# Lernkarten

Karten 88 Karten 1 Lernende English Universität 21.07.2018 / 27.08.2018 Keine Angabe
0 Exakte Antworten 88 Text Antworten 0 Multiple Choice Antworten

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.

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:

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)

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

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!