Questions

Answer using Sample Space

   Sample space = { (0,M), (1,M), (2,M), ......, (50,M)
                       (0,F), (1,F), (2,F), ......, (50,F) } 
               

We could pose our questions about probabilities related to this sample space, but there are over 100 probabilities to consider.

Since sample space is a set notion, it is very general. But numbers of cigarettes smoked and male/female are very specific, organized concepts.

Better Approach

Approach the data using the concept of a random variable

How many random variables do we want to define over this sample space?

Y = number of cigarettes smoked ( continuous r.v.)

X = 0 if male, 1 if female (dummy r.v.)

Use the notion of expectation to summarize the joint distribution of X and Y.

Possible Statistical Questions about the cigarette data

1. Does P(Y=0 | X=male) differ from P(Y=0 | X=female)?
2. Does E( Y | X=male) differ from E(Y| X=female)?
3. Does E( Y | Y > 0,X=female) differ E(Y | Y>0, X=male)?

Distribution of Cigarettes Smoked by Sex

Estimation

The expressions in the cigarette questions are population parameters. How does one estimate their values when only a sample from the population is available?

An estimator is a random variable, but it is not defined on the same sample space as the observations. For instance, in the cigarette example, Y is a random variable defined on the set of all young Canadians. You tell me which young person was contacted (that's the outcome of the random experiment), and I can find out how many cigarettes he/she claims to smoke each day (that's the mapping from outcomes to real numbers). I can also determine whether it is a he or she (that's the mapping for another random variable defined on the same sample space, namely X). However, for me to know what value the sample mean (that's a random variable too) takes on, it is not enough to tell me the name of one young person. If the sample size is N=3, then you need to tell me the names of 3 young people (that's a subset of S, which is not itself a member of S). So the sample mean is defined on events in S.

The Sample Space for an Estimator

We can calculate estimates from the data :

* log file from a Stata session analyzing the cigs.raw data
. use cigs
. gen byte any = cond(cigs>0,1,0)
. label define sex 1 "men" 2 "women"
. label values dvsex sex
. tab dvsex, summ(cigs)

            |           Summary of cigs
         Sex|        Mean   Std. Dev.       Freq.
------------+------------------------------------
        men |      6.5333     10.6017         300
      women |      4.8933      8.5900         300
------------+------------------------------------
      Total |      5.7133      9.6753         600

. tab dvsex any, row

           | any
        Sex|         0          1 |     Total
-----------+----------------------+----------
       men |       201         99 |       300 
           |     67.00      33.00 |    100.00 
-----------+----------------------+----------
     women |       214         86 |       300 
           |     71.33      28.67 |    100.00 
-----------+----------------------+----------
      Total|       415        185 |       600 
           |     69.17      30.83 |    100.00 

. sort dvsex
. by dvsex: summ cigs if any==1

-> dvsex= men  
Variable |     Obs        Mean   Std. Dev.       Min        Max
---------+-----------------------------------------------------
    cigs |      99     19.7980     8.8109     2.0000    50.0000  

-> dvsex=women  
Variable |     Obs        Mean   Std. Dev.       Min        Max
---------+-----------------------------------------------------
    cigs |      86     17.0698     7.0190     4.0000    35.0000  

The tables reveal:

In words:

An estimated 67 percent of young Cdn men don't smoke

An estimated 71 percent of young Cdn women don't smoke

Young Cdn men smoke an estimated average 6.53 cigarettes a day

Young Cdn women smoke an estimated average 4.89 cigarettes a day

Young Cdn men smoke who smoke at all smoke an estimated 19.8 cigarettes a day

Young Cdn women smoke who smoke at all smoke an estimated 17.1 cigarettes a day