stat-60_lect-02_April_02 - STATS 60, Spring 2008 April 2,...

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Unformatted text preview: STATS 60, Spring 2008 April 2, Lecture 02 1 Freedman Chapters 3 & 4 Populations and samples A population is the universe of objects/subjects we are interested in. If we want to know the average age of US males, the population is every living male in the US. Rather than ask every male his age, we instead choose a sample (a subset) from the population and obtain ages. From this we estimate the average age of all males. A common task in statistics is to use sample information to estimate population information. That’s what a sample is for! Types of variables A random variable in statistics can be thought of (informally) as a container of items holding all possible values. For example, a random variable of the names of US boys, where the name begins with B, holds ”Bob” and ”Burton” and ”Bruno”, each in many copies. Any one realization of the random variable has a specific value, such as ”Barack”. A random value is often denoted by a capital Roman letter, such as X or Y . A particular observation is usually denoted by a small Roman letter. So x = 3 . 2 might be a realization of X , and y = 7 . 1 might be a realization of Y . Random variables are not really containers, they are more akin to potentials. What potential they have for different values is described by their distribution . The distribution of a variable is the chance of obtaining any particular realized value when making an observation. How likely is a value of 5? How likely is a value less than 5.5? How likely is a value less than 3.25? A little more formally, a probability distribution describes the values that a random event can take, and the probabilities it takes them. A realization of a random variable is called a random event. Statistics and parameters Parameters are characteristics of a population . Statistics are characteristics of a sample . Thus, the average age of all US males is a (unknown) parameter. The average age of a sample of males is a statistic. A Figure 1. Types of variables 1 Figure 2. Example of a box plot for data with three groups (MAO data). guess at a parameter, usually based on sample information, is called a parameter estimate . A central theme of statistics is to use sample statistics to estimate population parameters. What are these parameters? Typically they are parameters that describe the distribution of the population. For example, a normal (bell-shaped) distribution is described by two parameters: the population average and the spread around that average. We are keenly interested in distributions of samples and populations, and so we are keenly interested in the parameters that describe these distributions. A parameter is a constant for any steady population. Every time you observe it, it is the same. A statistic is a random variable . Every time you observe a new sample, it can be different . Let’s define a population as the numbers 1 through 5. The population average is 3 every time you check. Now define a sample as any two numbers picked from the population (without replacement). The sample average is a random variable—itnumbers picked from the population (without replacement)....
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This note was uploaded on 04/28/2008 for the course STATS 60 taught by Professor Boik,j during the Spring '08 term at Stanford.

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stat-60_lect-02_April_02 - STATS 60, Spring 2008 April 2,...

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