# Methodology of Scientific Research

### 25 May 2023

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 $ℙ(\bar{X}_f ≠ \bar{X}_m | μ = 0, σ )?$