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Chapter Eight PSY201

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University of Toronto St. George
Kristie Dukewich

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) = = 0.75 + if odds were 3 to 1 against Fred's winning, p(Fred's winning) = = 0.25 Computing Probability addition and multiplication rule TheAddition Rule determining the probability of occurrence of any one of several possible events probability of occurrence of Aor 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
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