Original subtitles

Instructor: We are not done

with hypothesis testing just yet.

Remember how we started with confidence intervals

for a single population mean and then switched

to examples considering multiple populations?

Well, we are in the same situation here.

Single populations are just the beginning.

Time to do multiple means.

We will start with dependent samples.

The most intuitive examples of dependent samples

are the ones you have been through,

like weight loss and blood tests.

The sample is drawn from weight loss data

or concentration of nutrients data

but the subject of interest

is the same person before and after.

Okay, let's get to work.

Do you recall our example with the magnesium levels

in one's blood?

There was this drug company developing a new pill

that supposedly increased levels of magnesium of recipients.

There were 10 people involved in the study

that were taking the drug for some time

and we calculated confidence intervals

that helped us study the effects of that drug.

They indicated the range of plausible values

for the population mean.

This time, we wanna come to a single definite conclusion

about the effectiveness of the drug.

All right, let's state the null hypothesis.

The population mean before is greater

or equal than the population mean after.

The alternative is that the population mean before

is lower than the one after.

Once again, we wanna know

if the magnesium levels are higher.

We construct the null and alternative hypotheses

in such a way so that we are aiming

to reject the null hypothesis.

We expect the levels to be higher

so when the null hypothesis,

we state then to be lower or equal.

Okay, let's reorder a bit.

H0 is mu before, which is equal or higher than mu after.

This is equivalent to mu before minus mu after

is equal to zero or positive.

We can substitute this with capital D0.

It stands for the hypothesized population mean difference.

So we restate our hypotheses using D for simplicity.

Now we have our test designed, let's crunch some numbers.

Here's the dataset.

We have 10 observations people

have registered before and after.

Naturally, the difference is equal to before minus after.

We can calculate the sample mean of the difference.

We get -0.33.

The sample standard deviation is 0.45

and the standard error is 0.14.

The appropriate statistic to use here

is the t-statistic.

We have a small sample we assume normal distribution

of the population and we don't know the variance.

So the t-score is equal to the following expression.

Now we can simply carry out the calculations

and find that its value is -2.29.

Since we don't wanna choose a level of significance,

let's solve this problem with the p-value.

In order to find the p-value of this one-sided test,

we may go to the table

and see it as somewhere between 0.01 and 0.025.

As I told you earlier, using software is much easier.

So after using an online p-value calculator,

I can tell you that it is exactly 0.024.

What was the decision rule again?

If the p-value is lower

than the significance level we are interested in,

we reject the null hypothesis.

Okay, so if the level of significance is 0.05,

and the p-value is lower,

we will be able to reject the hypothesis at 5%.

If the level of significance is 0.01, however,

the p-value is higher,

so we cannot reject the null hypothesis

at a 1% level of significance.

The lowest value for which we can reject it is 0.0024,

which is exactly the p-value.

What does this tell us?

Well, it is up to the researcher

to choose the level of significance.

In the case of the magnesium pill,

we expect that the researcher

will be very cautious

as he would wanna know if this is an effective pill

that will be able to actually help people.

If we cannot say that the pill works

at a 1% significance level,

perhaps it is better to take it back to the laboratory.

An alternative would be to test again

and increase the sample size for better results.

A sample of 100 people would improve the level

of precision significantly.

All right, so we've done some more hypothesis testing

and we've explored some factors

that helped you determine the significance level

of the test.

Stay with us for our next lesson

where we will learn how to test independent samples.

Thanks for watching.