Statistics t Distributions

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Today we are going to talk about t-distribution.0001

Previously, we learn that there are different situations where we use z and when you use t.0004

Today we are going to talk about when to use z versus t.0011

We are going to break down and sort of reflect and recognize what is z and t?0015

What do they have in common and with is different about them?0022

For certain cases we are going to ask question, why not z why t instead?0024

What does not z have?0031

What is deficient about z?0033

We will talk about rules of t distribution, they follow certain patterns and t distributions0035

are a family of distributions separated by degrees of freedom.0044

Different t distributions have different degrees of freedom.0049

We are going to talk about what are degrees of freedom?0053

We are going to talk about how degrees of freedom relates to that family of t distribution, and then finally summarize how to find t.0056

First off, when do we use z versus t?0065

We covered in the previous sections where we look at whether we knew the population parameters or not.0072

In hypothesis testing, we frequently do not know the mu of the population, but sometimes we are given sigma for some reason or another.0080

In this case we use z in order to figure out how many standard errors away from the mean we are in our SDOM.0091

But in other situations, we do not know what sigma is.0102

In that case we use t in order to figure out how many standard errors away our x bar is from our mu.0107

Just to draw that picture for you remember we are interested in the SDOM because the SDOM tends to be normal given certain conditions.0118

Although mu sub x bar = mu given the CLT what we often want to know is if we have x bar that fall here or x bar that falls here.0126

We want to know how far away it is from the mu sub x bar.0147

In order to find that we would not just use the raw score and get the raw distance but we would want that distance in terms of standard deviation.0153

But because this is the SDOM, we call it the standard error.0165

We would either want a z or t and these numbers tell us how many standard errors away we are from this point right in the mu.0168

What is the z and t?0181

The commonality as we saw before is it tells us number of standard error away from mu sub x bar and that is common to both.0186

That is what the z score and t score both have in common.0208

Because of that their formula looked very much the same.0213

For instance, one way we can write the z formula is like this.0217

We have x bar - mu or mu sub x bar they are the same and this gives us the distance in terms of just the raw values.0231

Just how many whatever inches away, points away, whatever it is.0251

Whatever your raw score means, degrees away divided by standard error.0258

If we double-click on that standard error and look at what is inside than the standard error also written as sigma sub x bar0264

because it is the standard deviation of a whole bunch of mean = sigma ÷ √n.0275

If we look at the t score formula then we have almost the same formula.0284

We have that distance ÷ how big your little steps are, how big your standard deviations are.0294

But when we double-click on the standard error like something on the desktop, you double-click it and open it up what is inside?0302

Well, you could also write this one as s sub x bar and that would be s ÷ √n.0311

Here in lies this difference right there.0323

That is our difference.0325

Here the difference is that standard error found using the sigma, the true population standard deviation.0327

Obviously if you use the real deal that is better or more accurate than the standard error found using estimate population standard deviation.0345

That is s.0371

S is estimated from the sample and if we double clicked on s it would look like this.0375

It is that basic idea of all the squared deviations away from x bar, away from the mean of the sample.0383

X sub i - x bar².0395

We have all the squared deviations and we add them up ÷ n -1 because this is our estimate of the population standard deviations0402

and all of that under the square root sign in order to just leave us a standard deviation rather than variance.0414

This is an estimate of population standard deviation.0421

It is not the real deal, so it is not as accurate.0426

One thing you should know is that the z score is less variable and the t score is going to be more variable.0430

That is going to come in to bear on why we use which one.0438

Okay, so why not z?0443

When we have situations where we do not have the population standard deviation, why not z?0448

Why cannot just be like you are using s, why cannot we do that?0458

Why do we use t?0466

It is because we use s this is actually something a little bit weird.0468

The weirdness comes from the fact that this s is much more variable than sigma.0474

Sometimes when we get our estimate, our estimate is scat on.0481

Sometimes when we get our estimate it is off.0485

That is what we mean when it is more variable.0489

It is not going to hit the nail and head everything single time.0491