Original subtitles

-: This is the final lesson we will do on testing.

The last case we'll examine here is the one with

independent samples and unknown variances,

which are assumed to be equal.

I'll quickly brush up your memory

on the data set we did in the confidence interval section.

You were trying to see if apples in New York

are as expensive as the ones in LA.

You went to 10 grocery shops in New York

and your friend Paul,

who lives in LA,

went to eight grocery shops there.

You got all the prices and put them in a table,

with what the population variance of apple prices is.

But you assume it should be the same for New York and LA.

Let's state the null and alternative hypotheses.

H zero: Mu in New York is equal To Mu in LA.

Or Mu in New York minus Mu in LA

is equal to zero.

H one: Mu in New York is different to Mu in LA.

Mu in New York minus Mu in LA differs from zero.

All right, that's our data set.

We have also calculated the sample means,

standard deviations and sample sizes.

What can we do when the variance is unknown

but assumed to be equal?

Earlier, we use the pooled variance formula.

Well, here it is again.

Remember?

All right, it's all about plugging in numbers

so I'll save you the trouble.

The pooled variance is 0.05.

One last thing we need is the standard error

of the difference of means.

It is given by the following formula.

I'm going faster than usual,

as we've seen all of this before.

Moreover, testing is about understanding,

computation is routine.

So, let's start testing, shall we?

Small samples, variance unknown.

Which statistic do we need?

Exactly.

It's the T statistic again.

How many degrees of freedom?

You may recall it from earlier.

It was the combined sample size

minus the number of variables.

So 10 plus eight minus two, which gives us 16.

Let's see the T statistic formula.

Once again, the difference between sample means

minus the difference between hypothesized true means

divided by the standard error.

After plugging in everything,

we get a test statistic of 6.53.

Do we need to compare it?

This is by far the most extreme test statistic we have seen.

You will have a hard time finding it in the T table.

For common tests, a rule of thumb is

to reject the null hypothesis When T-score is

bigger than two.

Generally, for Z-score and T-score

a value that is higher than four is extremely significant.

Let's see the two-sided P value.

The P value of this test is lower than 0.000,

somewhere around 0.000001.

In our lesson about P value, we said that researchers

are always looking for those three zeros after the dot.

It means that the test is extremely significant

and the probability of making a type one error

is virtually zero.

Therefore, we reject the null hypothesis at all common

and uncommon levels of significance.

There is a strong statistical evidence that the price

of apples in New York differs from in LA.

But such an extreme result may also mean

that the hypothesis is pointless or poorly designed.

From the mean values of 3.94 and 3.25, and

with such small and close standard deviations of around 0.2,

we could easily say that the prices are different.

No testing needed.

A much more interesting question would be if the price

of apples in New York is 20% higher than that in LA.

I will leave you this exercise for homework.

All right, we are done with hypothesis testing.

Cheers, and thanks for watching.