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Is the sample mean and the sample variance unbiased?

  1. The sample mean is an unbiased estimator of the expected value
  2. The sample variance  \(\widehat{Var}(x)=\frac{1}{N}\sum\limits^N_{i=1}(x_i-\bar{x})^2\) is an not unbiased, but asymptotically unbiased estimator
  3. However, the sample variance \(\widehat{Var}(x)=\frac{1}{N-1}\sum\limits^N_{i=1}(x_i-\bar{x})^2\) is an unbiased estimator. 
Fenster schliessen

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. 

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What is the t-distribution and name one application.

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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. 

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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. 

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What kinds of significance testing are there?

Significance tests aim at verification of a hypothesis based on statistical data:

  1. Parametric tests consider hypothesis regarding parameters of the distribution 
  2. Non-parametric tests consider hypotheses not involving paramters (e.g. distributaions are the same or different)
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Steps of a significance test 

  1. Formulate Null hypothesis and an alternative hypothesis 
  2. Choose significance level \(\alpha\)
  3. Choose significance test and test statistic; clarify assumptions to be made 
  4. Calculate Null distribution and critical value 
  5. Calculate test statistic and/or p-value 
  6. Decide whether Null hypothesis is rejected or not 
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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 \(|t_{obs}|\geq t_{crit}\)

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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!