Motivation Suppose you have a population with known \(\mu\) and \(\sigma\). You then take a sample (perhaps not randomly) and discover...
For a Poission distribution with large \(n\), use \(Z=\frac{\bar{X}-\lambda}{\sqrt{\lambda/n}}\) (i.e. \(\lambda\) instead of \(S\)). It...
Things you should think about: What are the implications of your choice of \(\alpha\)? How much did intuition play a role in deciding...
The p-value is the smallest significance level (i.e. \(\alpha\)) at which \(H_{0}\) would be rejected. So if \(p\le\alpha'\), you reject...
Let \(X\) be the number of successes. If \(n<
Steps to Carry Out The Experiment Identify the parameter of interest. Determine the null value and state the null hypothesis. State the...
\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) \(\newcommand{\Sample}{X_{1},\dots,X_{n}}\) Let \(\Sample\) be...
The one-sample t-distribution confidence interval is robust for small or even moderate departures from normality, unless \(n\) is very...
What if you have \(n\) observations in a normal distribution and want to predict \(X_{n+1}\)? The prediction interval is \(\bar{x}\pm...
Say you take a sample where \(n\) is not large. Then the CLT doesn’t apply. We must then know/assume a distribution. Suppose we know it...
Let \(\Sample\) be independent and identically distributed from \(N(\mu,\sigma^{2})\). Define the following random variable:...
\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) \(\newcommand{\Sample}{X_{1},\dots,X_{n}}\) For any...
\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) \(\newcommand{\Sample}{X_{1},\dots,X_{n}}\) Assume you have a...
\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) Let \(X_{1}\dots X_{n}\) have means \(\mu_{i}\) and variances...
Let \(X_{1}\dots X_{n}\) be a random sample from a distribution with mean \(\mu\) and standard deviation \(\sigma\). Then:...
If \(Y=\ln X\) is a normal distribution, then \(X\) is log-normal. \begin{equation*} f(x;\mu,\sigma)=\frac{1}{\sqrt{2\pi\sigma}...
Chi-squared distribution Probability Density Function The parameter is \(\nu\) and it is a positive integer. It is the gamma...
Probability Distribution Function The parameters are \(-\infty<\mu<\infty\) and \(\sigma>0\). \begin{equation*}...