Is the sample mean and the sample variance unbiased?
What is the main idea for a confidence interval?
Given an estimate \(\widehat{\Theta}\) of \(\Theta\). An interval \((\widehat{\Theta}_L,\widehat{\Theta}_U)\) around \(\widehat{\Theta}\) is named a \((1-\alpha)\) confidence interval if
\(P(\Theta\in (\widehat{\Theta}_L,\widehat{\Theta}_U))=1-\alpha\)
A 95% confidence interval covers the true value in 95% of the cases.
What is the t-distribution and name one application.
t-Distribution:
\(f_X(x;\nu)=c(\nu)\left(1+\frac{x^2}{\nu}\right)^{-\frac{\nu+1}{2}}\)
with a constant \(c(\nu)\). \(\nu\in\mathbb{N}\) is called the degree of freedom. For \(\nu\rightarrow \infty\) it converges to the \(\mathcal{N}(0,1)\) distribution.
The t-Distribution is used to estimate the mean of a normally distributed population when the sample size is small and population standard deviation is unknown.
Derive the confidence intervals for the mean of a normal distributed variable with known variance!
The empirical mean \(\bar{x}\) has a distribution \(\mathcal{N}(\mu, \frac{\sigma}{\sqrt{n}})\) while
\(Z=\sqrt{n}(\bar{x}-\mu)/\sigma\)
has distribution \(\mathcal{N}(0,1)\) .
The value z is such that \(P(-z\leq Z\leq z)=1-\alpha\)
This yields: \(P(\bar{x}-z\frac{\sigma}{\sqrt{n}}\leq\mu\leq\bar{x}+z\frac{\sigma}{\sqrt{n}})=1-\alpha\)
The confidence interval can be easily found now.
What kinds of significance testing are there?
Significance tests aim at verification of a hypothesis based on statistical data:
Steps of a significance test
How are significance level and critical value defined?
Significance level and critical value are defined such that:
\(P(|T|\geq t_{crit})\equiv\alpha\)
i.e., the probability that T falls in the rejection region (Q) although the \(H_0\) (null hypothesis) is true (small probability).
Then the Null hypothesis is rejected in case \(p_{obs}\leq\alpha \) or \(|t_{obs}|\geq t_{crit}\)
significance testing: What is the error of the first kind and what is the error of the second kind?
error of the first kind or \(\alpha\)-error: Rection of the Null hypothesis \(H_0\) although it is true
Probability of this error is \(P(H_0 rejected\; | \;H_0 true)=\alpha\)
error of the second kind or \(\beta \)-error: No rejection of the Null hypothesis although it is wrong
Probability of this error is \(P(H_0\;not\;rejected\;|\;H_0\;false)=\beta \)
The reduction ofthe one error leads to increase of the other, unless we can increase sample size!