Analysis with Programming

Hypothesis testing have been extensively used on different discipline of science. And in this post, I will attempt on discussing the basic theory behind this, the Likelihood Ratio Test (LRT) defined below from Casella and Berger (2001), see reference 1. Definition . The likelihood ratio test statistic for testing $H_0:\theta\in\Theta_0$ versus $H_1:\theta\in\Theta_0^c$ is \begin{equation} \label{eq:lrt} \lambda(\mathbf{x})=\frac{\displaystyle\sup_{\theta\in\Theta_0}L(\theta|\mathbf{x})}{\displaystyle\sup_{\theta\in\Theta}L(\theta|\mathbf{x})}. \end{equation} A likelihood ratio test (LRT) is any test that has a rejection region of the form $\{\mathbf{x}:\lambda(\mathbf{x})\leq c\}$, where $c$ is any number satisfying $0\leq c \leq 1$. The numerator of equation (\ref{eq:lrt}) gives us the supremum probability of the parameter, $\theta$, over the restricted domain (null hypothesis, $\Theta_0$) of the parameter space $\Theta$, that maximizes the joint probability of the sample, $\math

Another powerful procedure of SAS, my favorite one, that I would like to share is the PROC IML (Interactive Matrix Language). This procedure treats all objects as a matrix, and is very useful for doing scientific computations involving vectors and matrices. To get started, we are going to demonstrate and discuss the following: Creating and Shaping Matrices; Matrix Query; Subscripts; Descriptive Statistics; Set Operations; Probability Functions and Subroutine; Linear Algebra; Reading and Creating Data; Above outline is based on the IML tip sheet (see Reference 1). So to begin on the first bullet, consider the following code:

In this post, I would like to share a simple problem about sampling analysis. And I will demonstrate how to solve this using Python and R. The first two problems are originally from Sampling: Design and Analysis book by Sharon Lohr. Problems Let $N=6$ and $n=3$. For purposes of studying sampling distributions, assume that all population values are known. $y_1 = 98$ $y_2 = 102$ $y_3=154$ $y_4 = 133$ $y_5 = 190$ $y_6=175$ We are interested in $\bar{y}_U$, the population mean. Consider eight possible samples chosen. Sample No. Sample, $\mathcal{S}$ $P(\mathcal{S})$ 1 $\{1,3,5\}$ $1/8$ 2 $\{1,3,6\}$ $1/8$ 3 $\{1,4,5\}$ $1/8$ 4 $\{1,4,6\}$ $1/8$ 5 $\{2,3,5\}$ $1/8$ 6 $\{2,3,6\}$ $1/8$ 7 $\{2,4,5\}$ $1/8$ 8 $\{2,4,6\}$ $1/8$