Statistics Introduction to Confidence Intervals

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Today we are going to be introduced to competence intervals.0002

Here is the roadmap for today, first we are going to do a brief overview of inferential statistics.0005

We have been trying to do some inferential statistics but there have been a couple of problems we keep running into.0013

So far I have fudged it.0022

We will address some of those problems head on and come up with 2 solutions.0024

One of those solutions is the competence interval and we are going to talk about competence intervals0031

when the sigma, population, standard deviation is known and when sigma is unknown.0039

Those are the two situations we are going to be focused on.0046

Let us go over inferential statistics.0049

We know the big picture idea there is some population represented by X and we wish we could know the population but we do not.0055

But instead what we can know is little samples.0065

We could know that but the problem is samples are biased.0071

Whenever we have samples and we summarize them using these mathematical summaries we call them statistics.0074

Just to give you an example of some statistics there things like x bar or s, those are all statistics.0084

What we would like to do is use these samples to understand something about the population.0093

Statistics, the field is about using these statistics to estimate parameters and0100

to give you ideas about parameters there are things like mu or sigma.0108

That is our whole goal.0112

Here we realize in order to jump from things like x bar and s to mu and sigma we are going to need more than just wishful thinking.0114

And that is where the sampling distributions come in.0132

Here we talk about sampling distribution often we are talking about some sort of statistic.0135

When we talk about sampling distribution of the mean we are talking about a whole bunch of x bars.0142

Here we have a whole bunch of x.0148

Here we have a whole bunch of x bar and that is the distribution.0150

When we summarize these statistics in the sampling distribution we call them expected values.0155

So it is not just mu, it is mu sub x bar.0164

It is not just sigma it is sigma sub x bar.0168

What we want to do is go from this to understand this but what we have learned0172

so far is how to see the relationship between parameters and expected values.0177

We know that these things have a relationship to each other.0186

And from doing that we could then make this jump.0189

It is like we use this to say something like this.0195

There are two problems with this picture although it seems rosy and there is still to nagging questions.0199

We would look at them a little bit before but we need to solve this more rigorously than we had before.0210

One question is this, what happens when we do not know what the population looks like?0217

Of course we could use the central limit theorem when we know mu and sigma from the population.0222

What if we do not know mu?0229

What if we do not know sigma?0231

Then what happens?0233

Also, how do we know whether a sample is sufficiently unlikely because remember the whole point0234

of the sampling distribution is for us to take sampling distributions from a known population and compare it to an unknown population.0240

If this sample does not match the sampling distribution enough that it is very unlikely to come from the sampling distribution.0254

We could say this is probably not the population that the sample came from.0261

How do we know when it sufficiently weird?0266

To answer these two questions there is going to be to solutions.0269

You can think of it as this one.0275

This first question roughly, they are both actually are answered in each of these but this one goes along better with that one.0281

This one goes along better with that one.0287

The two solutions are these, one is competence interval.0291

When we talk about competence interval here is what we are doing, we are going to figure out where mu might be from the sample.0302

We are going to try to figure out the population mu from the sample and0306

that is what we do when we do not know what the population looks like.0336

We try to figure it out from the sample.0342

Hypothesis testing actually takes another view.0344

The hypothesis testing, we come up with a hypothesis for what the population is like.0349

Hypothesize a population mu first.0355

In this case we are saying we are going to pull from something and figure out and pick a potential population mu.0363

And then we are going to test how weird the sample is.0376

We are going to come up with a number to tell us this is how weird the sample is.0387

We are going to decide is that weirdness weird enough?0393

That is going to be hypothesis testing.0398

But we are going to focus here on competence intervals.0401

Okay, when we talk about competence intervals we need to get an inventory of what we know so far.0404

Basically that is asking the question, which parameters are known or given to us?0413

What happens when we do not know what the population looks like?0418

Well we may not know what0422

The population looks like because we do not know anything about the population, or we know0424

Only a little bit about the population.0428

This is the case where we know a little.0431

Here we do not know mu but we do know sigma.0434

For some reason we have some partial information and that helps us out.0444

Here we know nothing.0450

Here nothing is helping us that we do not know mu and we are trying to figure it out but we do not know sigma either.0454

It is like nothing is helping us out here.0464

We just have to pull ourselves up from our own bootstraps.0466

These are the two situations that we are going to talk about it.0471

Here is the goal of competence interval.0475

The basic idea of the competence interval is going to be this.0480

We are going to try to figure out where mu might be but we do know x bar.0484

We know everything about the sample but we do not know anything about the population.0497

But in this case I am going to show you what happens when we already know sigma.0503

So we have a leg up.0508

We know sigma life is little that easier for us today.0509

Here is the thing, we do not know what the population looks like so cannot draw a normal or skewed or anything.0513

We have no idea what the population looks like and we have no idea what the population mu is.0524

But we for some reason know sigma is.0531

Sigma is given to us.0533

From there can we construct an SDOM?0534

Given that n is sufficiently large we can assume that it is normal.0540

We have no idea what mu is and so we do not know what mu sub x bar is.0548