## Learning Objectives

- Conduct a randomized experiment taking a sample from a larger population
- Model the mean of a random process using a t-test and describe the assumptions of the model
- Explain the meaning of a confidence interval

## A simple example

We want to take the observations of some numeric variable and provide
an estimate of its **true** mean. The techniques we
cover today will apply to two similar but conceptually different
cases. These are:

- a variable sampled independently from a larger population
- a variable observed from repeated trials of a random process

In the first case, the true mean is the mean of the entire population. For the second case, the true mean is the average value we would get from an infinite number of trials. As long as the sample from the population is taken at random and the output collected from each random trial is independent of prior trials, the same exact technique is used for estimating the mean of both situations.

Consider a random sample of coins from a cup similar to the one we have in class:

Our best guess for the average value of all of the coins in the cups might be the mean of the sample we took:

Letâ€™s do this is a different way that will allow us to extrapolate on this single number:

This says to construct a model for the variable `number`

from the data
set `coins`

. The `1`

indicates that we are fitting a
single mean to the dataset; we will see later how to fit more
complex models. To see the output of the model, run `reg_table`

:

The model calls the mean an intercept, for reasons that will become clear shortly, and it gives the exact same value as with our old technique. The other numbers above and below the table can be useful but are not our primary subject of interest at the moment.

Why bother with this more involved method for finding a mean? For
one thing, `reg_table`

provides an option called `level`

that can
be set to a number between 0 and 1. For example:

The table now includes two additional numbers of the mean: the
5th and 95th percentiles of a *confidence interval*. A confidence
interval provides a guess for where the **true** mean, defined in either
of the ways described as above, actually lies. The construction of
a confidence interval involves some surprisingly deep mathematics,
including the law of large numbers and the central limit theorem.
Using confidence intervals is, however, incredibly simple!

The confidence level, here 90%, gives the probability that the testing procedure will lead to a correct result if a sample or experiment is repeated many times. Common confidence levels include 90%, 95%, and 99%.

Taking a set of sampled flight times from paper helicopters:

We can run the exact same analysis:

Unless we have a specific reason to use a different level, we will usually use a 95% confidence interval in this course.