Nathan VanHoudnos

11/7/2014

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

```
mean(hw5/165)
```

```
[1] 0.7391053
```

```
median(hw5/165)
```

```
[1] 0.8484848
```

One of the harder homeworks.

Students did much better than on the test!

I have posted solutions, as promised.

(Place holder for postponed question.)

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)

If weâ€™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 weâ€™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 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!**

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

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.

- 95% of
Use this to our advantage:

**retain \( H_0 \)**if the CI contains the \( H_0 \) value, and-
**reject \( H_0 \)**when it does not.

Therefore, for a 95% CI when \( H_0 \) is true:

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

- 95% of

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

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

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

If \( H_0 \) is true, then

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

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!
==========|==============|===============
```

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 farhas focused on \( H_0 \) being true. We will get to Type II error in detail later.