Chapter 5 – Page 1AAMS1773 QUANTITATIVE STUDIES CHAPTER 5: ESTIMATION AND HYPOTHESIS TESTING Estimation of parameters The statistical technique of estimating unknown population parameters based on a value of the corresponding sample statistic. The estimation procedure involves the following steps: 1.Select a sample 2.Collect the required information from the members of the sample 3.Calculate the value of the sample statistic 4.Assign value(s) to the corresponding population parameter Estimate The value(s) assigned to a population parameter based on the value of a sample statistic. Estimator The sample statistic that is used to estimate a population parameter. Two types of estimates 1.Point estimate The value (single number) of a sample statistic that is used to estimate a population parameter. Example: ˆx77, ˆ 2 s26 2.Interval estimate /Confidence Intervals An estimate of a population parameter given by two numbers between which the parameter may be considered to lie on. An interval that is constructed with a given confidence level. Example:66 88 The confidence level associated with a confidence interval states how much confidence we have, that this interval contains the true population parameter. The confidence level is denoted by(1 )100%. Consider a population with unknown parameterθ. If we can find an interval (a, b) such that P(a <θ< b) = 0.95, we say that (a, b) is a 95% confidence interval forθ. In this case, 0.95 is the probability that the interval includesθ.

xZ2nxZ2n xZ 2 n n CONFIDENCE INTERVAL FOR THE POPULATION MEAN The 100(1- )% confidence interval for the population mean, when the populationvariance 2 or is knownis given by The maximum error of estimate for is Z α2 . Example: To determine the mean waiting time for his customers, a bank manager took a random sample of 50 customers and found that the mean waiting time was 7.2 minutes. Assuming that the population standard deviation is known to be 5 minutes, find the 90% confidence interval of the mean waiting time for all of the bank’s customers. 0.10 , ,

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xZ 2S nxZ2Sn xZ 2 S n S n The 100(1- )% confidence interval for the population mean, when the populationvariance 2 (n30 ) is given by is unknownand the sample sizenis large or whereSis the sample standard deviation The maximum error of estimate for is Z α2 Example: Measurements of the diameters of a random sample of 200 ball bearings made by a certain machine during 1 week showed a mean of 8.24 mm

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