10. Hypothesis Testing Methods and Confidence Regions 609
1.2 Experiment, Sample Space, Outcome and Event
The term experiment is used very generally in the field of probability and statistics, and is not at all limited to the colloquial interpretation of the term as referring to activities that scientists perform in laboratories.
Definition 1.1
Experiment Any activity for which the outcome or final state of affairs cannot be specified in advance, but for which a set containing all potential outcomes or final states of affairs can be identified.
Thus, determining the yield per acre of a new type of wheat, observing the quantity of a commodity sold during a promotional campaign, identifying the fat percentage of a hundredweight of raw farm milk, observing tomorrow’s closing Dow Jones Industrial Average on the NY Stock Exchange, or analyzing the underlying income elasticity affecting the demand for gasoline are all examples of experiments according to this definition of the term.
The final state of affairs resulting from an experiment is referred to as an outcome.
Definition 1.2 Outcome of an Experiment
A final result, observation, or measurement occurring from an experiment.
Thus, referring to the preceding examples of experiments, 80 bushels per acre, 2,500 units sold during a week of promotions, 3.7 percent fat per hundred- weight, a DJIA of 13,500, and an income elasticity of .75 are, respectively, possible outcomes.
Prior to analyzing probabilities of outcomes of an experiment, it is necessary to identify what outcomes arepossible. This leads to the definition of thesample spaceof an experiment.
Definition 1.3
Sample Space,S A set that contains all possible outcomes of a given experiment.
Note that our definition of sample space, which we will henceforth denote by S, does not necessarily identify a unique set since we require only that the sample spacecontainsall possible outcomes of an experiment. In many cases, the set of all possible outcomes will be readily identifiable and not subject to controversy, and in these cases it will be natural to refer to this set as the sample space. For example, the experiment of rolling a die and observing the number of dots facing up has a sample space that can be rather uncontroversially specified as {1, 2, 3, 4, 5, 6} (as long as one is ruling out that the die will not land on an edge!). However, defining the collection of possible outcomes of an experiment may also require some careful deliberation. For instance, in our example of measuring the fat percentage of a given hundredweight of raw farm milk, it is clear that the outcomes must reside in the setAẳfx:0x100g.However, the accuracy of our measuring device might only allow us to observe differences in fat percentages up to hundredths of a percent, and thus a smaller set containing all possible measurable fat percentages might be specified as Bẳ
0; :01; :02; :::;100
f gwhereBA. It might be argued further that fat percentages of greater than 20 percent and less than 1 percent will simply not occur in raw farm milk, and thus the smaller setCẳf1; 1:01; 1:02; :::; 20gwhereCBA could represent the sample space of the fat-measuring experiment. Fortunately, as the reader will come to recognize, the principal concern of practical impor- tance is that the sample space be specified large enough to contain the set of all possible outcomes of the experiment as a subset. The sample space need not be identically equal to the set of all possible outcomes. The reader may wish to suggest appropriate sample spaces for the remaining four example experiments described above.
Consistent with set theory terminology and the fact that the sample space is indeed a set, each outcome in a sample space can also be referred to as an elementormemberof the sample space. In addition, the outcomes in a sample
1.2 Experiment, Sample Space, Outcome and Event 3
space are also sometimes referred to as sample points. The reader should be aware of these multiple names for the same concept, and there will be other instances ahead where concepts are referred to by multiple different names.
The sample space, as all sets, can be classified according to whether the number of elements in the set is finite, countably infinite, or uncountably infinite.1Two particular types of sample spaces will figure prominently in our study due to the fact that probabilities will ultimately be assigned using either finite mathematics, or via calculus, respectively.
Definition 1.4
Discrete Sample Space A sample space that is finite or countably infinite.
Definition 1.5 Continuous Sample Space
An uncountably infinite sample space that consists of a continuum of points.
The fundamental entities to which probabilities will be assigned are events, which are equivalent tosubsetsin set theoretic terminology.
Definition 1.6
Event A subset of the sample space.
Thus, events are simply collections of outcomes of an experiment. Note that a technical issue in measure theory can arise when we are dealing with uncountably infinite sample spaces, such that certain complicated subsets can- not be assigned probability in a consistent manner. For this reason, in more technical treatments of probability theory, one would define the termeventto refer tomeasureablesubsets of the sample space. We provide some background relating to this theoretical problem in the Appendix of this chapter. As a practi- cal matter, all of the subsets to which an empirical analyst would be interested in assigning probability will be measureable, and we refrain from explicitly using this qualification henceforth.
In the special case where the event consists of a single element or outcome, we will use the special termelementary eventto refer to the event.
Definition 1.7
Elementary Event An event that is a singleton set, consisting of one element of the sample space.
1A countably infinite set is one that has an infinite number of elements that can be “counted” in the sense of being able to place the elements in a one-to-one correspondence with the positive integers. An uncountable infinite set has an infinite number of elements that cannot be counted, i.e., the elements of the set cannot be placed in a one-to-one correspondence with the positive integers.
One says that an event A has occurred if the experiment results in an outcome that is a member or element of the event or subsetA.
Definition 1.8 Occurrence of an Event
An event is said to have occurred if the outcome of the experiment is an element of the event.
The real-world meaning of the statement “the eventAhas occurred” will be provided by the real-world definition of the setA. That is, verbal or mathemati- cal statements that are utilized in a verbal or mathematical definition of setA, or the collection of elements or description of elements placed in brackets in an exhaustive listing of setA, provide the meaning of “the event A has occurred.”
The following examples illustrate the meaning of botheventand theoccurrence of an event.
Example 1.1 Occurrence of Dice Events
An experiment consists of rolling a die and observing the number of dots facing up. The sample space is defined to beSẳ{1, 2, 3, 4, 5, 6}. Examine two events in S:A1ẳ{1, 2, 3}andA2 ẳ{2, 4, 6}. EventA1has occurred if the outcome,x, of the experiment (the number of dots facing up) is such thatx∈A1. ThenA1is an event whose occurrence means that after a roll, the number of dots facing up on the die is three or less. EventA2has occurred if the outcome,x, is such thatx∈A2. Then A2is an event whose occurrence means that the number of dots facing up on the
die is an even number. □
Example 1.2 Occurrence of Survey Events
An experiment consists of observing the percentage of a large group of consumers, representing a consumer taste panel, who prefer Schpitz beer to its closest competitor, Nickelob beer. The sample space for the experiment is specified as Sẳ fx:0x100g. Examine two events in S:A1ẳ{x:x<50}, andA2 ẳ{x:x>75}. EventA1has occurred if the outcome,x, of the experiment (the actual percentage of the consumer panel preferring Schpitz beer) is such that x∈A1. ThenA1is an event whose occurrence means that less than 50 percent of the consumers preferred Schpitz to Nickelob or, in other words, the group of consumers preferring Schpitz were in the minority. EventA2has occurred if the outcomex∈A2. ThenA2is an event whose occurrence means that greater than 75 percent of the consumers preferred Schpitz to Nickelob. □ When two events have no outcomes in common, they are referred to as disjoint events.
Definition 1.9
Disjoint Events Events that are mutually exclusive, having no outcomes in common.
The concept is identical to the concept of mutually exclusive or disjoint sets, whereA1 andA2are disjoint eventsiff A1\A2ẳ ;. Examples 1.1 and 1.2 can be used to illustrate the concept ofdisjoint events. In Example 1.1, it is recognized that eventsA1andA2arenotmutually exclusive events, sinceA1\A2ẳf g 6ẳ ;2 .
1.2 Experiment, Sample Space, Outcome and Event 5
Events that are not mutually exclusive can occur simultaneously. EventsA1and A2will occur simultaneously (which cannot be the case for mutually exclusive events)iff x2A1\A2ẳf g. In Example 1.2, events2 A1andA2are disjoint events since A1\A2ẳ ;. EventsA1andA2 cannot occur simultaneously since if the outcome is such thatx∈A1, then it follows thatx=2A2, or ifx∈A2, then it follows thatx=2A1.
We should emphasize that in applications it is the researcher who specifies the events in the sample space whose occurrence or lack thereof provides useful information from the researcher’s viewpoint. Thus, referring to Example 1.2, if the researcher were employed by Schpitz Brewery, the identification of which beer was preferred by a majority of the participants in a taste comparison would appear to be of significant interest to the management of the brewery, and thus event A1 would be of great importance. Event A2 in that example might be considered important if the advertising department of Schpitz Brewery wished to utilize an advertising slogan such as “Schpitz beer is preferred to Nickelob by more than 3 to 1.”