Statistics Confidence Intervals for the Difference of Two Independent Means

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Today we are going to talk about confidence intervals for the difference of two independent means.0002

It is pretty important that there are for independent means because later we are going to go to non-independent or error means.0007

We have been talking about how to find confidence intervals and hypothesis testing for one mean.0013

We are going to talk about what that means for how we go about doing that for two means.0023

We are going to talk about what two means means?0029

We are going to talk a little bit about mu notation and we are going to talk about sampling distribution of the difference between two means.0032

I am going to shorten this, this is just means this is not like official or anything as SDOD0041

because it is long to say assembling distribution of the difference between two means, but that is what I mean.0048

We will talk about the rules of the SDOD and those are going to be very similar to the CLT (the central limit theorem) with just a few differences.0055

Finally, we all set it all up so that we can find and interpret the confidence interval.0066

One mean versus two means.0075

So far we have only looked at how to compare one mean against some population, but that is not usually how scientific studies go.0081

Most scientific studies involve comparisons.0091

Comparisons either between different kinds of water samples or language acquisition for babies versus babies who did not.0093

Scores from the control group versus the experimental group.0102

In science we are often comparing two different sets of the two different samples.0106

Two means really means two samples.0112

Here in the one mean scenarios we have one sample and we compare that to an idea in hypothesis testing0120

or we use that one sample in order to derive the potential population means.0132

But now we are going to be using two different means.0140

What do we do with those two means?0143

Do we just do the one sample thing two times or is there a different way?0145

Actually, there is different and more efficient way to go about this.0152

Two means is a different story.0155

They are related but different story.0159

In order to talk about two means and two samples, we have to talk about some new notation.0162

This is totally arbitrary that we use x and y.0170

You could use j and k or m and n, whatever you want.0176

X and y is the generic variables that we use.0182

Feel free to use your favorite letters.0189

One sample will just be called x and all of its members in the sample will be x sub 1, x sub 2, x sub 3.0191

When we say x sub I, we are talking about all of these little guys.0203

The other sample we do not just call it x as well because we will get confused.0208

We cannot call it x2 because x sub 2 has a meaning.0216

What we call it is y.0221

Y sub i now means all of these guys.0224

We could keep them separate.0229

In fact this x and y is going to follow us from here on out.0232

For instance when we talk about the mean of x we call it the x bar.0236

What would be the mean of y?0241

Maybe y bar right.0243

That makes sense.0246

And if you call this b, this will be b bar.0247

It just follows you.0253

When we are talking about the difference between two means we are always talking about this difference.0256

That is going to be x bar - y bar.0264

Now you could also do y bar - x bar, it does not matter.0267

But definitely mean by the difference between two means.0271

We could talk about the standard error of all whole bunch of x bars, standard error of x, standard error of y.0274

You could also talk about the variance of x and the variance of y.0285

You can have all kinds of thing they need something to denote that they are little different.0292

That standard error of x sort and another way you could write it is that we are not just talking about standard error.0298

When we say standard error, you need to keep in mind if we double-click on it that means the standard deviation of a whole bunch of means.0312

Standard deviation of a whole bunch of x bars.0322

Sometimes we do not have sigma so we cannot get this value.0328

We might have to estimate sigma from s and that would be s sub x bar.0334

If we wanted to know how to get this that would just be s sub x.0345

Notice that is different from this, but this is the standard error and this is the actual standard deviation of your sample ÷ √n.0353

Not just n the n of your sample x.0367

In this way we could perfectly denote that we are talking about the standard error of the x, the standard deviation of the x, and the n(x).0372

You could do the same thing with y.0387

The standard error of y, if you had sigma, you can just call it sigma sub y bar because it is the standard deviation of a whole bunch of y bars.0390

Or if you do not have sigma you could estimate sigma and use s sub y bar.0402

Instead of just getting the standard deviation of x we would get the standard deviation of y and divide that by √n Sub y.0411

It makes everything a little more complicated because now I have to write sub x and sub y after everything.0423

But it is not hard because the formula if you look remains exactly the same.0430