# Class 33: P-values

# Methodology of Scientific Research

## Andrés Aravena, PhD

### 25 May 2023

## Asking several persons

Now we will stop 𝑁 persons and take the mean of their age

We will *assume* that the average age follows a Normal
distribution

(Is this a reasonable hypothesis?)

Find an interval that will contain the outcome with 80% of
probability

## Guessing \(μ\), knowing \(σ\)

This time we assume we do not know the population average \(μ\) and we want to estimate it from a
sample

This time we know the sample average \(\bar{X}\) and the population standard
deviation \(σ\)

Build an interval for the population average

## Is the average 0?

What is the probability of observing \((\bar{X}≥y)\)

if the real average μ is 0?

More precisely \[ℙ(\bar{X}≥y | μ = 0, σ
)\]

Here \(y\) is the observed sample
average

## Are men same age as women, on average?

Are men older or younger or same age as women?

This time we have a sample of men and a sample of women

We assume that both populations are Normal, same variance

Then \((\bar{X}_f - \bar{X}_m)\)
follows a normal with \(μ=μ_f -
μ_m\)

We want to know if that number is 0. What is \[ℙ(\bar{X}_f ≠ \bar{X}_m | μ = 0, σ
)?\]