Original subtitles

Instructor: Hi again.

So you know what a hypothesis is,

and you have an idea of how to form

the null and alternative hypotheses.

By the end of this lesson

we will understand the reason why hypothesis testing works.

First, we must define the term significance level.

Normally, we aim to reject the null if it is false, right?

However, as with any test, there is a small chance

that we could get it wrong and reject

a null hypothesis that is true.

The significance level is denoted

by alpha and is the probability

of rejecting the null hypothesis if it is true.

So the probability of making this error,

typical values for alpha are 0.01, 0.05, and 0.1.

It is a value that you select based

on the certainty you need.

In most cases,

the choice of alpha is determined

by the context you are operating in

but 0.05 is the most commonly used value.

Let's explore an example.

Say you need to test if a machine is working properly

You would expect the test to make little

or no mistakes as you wanna be very precise,

you should pick a low significance level, such as 0.01.

The famous Coca-Cola glass bottle is 12 ounces.

If the machine pours 12.1 ounces

some of the liquid will be spilled

and the label would be damaged as well.

So in certain situations

we need to be as accurate as possible.

However, if we are analyzing humans or companies

we would expect more random or at least uncertain behavior

and hence a higher degree of error.

For instance, if we wanna predict how much

Coca-Cola it's consumers drink, on average,

the difference between 12 ounces and 12.1 ounces

will not be that crucial.

So we can choose a higher significance level

like 0.05 or 0.1.

Okay, now that we have an idea about the significance level,

let's get to the mechanics of hypothesis testing.

Imagine you were consulting a university and wanna carry out

an analysis on how students are performing on average.

The university dean believes

that on average students have a GPA of 70%.

Being the data driven researcher that you are

you can't simply agree with his opinion,

so you start testing.

The null hypothesis is the population mean grade is 70%.

This is a hypothesized value, and we denote it with mu zero.

The alternative hypothesis is

the population mean grade is not 70%,

so mu zero defers from 70%.

All right, assuming that the population

of grades is normally distributed,

all grades received by students should look this way.

That is the true population mean.

Now a test we would normally perform is the Z test.

The formula is, Z equals the sample mean,

minus the hypothesized mean, divided by the standard error.

The idea is the following,

we are standardizing or scaling the sample mean we got,

if the sample mean is close enough to the hypothesized mean,

then Z will be close to zero,

otherwise it will be far away from it.

Naturally, if the sample mean is

exactly equal to the hypothesized mean, Z will be zero.

In all these cases, we would accept the null hypothesis.

Okay, the question here is the following,

how big should Z be for us to reject the null hypothesis?

Well, there is a cutoff line.

Since we are conducting a two sided or a two tail test,

there are two cutoff lines, one on each side.

When we calculate Z, we will get a value.

If this value falls into the middle part

then we cannot reject the null.

If it falls outside, in the shaded region

then we reject the null hypothesis.

That is why the shaded part is called rejection region.

All right, the area that is cut off

actually depends on the significance level.

The level of significance, alpha, is 0.05.

Then we have alpha divided by 2,

or 0.025 on the left side and 0.025 on the right side.

Now, these are values we can check from the Z table.

When alpha is 0.025, Z is 1.96,

so 1.96 on the right side and -1.96 on the left side.

Therefore, if the value we get for Z

from the test is lower than minus 1.96 or higher than 1.96,

we will reject the null hypothesis.

Otherwise, we will accept it.

That's more or less how hypothesis testing works.

We scale the sample mean

with respect to the hypothesized value.

If Z is close to zero, then we cannot reject the null.

If it is far away from zero

then we reject the null hypothesis.

Okay, what about one-sided tests?

We have those too.

Let's take the example from last lecture.

Paul says, "Data scientists earn more than $125,000."

So, H zero, is mu zero, is bigger or equal to $125,000.

The alternative is that mu zero is lower than $125,000.

Using the same level of significance,

this time, the whole rejection region is on the left,

so the rejection region has an area of alpha.

Looking at the Z table

that corresponds to a Z score of 1.645,

and since it is on the left, it is with a minus sign.

Now, when calculating our test statistic Z,

if we get a value lower than -1.645,

we would reject the null hypothesis

as we have statistical evidence

that the data scientists salary is less than $125,000.

Otherwise, we would accept it.

All right, to exhaust all possibilities,

let's explore another one-tail test.

Say the university dean told you

that the average GPA students get is lower than 70%.

In that case, the null hypothesis is

mu zero is lower or equal to 70%,

while the alternative, mu zero is bigger than 70%.

In this situation, the rejection region

is on the right side,

so if the test statistic is bigger than the cutoff Z score,

we would reject the null.

Otherwise we wouldn't.

Cool. That's all for now.

In a lesson or two, we'll start testing.

Just hold on a bit and thanks for watching.