Homer White, Georgetown College

Always remember to make sure the necessary packages are loaded:

```
require(mosaic)
require(tigerstats)
```

We say that we are about 95% confident that

\[ \bar{x}-2SE(\bar{x}) < \mu < \bar{x}+2SE(\bar{x}), \]

**because** before the sample was taken

\[ P(\bar{x}-2SE(\bar{x}) < \mu < \bar{x}+2SE(\bar{x})) \approx 0.95. \]

\[ \bar{x}-2SE(\bar{x}) < \mu < \bar{x}+2SE(\bar{x}) \]

means the same as:

\( \mu \) is within 2 SE's of \( \bar{x} \),

which means the same as

\( \bar{x} \) is within two SE's of \( \mu \).

But this means the same thing as:

\[ -2 < \frac{\bar{x}-\mu}{SE(\bar{x})} < 2. \]

For short, let's define the *t-statistic* as:

\[ t=\frac{\bar{x}-\mu}{SE(\bar{x})}. \]

- This is the “t” in
`ttestGC()`

. - \( t \) says how many SEs \( \bar{x} \) is above/below the population mean \( \mu \).
- So \( t \) has “\( z \)-score style”!

Then all along we were saying that before the sample is taken:

\[ P(-2 < t < 2) \approx 0.95. \]

How good is this approximation? Can we do better?

\[ t=\frac{\bar{x}-\mu}{SE(\bar{x})}, \\ SE(\bar{x})=\frac{s}{\sqrt{n}}. \]

So \( t \) depends on:

- \( \bar{x} \) and \( s \), which depend on …
- the sample, which depends on …
- …
**chance!**

So \( t \) is a random variable!

If

- you take a random sample from a population and
- the population in perfectly normal and
- your sample is of size \( n \)

then

- probabilities for \( t \) are given by a
*t-density curve*with \( n-1 \)*degrees of freedom*.

- There is a \( t \)-curve for each degree of freedom \( df = 1,2,3,\ldots \)
- They are symmetric and centered around 0
- They have fatter tails than the standard normal curve does
- But the bigger the degree of freedom is, the more the \( t \)-curve resembles the standard normal curve

```
require(manipulate)
tExplore()
```

To find probabilities for \( t \)-random variables, use `ptGC()`

.

**Example:** Say that you are going to take a SRS of size \( n=40 \) from a population. What is

\[ P(-2 < t <2)? \]

```
ptGC(c(-2,2),region="between",
df=39,graph=TRUE)
```

```
[1] 0.9475
```

At size \( n=40 \), rough 95% intervals are not so bad!

**Example:** Say that you are going to take a SRS of size \( n=4 \) from a population. What is

\[ P(-2 < t <2)? \]

```
ptGC(c(-2,2),region="between",
df=3,graph=TRUE)
```

```
[1] 0.8607
```

At size \( n=4 \), rough 95% intervals are not reliable!

Our rough 95%-confidence intervals for are of the form:

\[ \bar{x} \pm 2SE(\bar{x}) \]

or more generally:

\[ \textbf{estimator} \pm 2SE(\textbf{estimator}) \]

The 2 is a *multiplier*. It makes the interval a rough 95%-confidence interval.

We used 1 as a multiplier to make rough 68%-confidence intervals:

\[ \bar{x} \pm SE(\bar{x}) \]

We used 3 as a multiplier to make rough 99.7%-confidence intervals:

\[ \bar{x} \pm 3SE(\bar{x}) \]

`ttestGC()`

also uses multipliers to make its confidence intervals.If the population is exactly normal, then they yield exactly the advertised level of confidence!

*How does R find the multipliers?*

Say that:

- you have taken a random sample of size \( n=16 \) from a population;
- you know the population is normally distributed,
- you don't know \( \mu \) or \( \sigma \);
- you want to make a 95%-confidence interval for \( \mu \).

Your interval will look like:

\[ \bar{x} \pm t^*SE(\bar{x}), \]

where \( t^* \) is the multiplier for a 95%-confidence interval for \( \mu \), at sample size \( n=16 \).

We want

\[ P(-t^* < t < t^*) = 0.95. \]

```
[1] 0.95
```

Looks like \( t^* \) should be about 2.13145.

So at sample size \( n=16 \), R computes a 95%-confidence interval using the formula:

\[ \bar{x} \pm 2.13145 \times SE(\bar{x}). \]

The multiplier depends on:

- the sample size \( n \)
- the desired level of confidence

If the population is not exactly normal

- then R's formula for confidence intervals is not exactly correct!
- (but as sample size \( n \) increases, it is closer and closer to correct)

Suppose you make intervals at a certain level of confidence, say 95%.

- For your method to be reliable, your intervals should contain \( \mu \) 95% of the time, in repeated sampling.
- If the population is normal and you took a random sample, this will happen!
- (No matter what the sample size is!)

… the population is not normal?

Then your intervals are only approximately 95%-confidence intervals.

- At very large sample sizes, this is not a problem (CLT).
- But at smaller sample sizes, you must have some reason to believe that the population is not “too far” from normal.

```
require(shiny)
runApp(system.file("CIMean",
package="tigerstats"))
```

or:

http://homer.shinyapps.io/CIMean

or:

- At smaller sample sizes (\( n < 30 \), say), make a histogram or boxplot of the sample.
- If you see much skewness, be worried.
- If you see outliers, be worried.
- The smaller the sample size, the more the skewness/outliers should worry you.

The other part of the safety check is:

Did we take a random sample from the population?

If our sample was not random, no amount of clever proability theory will help us make reliable confidence intervals.