03/01/2021
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The following topics will be covered in this lecture:
Our goal in this course is to use statistics from a small, representative sample to say something general about the larger, unobservable population or phenomena.
Recall, the measures of the population are what we referred to as parameters.
Parameters are generally unknown and unknowable.
Random variables and probability distributions give us the model for estimating population parameters.
Note: we can only “find” the parameters exactly in very simple examples like games of chance.
Generally, we will have to be satisfied with estimates of the parameters that are uncertain, but also include measures of “how uncertain”.
Courtesy of M. W. Toews CC via Wikimedia Commons.
The (arithmetic sample) mean is usually the most important measure of center.
Suppose we have a sample of \( n \) total measurements of some random variable \( X \).
Then, the (arithmetic sample) mean is defined
\[ \text{Sample mean} = \overline{x} = \frac{x_1 +x_2 +\cdots + x_n}{n}= \frac{\sum_{i=1}^n x_i}{n} \]
Q: is the sample mean a statistic or a parameter?
An important property of the sample mean is that it tends to vary less over re-sampling than other statistics.
However, the sample mean is very sensitive to outliers.
A statistic is called resistant if it doesn't change very much with respect to outlier data.