Sampling Strategies for Error Rate Estimation and Quality ...

[Pages:56]Project Number: JPA0703

Sampling Strategies for Error Rate Estimation and Quality Control

A Major Qualifying Project Report Submitted to the faculty of the

Worcester Polytechnic Institute in partial fulfillment of the requirements for

the Degree of Bachelor of Science in Actuarial Mathematics by

__________________________ ____________________________

Nicholas Facchiano

Ashley Kingman

__________________________ ____________________________

Amanda Olore

David Zuniga

Date: April 2008

Approved by:

____________________________ Jon Abraham, Project Advisor

_____________________________ Jayson Wilbur, Project Advisor

Table of Contents

Table of Figures ............................................................................................................................. 3 Table of Tables .............................................................................................................................. 4 Abstract ......................................................................................................................................... 5 Executive Summary...................................................................................................................... 6 Introduction ................................................................................................................................... 8 I. Sampling Optimization.............................................................................................................. 9

1. Background....................................................................................................................... 9 2. Methodology .................................................................................................................... 9

Confidence............................................................................................................................ 9 Sample Proportion............................................................................................................... 11 Margin of Error...................................................................................................................... 11 Initial Sampling Function .................................................................................................... 11 3. Analysis & Discussion ...................................................................................................... 13 Cost Function Derivation.................................................................................................... 13 Optimization Function Derivation ..................................................................................... 14 MS Excel Tool Development .............................................................................................. 15 4. Results and Conclusions ................................................................................................ 16 II. Universal Life Policies .............................................................................................................. 18 1. Background..................................................................................................................... 18 T Distribution ......................................................................................................................... 19 Bootstrapping ...................................................................................................................... 19 2. Methodology .................................................................................................................. 21 3. Analysis and Discussion.................................................................................................. 24 4. Results and Conclusions ................................................................................................ 26 III. NWI Tracking ........................................................................................................................... 28

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1. Background..................................................................................................................... 28 Quality Assurance ............................................................................................................... 28 Data ...................................................................................................................................... 29 Hypergeometric Distribution.............................................................................................. 29 Binomial Distribution ............................................................................................................ 30

2. Methodology .................................................................................................................. 31 NWI Tracking MS Tool Descriptions.................................................................................... 32

IV. Variable Surrender ................................................................................................................ 36 1. Background..................................................................................................................... 36 Definition of the Problem: .................................................................................................. 36 Statistical Properties ............................................................................................................ 36 2. Methodology .................................................................................................................. 37 Errors ...................................................................................................................................... 37 Process .................................................................................................................................. 38 3. Analysis and Discussion.................................................................................................. 38 4. Results and Conclusions ................................................................................................ 41

Appendices................................................................................................................................. 42 Appendix 1: Sampling Optimization Worksheet Instructions ............................................ 42 Appendix 2: Bootstrapping Program Instructions............................................................... 42 Appendix 3: Bootstrapping Program VBA Code................................................................ 44 Appendix 4: NWI Tracking Sheet 11-26 Descriptions.......................................................... 52

Works Cited ................................................................................................................................. 55

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Table of Figures

Figure 1.1: Depiction of data range used in a two-sided confidence interval Figure 1.2: Depiction of data range used in a one-sided confidence interval Figure 1.3: Screenshot of Sampling Optimization Worksheet Sheet 1 Figure 1.4: Screenshot of Sampling Optimization Worksheet Sheet 2 Figure 1.3: Sample Size versus Total Cost with Cs = $5 and Ct = $1000 Figure 1.4: Three-Dimensional Projection of Resultant Sample Size as a function of varying per sample and per trait costs Figure 2.1: Histogram of Company Shortfall Amounts Figure 2.2: Bootstrapping Steps Figure 2.3: Bootstrapping Program Input Worksheet Figure 2.4: Input Worksheet ? John Hancock Data

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Table of Tables

Table 1.1: Binomial sample sizes with corresponding margins of error Table 2.1: Example of Bootstrapping Program Output Table 2.2: Bootstrap Program Results Table 3.1: Defective vs. Non-Defective Table 3.2: MS Tool Descriptions Table 4.1: Worker Differences Table 4.2: Level of Confidence Variation Table 4.3: Population Variation

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Abstract

John Hancock must utilize processes to monitor departmental performance. Our group was presented with several problems whose solutions will benefit the company's quality assurance capabilities. This MQP analyzes the optimization of a sampling function for a book of insurance policies, the Bootstrapping method for the estimation of policy shortfall confidence intervals, and a sampling procedure to aid the company in customer satisfaction screening.

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Executive Summary

John Hancock is a leader in the insurance and financial service industry. The company must utilize quality control procedures to monitor departmental performance. The objective of this project was to develop sampling strategies for the estimation of error rates and means in an effort to improve current quality control processes.

John Hancock requested the development of a sampling methodology to estimate the proportion of defects within a book of life insurance policies. Defective policies carry a liability which must be paid to the policy holder. Our group was able to optimize a preexisting sampling function by introducing constraints for liability cost and sampling cost. This new function will allow John Hancock to take a properly sized sample which minimizes the total project cost for the given parameters.

The second initiative involves the estimation of a confidence interval for mean shortfall amount. Statisticians often use a method known as Bootstrapping to create larger data sets by taking random samples from initially small samples. The key assumption in this process is that the sample data is representative of the entire data set. With the newly created data set, we were able to estimate an interval for mean shortfall amount given a desired confidence level.

Lastly, John Hancock consults with a call center to monitor customer satisfaction. Each call center employee has his/her work screened on a monthly basis; currently, each employee is sampled five times per month. However, this sample size has no statistical significance. The goal is to create a tool which will provide John Hancock with a properly sized sample of work items to screen for each employee. The resultant sample size is a function of historical employee performance and desired confidence.

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