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-: Hi again.

As you probably expected, in this lesson

we will learn about independent samples with known variants.

Let's get into the example right away.

You may remember this one.

We are about to test the average grades of students

from two different departments in a UK university.

I would like to remind you

that in the UK grades are expressed in percentages.

The two departments are engineering and management.

We were told by the dean

that engineering is a tougher discipline

and people tend to get lower grades.

He believes that on average management students

outperform engineering students by four percentage points.

Now it is our job to verify if that is the case.

Let's state the two hypotheses.

H zero is the difference

between the means of the two populations is minus four.

By the way, notice that I can make H zero engineering

minus management and get a negative difference

or I can make H zero management

minus engineer and get a positive difference.

Either way works.

Just so we can see as many different situations as possible.

I will keep the difference negative.

So, H one is the population mean difference is different

than four.

Once again, this is a two-sided test.

Our research question is not to find the difference

but to check if it is exactly four.

Right. Let's get our hands dirty.

Here's the table that summarizes the data.

The sample sizes are 170 respectively.

The sample means our 58% and 65%

and the population's standard deviations are 10%

and 6% and are known from past data.

If you remember, when the population is known

for independent samples, the standard error

of the difference is equal to the square root

of the sum of the variance of engineering divided

by its sample size and the variance of management,

again divided by its sample size.

All we have left is to compute the test statistic.

We have big samples and known variances.

Therefore, we can use the Z statistic.

I hope you are getting the point.

Small samples and unknown variances means T large sample

and known variances mean Z.

When we have large samples and unknown variances

it is up to the researcher

but generally it is okay to use Z in that case as well.

All right, here's the formula for the test statistic.

Sample difference mean

minus hypothesized difference mean divided

by the standard error.

We plug in the numbers and get a Z score of minus 2.44.

Let's calculate the P value.

Once again, I'll just tell you the P value

as usually you will obtain it using a software.

The P value of the two-sided test is 0.015.

What we can say is that at 5% significance, which is common

for such a study, the P value of 0.015 is lower than 0.05.

Thus, we reject the null hypothesis.

There is enough statistical evidence

that the difference of the two means is not 4%.

All right, cool.

Here's a trick.

What if you wanna know

if the difference is higher or lower than four?

The sign of the test statistic

can give you that information.

A minus sign of the test statistic means it's smaller.

If you reverse engineer the standardization process

you'll find that true value is likely to be lower

than the hypothesized value.

In our case, this translates

into the true mean is likely to be lower than minus four.

Lower than minus four entails

that possible values are minus five, minus six, and so on.

This is additional information that you can give

to the dean.

All right, done with that lesson too

let's proceed to the final topic.

Independent samples and unknown variances.