interval one of the 95%, or one of the 5%? Does the true population mean lie between 15.7 and 16.1? Does the sample mean lie between 15.7 and 16.1? What is the probability that m lies between 15.7 and 16.1? Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 24 Concept questions. Our 95% confidence interval is 15.7 to 16.1. X = 15.9 Yes or No or ? Do 95% of the sample data lie between 15.7 and 16.1? Is the probability .95 that a future sample mean will lie between 15.7 and 16.1? Do 95% of all possible sample means lie between m - e.m. and m + e.m.? If the confidence level is higher, will the interval width be wider? Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 25 End of File M33 Department of ISM, University of Alabama, 1995-2003 M33 Confidence intervals 26