Day 20: STA205 - Introduction to Statistical Methods

The usual Announcements:
- During class time I'll be on Zoom, at https://nku.zoom.us/j/7057440907.
  Feel free to check in with questions or concerns.
- I hope that you're doing well, in spite of the difficulties.
- Math/Stat lab is now on-line.
- You have an Imath assignment due next Wednesday. Please keep up with those.
- Perhaps more importantly, we have an exam coming up next Thursday. So you'll probably want to start reviewing. The last packet it covers will be the one we cover today (Packet 9) -- so packets 6-9.
  - I'll expect folks to take the exam during the standard class meeting time. Let me know if that's a problem for you.
  - You'll also need a Z-table and a formula sheet, but I'll have a link on your exam for those.

Today:

Today is an opportunity to re-examine One-Sample hypothesis test for the population mean.

We'll be using the sample standard deviation $s$ to estimate the population standard deviation $\sigma$, which means that the normal distribution assumption won't fly anymore. $z$-scores require a $\sigma$ in their calculation: no $\sigma$, no $z$!

We'll have to swap in the t-distribution for the normal distribution. But don't panic.

They look a lot alike, but the problem is that, as the sample size changes, the t-distribution changes: so you'd need a table for each sample size. That would be a lot of tables! The good news is that software takes care of all of that, so, once again, we'll be relying on StatCrunch output to do our work.

Here we have several $t$-distributions plotted against a normal (which is what you get when you let the number of degrees of freedom $\nu$ go to $\infty$. They're squatter -- shorter and wider -- than the normal.

Here's the $t$-distribution for $\nu=17$. It's already hard to tell the difference between it and the normal.

This one looks at the difference between nine levels of degree of freedom, and it compares the t-distribution to the normal. If you see a straight line, then there's no difference. The point that the author makes is that by 20 df the line looks essentially straight. So even for relatively small samples, a t-distribution is basically replaceable by a normal....

I've prepared some video lectures, using an iPad, and I hope that you'll take the time to listen and follow along with your packet.
- Intro
- First Examples
- Second Examples
- The final filled-in packet, and the summary.

Links:
- The Standard Normal Table of probabilities.

Website maintained by Andy Long. Comments appreciated.

	Here we have several $t$-distributions plotted against a normal (which is what you get when you let the number of degrees of freedom $\nu$ go to $\infty$. They're squatter -- shorter and wider -- than the normal.
	Here's the $t$-distribution for $\nu=17$. It's already hard to tell the difference between it and the normal.
	This one looks at the difference between nine levels of degree of freedom, and it compares the t-distribution to the normal. If you see a straight line, then there's no difference. The point that the author makes is that by 20 df the line looks essentially straight. So even for relatively small samples, a t-distribution is basically replaceable by a normal....