probability and statistics - montefiore...probability and statistics chapter 5: sampling to...

Probability and Statistics

Kristel Van Steen, PhD2

Montefiore Institute - Systems and Modeling

GIGA - Bioinformatics

ULg

[email protected]

Probability and Statistics Chapter 5: Sampling to approximate the true world

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CHAPTER 5: SAMPLING TO APPROXIMATE THE TRUE WORLD

1 Introduction

2 Generating data

2.1 Design of experiments

2.2 Sampling designs and towards inference

2.3 Ethics


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3 Sample versus population

3.1 Introduction

3.2 Distribution of a sample

3.3 Statistics and sample moments


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4 Sample mean

4.1 Mean and variance

4.2 Law of large numbers revisited

4.3 Central-limit theory revisited

4.4 Bernoulli and Poisson distribution

4.5 Exponential distribution

4.6 Uniform distribution


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5 Sampling from the normal distribution

5.1 The role of normal distributions in statistics

5.2 Sample mean

5.3 The chi-square distribution

5.4 The F distribution

5.5 The Student’s t distribution


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6 Future highlights

6.1 Estimating parameters

6.2 Order statistics

6.3 Sample size calculations


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1 Introduction

Probability theory versus statistics


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Inductive versus deductive inference


Inductive versus deductive inference

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2 Generating data

2.1 Design of experiments

Obtaining data


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Population versus sample


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Observational studies versus experiments


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Some terminology


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Comparative experiments


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Placebo effects


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Caution about experimentation


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Other ways to remove bias


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Lack of realism


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Randomization


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Principles of experimental design


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Completely randomized designs


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Block designs


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Matched pairs designs


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Why experimental designs ?


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2.2 Sampling designs and towards inference

Sampling methods


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Simple random samples


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Stratified samples


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Caution about sampling surveys


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2.3 Ethics

Institutional Review Boards


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Informed Consent


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Confidentiality


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Clinical trials


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Behavioral and social science experiments


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3 Sample versus population

3.1 Introduction


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Towards statistical inference


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Sampling variability


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3.2 Distribution of a sample


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3.3 Statistics and sample moments


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Practical note


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Capture-recapture sampling


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4 Sample mean

4.1 Mean and variance


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For normally distributed populations


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On a practical note


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4.2 Law of large numbers revisited

• Question: using only a finite number of values of X (a random sample

of size n, say), can any reliable inference be made about E[X], “the

average of an infinite number of values of X”?

• Answer: YES!


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• A positive integer n can be determined such that if a random sample

of size n or larger is taken from a population with density f(.) (with

E[X])=µ), the probability can be made as close to 1 as desired that the

sample mean X bar will deviate from µ by less than an arbitrarily

specified small quantity:

(cfr. weak law of large numbers)

• Proof: Use Chebyshev inequality


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4.3 Central-limit theory revisited


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How large a sample size ?


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4.4 Bernoulli and Poisson distribution

See before


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Binomial distributions for sample counts


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Binomial distribution in statistical sampling


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Reminder: sampling variability


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Binomial mean and standard deviation


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Sample proportions


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Normal approximation


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Sampling distribution of the sampling proportion


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Normal approximation: continuity correction


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4.6 Uniform distribution


on


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5.2 Sample mean



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5.3 The chi-square distribution


square distribution


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• Recall: that if two moment generating functions both exi

equal (they agree), then the corresponding cumulative distribution

functions are the same (agree)


Recall: that if two moment generating functions both exi


functions are the same (agree)


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Recall: that if two moment generating functions both exist and are



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5.4 The F distribution



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(**)


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(**)


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This theorem can be very useful in sampling


This theorem can be very useful in sampling


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5.5 The Student’s t distribution


dent’s t distribution


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(^^)


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(^^)


(^^)


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