R

Bootstrap-t Confidence Interval We want to approximate the sampling distribution of a pivotal quantity with bootstrap distribution so that we could construct a confidence interval. The method is similar to approximating a sampling distribution of a pivotal quantity using a t-distribution. The Step-By-Step Approach Given a sample 𝒮, an attribute a(𝒮), and standard error $$\widehat {SD}[\tilde a(\mathcal S)]$$, calculate a(𝒮) and standard error $$\widehat {SD}[\tilde a(\mathcal S)]$$ based on the sample. ...

Stardardized Test Scores of Grade 5 Students in California Describing the Data The data contains the average standardized test scores of grade 5 students in each school in California in the school year of 1998 through 1999. On top of the scores, information are collected on areas such as the number of students enrolled in the school, number of computers per classroom, the percentage of students in school that qualify for a reduced price lunch, and the percentage of students whose first language is not English in the school. ...

Sample Sample Definitions A sample S is a subset of the population. A sample has n < < N units. An sample attribute a(𝒮) is an estimate of the population attribute a(𝒫) $$ a(\mathcal S) = \widehat{a(\mathcal P)} = a(\hat{\mathcal P})$$ Sample error is the difference between the sample estimate a(𝒮) and the population attribute a(𝒫) (the estimand). For numerical attributes , sample error is determined mathematically. For graphical attributes, sample error is not determined precisely but it is still conceptually applicable. error = a(𝒮) − a(𝒫) Fisher consistency happens if the sample 𝒮 is equal to the population 𝒫 so the sample error is zero, meaning the estimation is sometimes consistent. ...

Influence Influence Definition Influence of a variate yu on the population attribute when a variate yu is removed from the population is measured by the following expression: Δ(α,u) = α(y1, …, yu − 1, yu, yu + 1, …, yN) − α(y1, …, yu − 1, yu + 1, …, yN) for each unit u in the population, and α is an arbitrary population attribute. Note the first part of the equation contains the unit u, and the unit u is removed in the second part of the equation. ...