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381 Inferences About the Mean-I (Large Samples) QSCI 381 – Lecture 21 (Larson and Farber, Sect 6.1) Statistical Inference 381 Sample statistics can be used to the value(s) of unknown population parameter(s). We are making inference about the unknown population parameter(s) based on data. We start by making estimates of the population parameter : when the sample size is large (30) when the sample size is small. Point Estimation 381 A is a single value for a population parameter. The most unbiased point estimate of the population mean is the sample mean x . Warning: x is an estimate and not the population mean – the difference between the two is the uncertainty of the estimate. Example 381 An acoustic survey involved three passes of an area. The average density during the first pass was 52.3. 12 Results for passes 2 and 3 10 x 50.4 x 51.0 Frequency 8 6 4 2 0 30 35 40 45 50 55 Density 60 65 70 75 80 Interval Estimation 381 An is an interval, or range of values, used to estimate a population parameter. 41.0 44.6 48.2 51.8 55.4 Point estimate=52.3 Interval = (48.5, 58.1) 59.0 Level of Confidence 381 The is the probability that the interval estimate contains the population parameter. For n30, the sampling distribution of sample means is a normal distribution. The level of confidence, c, is the area under the standard normal distribution between –zc and zc. Hint: for zc=1.645, there is a 90% probability that the interval estimate contains the population mean. Why do I say this? c -zc z=0 ½(1-c) zc Extent of Error 381 Given a level of confidence, c, the (margin of error or error tolerance) is the greatest possible difference between the point estimate and the value of the parameter being estimated: E zc x zc n Example-I 381 1. 2. 3. 4. Use the data for the first pass of the acoustic survey and a 95% level of confidence (i.e. zc=1.96) to find the maximum error of estimate for the mean density. The steps to calculate the maximum error of estimate are: Find the sample statistics n and x . Specify if known. Otherwise, if n30, find the sample standard deviation, s, and use this as an estimate of . Find the level of z that corresponds to the confidence level. Find the maximum error of estimate of E. Example-II 381 Application of these steps gives: 1. n=50; x 52.28 2. is not known so we estimate it from the sample: 1 n 2 s x x n 1 i 1 i 10.68 The critical value of z is 1.96. 2. The maximum error of estimate E is: 10.68 E zc 1.96 2.95 n 50 We are 95% confident that the maximum error of estimate for the population mean is about 2.95 units of density. 1. Confidence Intervals for the Population Mean 381 A population mean is for the x E x E The probability that the confidence interval contains is c. The 95% confidence interval for the mean density is (52.28-2.95, 52.38+2.95) = (49.3, 55.2) Confidence Intervals for the Population Mean 381 To compute a (95%) confidence interval using EXCEL: =AVERAGE(D2:D51)+STDEV(D2:D51)/SQRT(COUNT( D2:D51))*NORMINV(0.025,0,1) =AVERAGE(D2:D51)+STDEV(D2:D51)/SQRT(COUNT( D2:D51))*NORMINV(0.975,0,1) Note: The calculation is based on STDEV and not STDEVP. Example-III ( known to be 10) 381 Application of the steps gives: 1. n=50; =10; x 52.28 2. The critical value of z is 1.96. 3. The maximum error of estimate E is: 10 E zc 1.96 2.77 n 50 We are 95% confident that the maximum error of estimate for the population mean is about 2.77 units of density. Interpretation 381 40 45 50 55 60 100 simulated confidence intervals for the mean of acoustic survey density Summary 381 Given c find zc Compute x xi Is known? Yes E zc n i 1 1 n 2 s xi x n 1 i 1 n x E x E No E zc s n