Stat 202: Lecture 18/19 (covers pp. 192-211)

Nathan VanHoudnos
11/7/2014

Agenda

  1. Homework comments
  2. Lecture 18/19 (covers pp. 192-211)

Homework #5 Comments

plot of chunk unnamed-chunk-1

mean(hw5/165)
[1] 0.7391053
median(hw5/165)
[1] 0.8484848

One of the harder homeworks.

Homework #5 comments

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Students did much better than on the test!

I have posted solutions, as promised.

(Place holder for postponed question.)

Homework #5 comments

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Here the most points were lost on Question 5.

  • Many people missed the final question:

What is the probability that the true value of 50 is in a 95% confidence interval? (10 pts)

A perfect answer

If we’re looking at a specific 95% confidence interval, the probability that the true value of 50 g is in there is either 0% or 100%.

If we’re talking about 95% confidence intervals as a whole, we know that 95% of the time they contain the true population mean. If we drew a 95% confidence interval from a hat that contained 100 95% confidence intervals, 95% of the time we will draw a mean-containing interval.

However, the probability that the mean is in the specific 95% confidence interval, due to the mean being a non-random value, is either 0% or 100%.

A common misconception

A student wrote this:

The probability that the true value of [the] mean is in a 95% confidence interval is 95% or 0.95.

Although the way the question was asked makes this response tempting, it is still wrong.

This question is a lot like the Obama question from Homework 2.

If I tempt you to say something false it is your duty to hold the line.

You will see this again!

Agenda

  1. Homework comments
  2. Lecture 18/19 (covers pp. 192-211)

Hypothesis testing overview

  1. If \( H_0 \) is true, then over many experiments

    • 95% of 95% CIs will contain the value assumed by \( H_0 \), and
    • 5% of 95% CIs will not contain it.
  2. Use this to our advantage:

    • retain \( H_0 \) if the CI contains the \( H_0 \) value, and
    • reject \( H_0 \) when it does not.
  3. Therefore, for a 95% CI when \( H_0 \) is true:

    • 95% of experiments will correctly retain \( H_0 \)
    • 5% of experiments will incorrectly reject \( H_0 \).

The mistakes we can make

We must either Retain \( H_0 \) or Reject \( H_0 \).

             Retain H0   |  Reject H0
==========|==============|===============
          |              |  
==========|==============|===============
          |              |
==========|==============|===============

The mistakes we can make

We must either Retain \( H_0 \) or Reject \( H_0 \).

If \( H_0 \) is true, then

             Retain H0   |  Reject H0
==========|==============|===============
H0 true   |   Good!      |  MISTAKE!
==========|==============|===============
          |              |
==========|==============|===============

The mistakes we can make

We must either Retain \( H_0 \) or Reject \( H_0 \).

If \( H_0 \) is false, then

             Retain H0   |  Reject H0
==========|==============|===============
H0 true   |   Good!      |  MISTAKE!
==========|==============|===============
H0 false  |   MISTAKE!   |  Good! 
==========|==============|===============

The mistakes we can make

We must either Retain \( H_0 \) or Reject \( H_0 \).

The two mistakes have special names:

             Retain H0    |  Reject H0
==========|===============|===============
H0 true   |  Good!        |  Type I error
==========|===============|===============
H0 false  | Type II error |  Good! 
==========|===============|===============

Note: Everything thus far has focused on \( H_0 \) being true. We will get to Type II error in detail later.

Everyone makes mistakes, so why can't you?