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Statistics: observational studies, consistency, 4 rules of p

1) Describe why observational studies are good for surveys and polls but not for showing causality

2) Define consistency as it related to standard deviation

3) List the 4 rules of probability-

4) Name the type of event probability can be applied to

5) List the 4 rules to gaming and explain why the house always wins

Solution Preview

1) Describe why observational studies are good for surveys and polls but not for showing causality.

In statistics, an observational study draws inferences about the possible effect of a treatment on subjects, where the assignment of subjects into a treated group versus a control group is outside the control of the investigator. This is in contrast with controlled experiments, such as randomized controlled trials, where each subject is randomly assigned to a treated group or a control group before the start of the treatment.

Observational studies are good for finding associations between independent variables and outcome of interest. But this does not imply causation and other studies are necessary to confirm a cause and effect relationship. Mainly, it is difficult to control the assignment of people into treated and control groups since this is already predetermined in an observational study. This leads to overt biases due to non-randomness. Thus, the differences between groups could be due to these biases or other hidden biases instead of the causal variable.

An observer of an uncontrolled experiment (or process) records potential factors and the data output: the goal is to determine the effects of the factors. Sometimes the recorded factors may not be directly causing the differences in the output. There may be more important factors which were not recorded but are, in fact, causal. Also, recorded or unrecorded factors may be correlated which may yield incorrect conclusions. Finally, as the number of recorded factors increases, the likelihood increases that at least one of the recorded factors will be highly correlated with the data output simply by chance.

Observational studies are good for surveys and polls because it gives good summaries of how we expect the sample from the population to be (For example, what fraction of people vote for democratic party, what percentage of people smoke). They are good at telling us how the population is rather than what is causing a particular outcome of interest.

2) Define consistency as it related to standard deviation.

In statistics, a sequence of estimators for parameter θ is said to be consistent (or asymptotically consistent) if this sequence converges in probability to θ. Intuitively, this means that estimators taken far enough in the sequence are more likely to be in the vicinity of the parameter being estimated, and in the limit they will be arbitrarily close to θ with probability one.

In practice one usually constructs a single estimator as a function of an available sample of size n, and then imagines being able to keep collecting data and expanding the sample ad infinitum. In this way one would obtain a sequence of estimators indexed by n and the notion of consistency will be understood as the sample size "tends to infinity". If this sequence converges in probability to the true value of the parameter being estimated, we call it a consistent estimator; otherwise the estimator is said to be inconsistent.

In terms of standard deviation, as the sample size tends to infinity, the standard deviation of the estimator tends to 0.

That, is if you have very large sample sizes, there is little variance in your estimates from different samples and all the estimates are very close to their true values.

3) List the 4 rules of probability.

RULE 1 - All Probability is between 0 and 1 (or 100%)
There are two possibilities:

a. An impossible event
b. A certain event

a. When P(A) = 0, the event will not happen (an impossible event).

Example

Since Bill Clinton cannot run for a third Presidential term:
P(Clinton is President in 2001) = 0

b. When P(A) = 1, the event will always happen (a certainty). A consequence of this rule is if A is the event something in the sample space will happen, then P(A) = 1.

RULE 2 - The Complement Rule
The complement of event A is the event (A does not occur). All simple events in the sample space must either be part of event A or part of the complement of event A.

In words:
The probabilities of an event and its complement add to 1.

In symbols: P(A) + P(not A) = 1

Example
What is the chance a Caucasian will not have type O blood?
If 37% of Caucasians have type O blood,
then 100% - 37% = 63% do not have type O blood.

Note:
The complement is defined by applying NOT to a group.
The complement of Republicans is NOT-Republicans as opposed to Democrats.
There may be Independents or another political party.

RULE 3 - The Either/Or Rule
The union of events A and B are ...

Solution Summary

4 Rules of probability
Consistency of standard deviation
Pros and Cons of Observational studies

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