CHAPTER 9 ESTIMATION Outline Estimation Point estimator Interval
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CHAPTER 9 ESTIMATION Outline • Estimation – Point estimator – Interval estimator – Unbiased estimator • Confidence interval estimator of population mean when the population variance is known • Selecting the sample size 1
ESTIMATION • Point estimator: A point estimator draws inferences about a population by estimating the value of an unknown parameter using a single value or point. • Interval estimator: An interval estimator draws inferences about a population by estimating the value of an unknown parameter using an interval. • Example: A manager of a plant making cellular telephones wants to estimate the time to assemble the telephone. A sample of 30 assemblies show a mean time of 400 seconds. The sample mean time of 400 seconds may be considered a point estimate of the population mean. Chapter 9 will provide a method for estimating an interval. 2
ESTIMATION • Unbiased estimator: an unbiased estimator of a population parameter is an estimator whose expected value is equal to that parameter. • In Chapter 4, the sample variance is defined as follows: • The use of n-1 in the denominator is necessary to get an unbiased estimator of variance. The use of n in the denominator produces a smaller value of variance. 3
CONFIDENCE INTERVAL ESTIMATOR OF POPULATION MEAN WHEN THE POPULATION VARIANCE IS KNOWN • For some confidence level 1 - , sample size n, sample mean, and the sample standard deviation, the confidence interval estimator of mean, is as follows: • Recall from Chapter 7 that area on the right is /2 is that value of z for which • Lower confidence limit (LCL) • Upper confidence limit (UCL) 4
CONFIDENCE INTERVAL 5
AREAS FOR THE 82% CONFIDENCE INTERVAL 6
AREAS AND z AND x VALUES FOR THE 82% CONFIDENCE INTERVAL 7
CONFIDENCE INTERVAL ESTIMATOR OF POPULATION MEAN WHEN THE POPULATION VARIANCE IS KNOWN • Interpretation: – There is (1 - ) probability that the sample mean will be equal to a value such that the interval (LCL, UCL) will include the population mean – If the same procedure is used to obtain a confidence interval estimate of the population mean for a sufficiently large number of k times, the interval (LCL, UCL) is expected to include the population mean (1 - )k times See Table 9. 2 on p. 310 for an example • Wrong interpretation: There is (1 - ) probability that the population mean lies between LCL and UCL. Population mean is fixed, not uncertain/probabilistic. 8
CONFIDENCE INTERVAL ESTIMATOR OF POPULATION MEAN WHEN THE POPULATION VARIANCE IS KNOWN • Interpretation of the 95% confidence interval: – There is 0. 95 probability that the sample mean will be equal to a value such that the interval (LCL, UCL) will include the population mean – If the same procedure is used to obtain a confidence interval estimate of the population mean for a sufficiently large number of k times, the interval (LCL, UCL) is expected to include the population mean 0. 95 k times See Table 9. 2 on p. 310 for an example • Wrong interpretation: There is 0. 95 probability that the population mean lies between LCL and UCL. Population mean is fixed, not uncertain/probabilistic. 9
CONFIDENCE INTERVAL Example 1 (Text 9. 3): The following data represent a random sample of 10 observations from a normal population whose standard deviation is 2. Estimate the population mean with 90% confidence: 7, 3, 9, 11, 5, 4, 8, 3, 10, 9 10
SELECTING SAMPLE SIZE • A narrow confidence interval is more desirable. • For a given a confidence level, a narrow confidence interval can be obtained by increasing the sample size. • Bound on error of estimation: If the confidence interval has the form of then, B is the bound on the error of estimation. • For a given confidence level (1 - ), bound on the error of estimation B and the population standard deviation the sample size necessary to estimate population mean, is An approximation for : = Range/4 11
SELECTING SAMPLE SIZE Example 2 (Text 9. 11): Determine the sample size that is required to estimate a population mean to within 0. 2% units with 90% confidence when the standard deviation is 1. 0. 12
READING AND EXERCISES • Reading: pp. 303 -322 • Exercises: 9. 2, 9. 4, 9. 6, 9. 12, 9. 14 13
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