Statistical Reasoning

Null Hypothesis Testisn

Author

Affiliation

Klinkenberg

University of Amsterdam

Published

September 6, 2023

Course structure

Course parts

Lectures: On campus / Online / Video recording
Preparatory Assignment: Submit in Testvision and Reflection in canvas
Tutorials: For your support and inclass assessment
Exam: Knowledge and skills

Grading

\[\text{Final grade} = 0.9 \times \text{exam grade} + 0.1 \times \text{credit points}\]

Exam
Credits for
- Tutorial participation
- Preparatory assignment

Credits

You get 10 at the start of the course.

Number of times missed	Deduction for TU	Deduction for PA
1x	0	0
2x	0	2
3x	2	3
4x	4	4
5x or more	5	5

See canvas for all the details.

Learning

Reasoning in statistics

Source: ARTIST

Statistical Literacy

Knowledge (Basic understanding of concepts)
- Identify
- Describe
Skils (Ability to work with statistical tools)
- Translate
- Interpret
- Read
- Compute

Statistical Reasoning

Understanding
- Explain why
- Explain how

Statistical thinking

Apply
- What methods to use in a specific situation
Critique
- Comment and reflect on work of others
Evaluate
- Assigning value to work
Generalize
- What does variation mean in the large scheme of life

Empirical Cycle

By Adriaan de Groot

The components

Observation
- Idea for hypothesis
Induction
- General rule
- Hypothesis
Deduction
- Expectation / Prediction
- Operationalization
Testing
- Test hypothesis
- Compare data to prediction
Evaluation
- Interpret results in terms of hypothesis

Explained by Annemarie Zandscholten

Experiment

Video registration

Heads

bit.ly/2j54A2U

Emperical Cycle

Observation Patiënt is showing post traumatic symptoms
Induction Can we diagnose PTSD
Deduction \(H_0\): P: fair coin → C: patiënt is balanced
Deduction \(H_A\): P: Unfair coin → C: patiënt is unbalanced
Deduction \(H_A\): P: data \(\neq\) EV → C: is unbalanced
Testing Choose \(\alpha\) and Power
Evaluation Make a decision

Null distribution

Let’s analyse the null distribution of the results.

Google sheet

Distributions

What is the difference between

Population distribution
Sample distribution
Sampling distribution

Binomial distribution

\[ {n\choose k}p^k(1-p)^{n-k}\]

\[ {n\choose k} = \frac{n!}{k!(n-k)!} \]

With values:

n = 10   # Sample size
k = 0:10 # Discrete probability space
p = .5   # Probability of head

Probabilities

Testing

I landed 2 times head. Can we conclude PTSD?

As you can see from the distribution of healthy coins, we cannot conclude that by definition.
What we can do is indicate how rare 2 is in a healthy population.

Testing

Based on the null distribution we can see that the expected value (EV is 5.)

We can now define the \(H_0\) hypothesis: \(H_0 = 5\)

What is the alternative hypothesis?

The alternative hypothesis describes a situation where PTSD is pressent.

We could say that the alternative hypothesis is not 5.

\(H_A \ne 5\)

We could also formulate our \(H_0\) and \(H_A\) more abstract:

\(H_0:\) the patient is balenced

\(H_A:\) the patient is unbalenced

What criterium should we use to conclude that one would be unbalenced?

In the social sciences this \(\alpha\) criteria is often 5%.

I tossed 2 times head. That is more frequent than 5%.

Therefore, we conclude that our patient is probably healthy but we can never be sure.

My coin could still be part of the unbalenced population.

Null distribution

Alternative Distribution

But we have no clue of what this distribution could look like.

For now let’s assume the probability of landing heads for my coin is .25

\(H_0\) and \(H_A\) distribution

End

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