Distributions, Probability, and the Normal Curve

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Distributions, Probability, and the Normal Curve

• Session 8

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• In research we often work from a sample to a
  population
• We draw conclusions from the sample about the
  population using probabilities
• Example Jar of marbles 50 white, 50 black
• Example Jar of marbles 90 black, 10 white
• By knowing the makeup of a population, we can
  determine the probability of obtaining specific
  samples

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• With the marbles examples, we begin with a
  population and then use probability to describe
  the samples that could be obtained. This is
  exactly backward from what we will do with
  inferential statistics.
• Example Jar of marbles which jar was it,
  based on the marbles you picked?

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• The inevitable, unavoidable definition
• In a situation where several different outcomes
  are possible, we define the probability for any
  particular outcome as a fraction or proportion.
  If the possible outcomes are identified as A, B,
  C, D, and so on, then
• probability of A number of outcomes classified
  as A
• total number of possible outcomes

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• Example p (ace) 4/52
• p (king of hearts) 1/52
• note that probability is defined as a proportion
  as in, out of the whole deck, what proportion
  are kings?
• p (spade) 13/52 1/4
• p (heads) 1/2 in a coin toss
• decimals or percentages can be used as well
• p 1/4 0.25 25
• probability values are most often expressed as
  decimal values, but any of the forms (fraction,
  proportion, percentage) is acceptable

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• Note that probability values are bounded between
  0 and 1
• Example A jar with all white marbles
• p (black) 0
• p (white) 1
• Two requirements for a random sample
• 1. Each individual in the population must have an
  equal chance of being selected p (i) 1/N
• 2. If more than one individual is to be selected
  for the sample, there must be constant
  probability for every selection p (jack of
  diamonds)

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Probability and Frequency Distributions

• ? Populations can be displayed as a graph
• ? Portions of the graph represent portions of the
  population
• ? Because probability and proportion are
  equivalent a particular proportion of the graph
  corresponds to a particular probability
• ? Whenever a population is presented in a
  frequency distribution graph, it will be possible
  to represent probabilities as proportions of the
  graph.

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The Normal Distribution

• Symmetrical
• Can be described by the proportions of area
  contained in each section of the distribution
• Sections of the normal distribution can be
  identified by z-scores

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• Example We have a feeling thermometer for
  President Bush. Scores have a mean of 68 and are
  normally distributed. What is the probability of
  randomly selecting an individual who scores
  President Bush at 80?
• p (X gt 80) ?
• Translate to a proportion question what
  proportion is greater than 80? In other words,
  what proportion is to the right of 80?
• Identify the position of 80 by computing a
  z-score
• z 80 68/6 12/6 2.00

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Looking Ahead to Inferential Statistics
Original Population
Treatment
Sample
Treated Sample