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

kristel.vansteen@ulg.ac.be

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

Chapter 5: Sampling to approximate the true world

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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

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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)

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

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5.4 The F distribution

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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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