Reading Property Data Analysis – A Primer, Ch.9 OMGT1117 - LECTURE 9: Parameter Estimation and Sample Size Determination Reading Property Data Analysis – A Primer, Ch.9

Objectives To summarise the desirable properties of point estimators. To highlight the factors determining the width of confidence interval estimates of true unknown parameters. Outline the mechanics of Confidence Interval Estimation for the true unknown population mean and the true unknown population proportion. Describe and Contrast the concepts of Precision and Error Tolerance (Reliability) Learn how to choose an appropriate sample size to achieve nominated precision and/or error-tolerance (reliability) requirements

Point Estimation Point Estimation: This arises when one uses a sample measure (a single value or “point” rather than a range of values or interval) to estimate the true unknown parameter (say the population mean m or the population proportion p) Distinction between Point Estimator and Point Estimate: Statisticians distinguish between the formula or rule (point estimator) for obtaining a statistic (sample measure) and the actual value (point estimate) generated by the evaluated formula or rule.

Point Estimation Continued Desirable Properties of Point Estimators: There are four desirable properties of point estimators: Unbiasedness: This occurs when the estimator generates values which on average equal the true unknown parameter. The formulae for the sample mean, sample proportion and sample variance each enjoy this property. Consistency: As the size of the sample increases the estimator will generate values that converge onto the corresponding population parameter Efficiency: An estimator is more efficient than another if for a given sample size its standard error (average sampling error) is smaller than that of any other competing estimator. Sufficiency: An estimator is said to be sufficient when it uses all information in the sample

Interval Estimation Intuitive Discussion of Confidence Interval Estimation: The diagram below represents a symmetrical interval constructed around a statistic (say the sample mean). That given degree of confidence may be attached to the proposition that the true unknown population parameter (say the population mean) lies in this interval helps us to understand why it is called a confidence interval estimate. Statistic d Statistic + d Statistic - d It is noted that the two end-points of the interval are equidistant from the statistic. As we shall see shortly the distance d: is +vely related to the chosen confidence level CL (where 0% < CL < 100%). is +vely related to the standard error sample statistic and negatively related to the sample size n. Also for a given level of confidence there is a trade-off between cost (which is +vely related to sample size n) and the precision (narrowness) of the resultant confidence interval.

Confidence Interval Estimation for the Population Mean m Choice of Confidence Level CL: The choice of CL depends on the choice of a which refers to the combined areas in the tails of a z-distribution or Student t-distribution whichever is applicable for the Task at hand. In particular one chooses: =.01 to construct a (1 - a)100% = 99% confidence interval a =.02 to construct a (1 - a)100% = 98% confidence interval a =.05 to construct a (1 - a)100% = 95% confidence interval a =.10 to construct a (1 - a)100% = 90% confidence interval The following notation will be used in what follows: za/2 is the z-value with a/2(100%) of the area of the z - distribution above it tn,a/2 is the t-value with a/2(100%) of the area of the t – distribution* above it * The student t-distribution is distributed with n = n - 1 degrees of freedom

Confidence Interval Estimation for the Population Mean m Continued

Confidence Interval Estimation for the Population Mean m Continued

Confidence Interval Estimation for the Population Mean m Continued

Confidence Interval Estimation for the Population Proportion p

Confidence Interval Estimation for the Population Proportion p

Precision and Reliability Requirements for Interval Estimation (1-a) confidence interval for true unknown parameter Statistic e Statistic + e Statistic - e w = 2e

Sample Size determination for Interval Estimates of m Various formulae are available for the determination of the appropriate sample size n* for confidence interval estimation of the population mean m. The applicable formula in any given instance depends on: The nature of the sampling-process (eg sampling with or without replacement and the implications for the size of the FCF) Whether or not s is known

Sample Size Determination for Interval Estimates of m Continued

Sample Size Determination for Interval Estimates of m Continued

Sample Size Determination for Interval Estimates of m Continued

Sample Size Determination for Interval Estimates of m Continued

Sample Size Determination for Interval Estimates of p

Sample Size Determination for Interval Estimates of p Continued

Sample Size Determination for Interval Estimates of p Continued

Sample Size Determination for Interval Estimates of p Continued

Sample Size Determination for Interval Estimates of p Continued Concluding Warning : For each of the procedures available for determining the appropriate sample size n* when undertaking interval estimation of p, it is really required that p  ND(p,sp). The procedures should avoided as a guide for determining appropriate sample size if the suggested sample size is too small.