# Parameter Estimation and Hypothesis Testing for the Truncated Normal Distribution with Applications to Introductory Statistics Grades

## Abstract

The normal distribution is a commonly seen distribution in nature, education, and business. Data that are mounded or bell shaped are easily found across various fields of study. Although there is high utility with the normal distribution; often the full range can not be observed. The truncated normal distribution accounts for the inability to observe the full range and allows for inferring back to the original population. Depending on the amount of truncation, the truncated normal has several distinct shapes. A simulation study evaluating the performance of the maximum likelihood estimators and method of moment estimators is conducted and a comparison of performance is made. The α Likelihood Ratio Test (LRT) is derived for testing the null hypothesis of equal population means for truncated normal data. A simulation study evaluating the power of the LRT to detect absolute standardized differences between the two population means with small sample size was conducted and the power curves were approximated. Another simulation study evaluating the power of the LRT to detect absolute differences for testing the hypothesis with large unequal sample sizes was conducted. The α LRT was extended to a k population hypothesis test for equal population means. A simulation study examining the power of the k population LRT for detecting absolute standardized differences when one of the population means is different than the others was conducted and the power curve approximated. Stat~221 is the largest introductory statistics course at BYU serving about 4,500 students a year. Every section of Stat 221 shares common homework assignments and tests. This controls for confounding when making comparisons between sections. Historically grades have been thought to be bell shaped, but with grade inflation and other factors, the upper tail is lost because of the truncation at 100. It is reasonable to assume that grades follow a truncated normal distribution. Inference using the final grades should be done recognizing the truncation. Performance of the different Stat 221 sections was evaluated using the LRTs derived.

MS

## College and Department

Physical and Mathematical Sciences; Statistics

2010-03-09

Selected Project

## Handle

http://hdl.lib.byu.edu/1877/etd3412

## Keywords

maximum likelihood estimators, method of moments estimators, likelihood ratio test, assessment of student learning, Stat 221

English

COinS