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Copyright unreserved and unlimited 1 Interval Estimation UNIVERSITY OF PRETORIA DEPARTMENT OF INDUSTRIAL AND SYSTEMS ENGINEERING ENGINEERING STATISTICS BES221 INTERVAL ESTIMATION Überhaupt ist es für den Forscher ein guter Morgensport, täglich vor dem Frühstück eine Lieblingshypothese einzustampfen – das erhält jung. Konrad Lorenz (1903 - 1989) Austro-German zoologist Histograms Probabilities Distributions Distributions Probability Models Sampling Sampling Estimation Inference BASIC CONCEPTS Point estimates such as the sample estimates : x for and s for Interval estimates (confidence intervals such as) : P( x - x + ) = 1 - P S Kruger/2011 Engineering Statistics BES221 2 Copyright unreserved and unlimited Interval Estimation Sampling distributions : Normal distribution and t-distribution (small sampling theory) for means, the difference between means, totals and proportions The 2- and F-distributions for variances and the difference between variances {William Sealy Gossett, Guinness Beer and the Student’s tdistribution} CONFIDENCE INTERVALS (interval estimates) General concepts given a sample distribution (assumption) 1- /2 a /2 X b Sample mean The probability that the true population mean will be within an interval (estimate of sample mean plus or minus delta, i.e. between a and b) may be calculated or if the probability is specified the confidence limits may be calculated (delta will be a function of the sample variance and sample size) P( x - x + ) = 1 - is known as the level of significance and is usually user specified P S Kruger/2011 Engineering Statistics BES221 Copyright unreserved and unlimited 3 Interval Estimation In similar ways confidence intervals for many other sample quantities may be calculated, for example the sample variance, the difference between two sample means, a sample proportion, etc. Please note : The inherent assumption is usually that the observations (sample units) are independent. If significant autocorrelation exists between the observations it should be taken into account, otherwise the confidence limits may be grossly optimistic. The effect on the “accuracy” of the estimates of : Increasing or decreasing the sample size Increasing or decreasing the level of significance The population variance CONFIDENCE INTERVALS FOR THE MEAN For large sample size (n 30) Maximum error of the estimate of the mean = z/2 ( / n ) with z/2 the standardized normal deviate The sample standard deviation, ( / n), is also often known as the standard error of the mean Most often used values for z/2 (from the Standard Normal Table) z0,005 z0,01 z0,025 z0,05 P S Kruger/2011 = 2,576 = 2,326 = 1,960 = 1,645 Engineering Statistics BES221 Copyright unreserved and unlimited 4 Interval Estimation Example : Sample mean x = 6,25 Sample standard deviation s = 1,60 Sample size n = 40 (“large sample” since n 30) Calculate a 95% confidence interval = 0,05 /2 = 0,025 z/2 = 1,96 = z/2 ( / n ) = 1,96 (1,60 / 40) = 0,4958 Confidence interval : P(5,7542 6,7458) = 0,95 Interpretation of the confidence interval : It should not be interpreted as : “The probability is 0,95 that the true mean lies within the interval” but rather as : “95% of all possible intervals calculated according to this procedure will contain the true mean” ! Calculation of required sample size n = [ ( z/2 . ) / ]2 P S Kruger/2011 Engineering Statistics BES221 5 Copyright unreserved and unlimited Interval Estimation Example : Sample standard deviation s = 1,60 Determine the minimum sample size if the required half-length of a 95% confidence interval, = 0,3 Use s as an estimate of n = [ ( z/2 . ) / ]2 = [ 1,96 . 1,60 / 0,3]2 = 109,2 = 110 For small sample size (n 30) = t/2 ( / n ) n = [ ( t/2 . ) / ]2 Degrees of freedom of the t-distribution : = n - 1 Extract from a t-table (values of t) 1 5 10 15 20 25 29 30 P S Kruger/2011 0,1 3,078 1,476 1,372 1,341 1,325 1,316 1,311 1,282 0,05 6,314 2,015 1,812 1,753 1,725 1,708 1,699 1,645 0,025 12,706 2,571 2,228 2,131 2,086 2,060 2,045 1,960 0,01 31,821 3,365 2,764 2,602 2,528 2,485 2,462 2,326 0,005 63,657 4,032 3,169 2,947 2,845 2,787 2,756 2,576 1 5 10 15 20 25 29 30 Engineering Statistics BES221 Copyright unreserved and unlimited 6 Interval Estimation Confidence Intervals for a Total Confidence Intervals for a Proportion Confidence Intervals for a Standard Deviation Confidence Intervals for the Difference Between Means Confidence Intervals for the Difference Between Proportions Confidence Intervals for the Difference Between Means Sample Size Determination for other Parameters Independent Samples Paired Samples *********************** P S Kruger/2011 Engineering Statistics BES221