Statistik 1st Semester

Markus Loecher, Nov 2, 2016

Table of Contents

  1. Kleinste Quadrate
    1. Regression Line Demo
    2. Regression
  2. Wahrscheinlichkeitsverteilungen
    1. Binomialverteilung: n=20 und n variabel
    2. The Binomial Distribution
    3. Shape of the Poisson Distribution
    4. Binomial Distribution Image Generator
  3. Hypothesis Testing
    1. Type I & Type II Errors
    2. Power of a hypothesis test
  4. Sampling
    1. Sampling from a population of ordered pairs
    2. The Sampling Distribution
  5. Confidence Intervals
    1. Confidence Interval for a Proportion
    2. Proportion Confidence Interval
    3. Confidence Interval for µ (σ unknown)
  6. Estimation
    1. Maximum likelihood estimators

1.

Kleinste Quadrate

Wenn man die Regression als Optimierung ansieht, ergibt sich die Regressionsgerade, indem man die quadrierten Residuen minimiert. Die Residuen sind einfach die vertikalen Abstände der Daten von der Gerade.

  • 1. Regression Line Demo
  • 2. Regression

1.1.

Regression Line Demo

Austad AHS Least Square Regression
– John Smethurst, stighenriksen, Austad

1.2.

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2.

Wahrscheinlichkeitsverteilungen

Binomialkoeffizient: [math]\frac{n!}{(n-k)! \cdot k!} = {n \choose k}[/math] Binomialverteilung: [math]P(x=k;n,p) = {n \choose k} p^k (1-p)^{n-k}[/math]

  • 1. Binomialverteilung: n=20 und n variabel
  • 2. The Binomial Distribution
  • 3. Shape of the Poisson Distribution
  • 4. Binomial Distribution Image Generator

2.1.

Binomialverteilung: n=20 und n variabel

Das Applet zeigt die Wahrscheinlichkeitsfunktion einer Binomialverteilung mit [color=#1551b5]n = 20 (fest)[/color] und variablem [color=#0a971e]n[/color].[br][br][b]Aufgabe[/b][br]Beschreibe, wie sich die Wahrscheinlichkeitsfunktion mit größer werdendem n verändert.[br]Begründe, warum der Flächeninhalt in allen Fällen immer unverändert gleich 1 bleibt?
– Andreas Lindner

2.2.

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2.3.

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2.4.

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3.

Hypothesis Testing

  • 1. Type I & Type II Errors
  • 2. Power of a hypothesis test

3.1.

Type I & Type II Errors
– Eric Usher

3.2.

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4.

Sampling

  • 1. Sampling from a population of ordered pairs
  • 2. The Sampling Distribution

4.1.

Sampling from a population of ordered pairs

– George Sturr

4.2.

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5.

Confidence Intervals

  • 1. Confidence Interval for a Proportion
  • 2. Proportion Confidence Interval
  • 3. Confidence Interval for µ (σ unknown)

5.1.

Confidence Interval for a Proportion

The applet below illustrates the concept of constructing confidence intervals for proportions (a 1-proportion z-interval). Set the population proportion, [br]sample size, and confidence level using the sliders. There are 100 confidence intervals at your chosen confidence level, based on 100 random [br]samples of your chosen sample size, from a population with your chosen proportion.[br][br]On average, the percentage of confidence intervals that contain the population proportion will be equal to the chosen confidence level.[br][br]You will notice that larger sample sizes produce shorter confidence intervals (why?).[br]You will notice that larger confidence levels produce longer confidence intervals (why?)
– Steve Phelps

5.2.

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5.3.

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6.

Estimation

  • 1. Maximum likelihood estimators

6.1.

Maximum likelihood estimators

This applet shows the (log of the) likelihood function, and the maximum likelihood estimator, for a sample and various statistical models.

This applet demonstrates the principle of maximum likelihood estimation. The blue curve represents a possible population generating the data, with parameter [color=#1551b5]θ[/color]. You can change population types by using the buttons at top-right. An actual observed sample [color=#000000][math]x_1, x_2,[/math][/color] ... is shown along the horizontal axis. You can adjust the sample size n using the slider at the top-left. You can move individual sample observations by dragging them. The likelihood of each [color=#000000][math]x_i[/math][/color] for the current [color=#1551b5]θ[/color] value is shown as a dotted blue line. You can drag the blue point to adjust [color=#1551b5]θ[/color], and observe how the likelihoods of the sample points are affected. You can also key in a new value for [color=#1551b5]θ[/color] in the text box. Try to guess the MLE by dragging [color=#1551b5]θ[/color] and observing the likelihoods, particularly for sample sizes 1 and 2. You can show the likelihood of the whole sample, and the MLE, using the 'Likelihood function' checkbox. You can switch to the log-likelihood of the whole sample using the 'Log-likelihood' checkbox. You can also use the zoom buttons and arrows to adjust the views of the windows.
– University of Melbourne School of Mathematics and Statistics

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