PSY201H1 Chapter Notes -Estimation Theory, Statistical Inference, Statistical Hypothesis Testing

4 views13 pages
20 Nov 2012
School
Department
Course
Page:
of 13
Chapter Eight: Random Sampling and Probability
inferential statistics
+ use sample scores to make a statement about a characteristic of the population
+ two kids of statements
- 1. hypothesis test
- 2. parameter estimation
in hypothesis testing, experimenter collecting data in an experiment on a sample set of subject
in an attempt to validate some hypothesis involving a population
+ example: an educational psychologist believes a new method of teaching math to third graders
in her school district (population) is superior to the usual way of teaching the subject
+ in her experiment, employs two samples of third graders – one is taught using new method,
other the old
+ each group tested on the same final exam
+ psychologist not satisfied with just supporting the mean of the group that received the new
method was higher than the mean of the other group – wants to make a statement like “the
improvement in final exam scores was due to the new teaching method and not chance factors
→ improvement does not apply just to the particular sample tested. Rather, improvement would
be found in whole population of third graders if they were taught by the new method”
in parameter estimation experiments, experimenter is interested in determining the magnitude of
a population characteristic
+ example: an economist interested in determining the average monthly amount of money spent
last year on food by single college students
+ using sample data, can estimate the mean amount spent by the population
+ statements: “the probability is 0.95 that the interval of $250-300 contains the population
mean”
topics of random sampling & probability are central to the methodology of inferential statistics
Random Sampling
to generalize validity from sample to population, both in hypothesis testing and parameter
estimation experiments, the sample cannot be just any subset of the population
+ critical that the sample is a random sample
random sample selected from the population by a process that ensures that each possible
sample of a given size has an equal size of being selected and all the members of the population
have an equal chance of being selected into the sample
sampling with replacement
+ sampling from population one score at a time then placing it back into the population before
drawing again (example)
sample should be a random sample for two reasons
+ 1. to generalize from a sample to a population, it's necessary to apply the laws of probability
to the sample. If sample hasn't been generated by a process ensuring that each possible sample
of that size has an equal size of being selected, can't apply the laws of probability to the sample.
+ 2. to generalize from a sample to a population, it's necessary that the sample be representative
of the population. One way to achieve representativeness is to choose the sample by a process
that ensures that all the members of the population have an equal chance of being selected into
the sample → requiring the sample to be random allows the laws of probability to be used on
the sample and at the same time results in a sample that should be representative of the
population
Techniques for Random Sampling
-
pg 182
Sampling with or without Replacement
sampling with replacement
+ want to form a sample of two scores from a population composed of scores 4, 5, 8, 10
+ randomly draw one score from the population, record its value, and then place it back in the
population before drawing the second score
sampling without replacement
+ randomly draw one score from population and not replace it before drawing the second one –
the same member of the population could appear in the sample only once
sampling with replacement: method of sampling in which each member of the population
selected for the sample is returned to the population before the next member is selected
sampling without replacement: method of sampling in which the members of the sample are
not returned to the population before subsequent members are selected
when subjects selected to participate in replacement, sampling without replacement must be
used because the same individual can't be in the sample more than once
Probability
may be approached in two ways
+ 1. from an a priori, or classical viewpoint
+ 2. from a posteriori, or empirical viewpoint
a priori means that which can be deduced from reason alone, without experienced
+ symbol p(A) is read “the probability of occurrence of event A.”
+ equation states that the probability of occurrence of event A is equal to the number of events
classifiable as A divided by the number of possible events
ie: a pair of dice.
+ each die has six sides with different numbers of spots printed on each sides – 1-6
+ a priori probability, going to roll a die once
+ what's the probability it'll come to rest with a 2 facing upward?
+ since there are six possible numbers that might occur and only one of these is two...
+ two problems solved without resource to any data collection
+ contrasted with the a posteriori, or empirical, approach to probability
+ a posteriori → “after the fact” ;; after some data have been colected
a posteriori, or empirical, viewpoint, probability is defined as
+ to determine the probability of a 2 in one roll die using the empirical approach, we would
have to take the actual die, roll it many times, and count the number of times a 2 has occurred
+ the more we roll the die, the better
+ ie: we roll the die 100.000 times and that a 2 occurs 16,000 times
+ probability of a 2 occurring in one roll of a die is found by
+ necessary to have actual die and to collect some data before determining the probability
+ if the die is evenly balanced (fair die), when we roll the die many, many times, the a
posteriori probability approaches the a priori probability
+ if we roll an infinite number of times, the two probabilities will equal each other
+ if the die is loaded (weighted so that one side comes up more often than the others), the a
posteriori probability will differ from the a priori dtermination
+ ie: if the die is heavily weighted for a 6 to come up, a 2 might never appear
+ a priori equation assumes that each possible outcome has an equal chance of occurrence
Some Basic Points Concerning Probability Values
since probability is fundamentally a proportion, it ranges in value from 0.00 to 1.00
+ if probability of an event equals 1.00, then event is certain to occur
+ if probability equals 0.00, then event is certain not to occur
probability of occurrence of an event is expressed as a fraction or a decimal number
+ ie: probability of randomly picking ace of spades from one draw from a deck of ordinary
playing cards is 1/52 or 0.0192
sometimes probability expressed as “chances in 100”
+ some say that probability of event A will occur is 5 chances in 100 (p(A) = 0.05)
probability also expressed as odds for or against an event occurring
+ a betting person might say that the odds are 3 to 1 favoring Fred to win the race
+ p (Fred's winning) = ¾ = 0.75
+ if odds were 3 to 1 against Fred's winning, p(Fred's winning) = ¼ = 0.25
Computing Probability
addition and multiplication rule
The Addition Rule
determining the probability of occurrence of any one of several possible events
probability of occurrence of A or B is equal to the probability of occurrence of A plus the
probability of occurrence of B minus the probability of occurrence of both A and B
+ assume there's only two possible events, A and B
in equation form, addition rule:
+ want to determine the probability of picking an ace or a club in one draw from a deck of
ordinary playing cards
+ first way: enumerating all the events classifiable as an ace or a club and using the basic