# Stat 202: Lecture 18 (covers pp. 192-198)

Nathan VanHoudnos
11/3/2014

### Agenda

3. Lecture 18 (covers pp. 192-198)

### Confidence intervals

• FALSE: The 95% confidence interval I just calculated has a 95% chance of containing the true value. FALSE

• TRUE: Over many repeated experiments, 95% of the confidence intervals you construct will contain the true value. TRUE

### Inference

• Point estimation
• Interval estimation
• Confidence Intervals – If we repeat the experiment over and over, 95% of intervals will contain the true value.
• Credible Intervals – If we do the experiment once, the probability that the true value is contained within the interval is 95%.
• Hypothesis Testing
• can be useful
• and, yet, as it stands, is a literal pox upon science.

### Mechanics before politics

A case of suspected cheating on an exam is brought in front of the disciplinary committee at a certain university.

There are two opposing claims in this case:

The student's claim: I did not cheat on the exam.

The instructor's claim: The student did cheat on the exam.

### Mechanics before politics

“innocent until proven guilty”

• the instructor must give sufficient evidence that the claim of innocence is unlikely

The instructor says:

• The exam had two versions with different values of $$\mu$$, $$\sigma$$, and $$n$$ between the two versions.
• This student used the values of $$\mu$$, $$\sigma$$, and $$n$$ from the other exam version to get the answer.
• The student did this for three out of the four exam questions.

### Mechanics before politics

“innocent until proven guilty”

• the instructor must give sufficient evidence that the claim of innocence is unlikely

The instructor says the student used the other set of numbers for three out of the four exam questions.

Is this sufficient evidence to reject the claim of innocence ?

Yes. We can reject the claim of innocence.

### Hypothesis testing

Step 1: State the claims. \begin{aligned} H_0 & \text{ : null hypothesis} & H_A & \text{ : alt. hypothesis} \end{aligned}

• $$H_0$$ the student is innocent
• $$H_A$$ the student is guilty

Step 2: Present evidence against $$H_0$$

• The instructor presents evidence that the student cheated.

Step 3: Decide if $$H_0$$ should be rejected or retained.

• $$H_0$$ is rejected. The student cheated.

### Why do it this way?

Karl Popper
1902 - 1994

Claim:

All swans are white.

• Finding millions upon millions of white swans does not prove this claim.

• If, there exists a single black swan, then the claim is false.

Popper argued that, if a theory is falsifiable, then it is scientific.

### Why do it this way?

Louis de Broglie
1892 - 1987

Before quantum mechanics, scientists thought that electrons, protons, neutrons and the like were essentially little billiard balls.

• $$H_0$$ (prevailing model) The electron is a particle.
• de Broglie's (1924) Ph.D. thesis showed that electrons act like waves.
• $$H_0$$ is rejected, and quantum mechanics gains strength

### Neyman-Pearson Theory

Karl Pearson 1857-1936