Probability

• the relative likelihood of occurrence of any given outcome or event
• P = # of times outcome or event can occur / total # of possible outcomes or events

Probability Rules

• converse rule: if we know the likelihood of
  success of a given outcome, we can calculate the likelihood of failure (or non-success):
  100%-30% (probability of getting a red) = 70% probability of not getting red
• addition rule: "or" likelihood of an outcome when there is more than one result that will be considered a success.
  
  • In such situations the probability of success is equal to the sum of the respective probabilities.
  • probability of getting green (2) or red (3) =>   2/10 + 3/10 = 5/10 or 50%
• multiplication rule: This rule is applied when we want to calculate the likelihood of outcomes in succession.
  
  • In such situations the probability of success is equal to the product of the respective probabilities.
  • probability of getting green and red => 3/10 x 2/10 = 6/10 or 60%

Probability Distribution

• P(x) ≤ 1 where x assumes all possible values
• 0 < P(x) ≤ 1 for every individual value of x

Standard Scores

• it's possible to determine area under the curve for any sigma difference from the mean
• this distance is called a z score or standard score
• z score - indicates direction and degree that any raw score deviates from the mean in sigma units
  • z = (x - μ) / σ
  • μ = mean of a distribution
  • σ = st. dev. of a distribution

Confidence Intervals

• the probability that population kmean falls within range of mean values

Probability Theory

• probabilities always range btwn 0 and 1
• the probability of alternative events is equal to the sum of the probabilities of the individual events
• the proability of a compound event is equal to the multiple of the proabilities of the separate parts of the event

Index

• composite measure that summarizes and ranks - orders several specific observations and represents some more general dimension
• accumulating scores assigned to individual attributes
• Represents a more general dimension – not as specific as a scale

Index Construction

• selecting items
  • logical validity
  • unidimensionality
  • variance
• empirical relationships (relationships among variables)
• index scoring

Scales

• scores are assigned to patterns of responses
• Scores are composites of several items that are linked logically
• more pinpoint picture of individual attributes
• Scales take into account the intensity with which various items reflect the variable
  under study
  

Scale Construction

• Social Distance Scale - willingness to participate in social relations (who'd you not want as your neighbor)
• Thurstone Scale - group of indicators w/ empirical structure among them (answers assigned numerical value - find mean)
• Linkert Scale - standardized response categories (agree, disagree)
• Semantic Differential - rate items of two opposite adjectives

Hypothesis

• basic statement that is tested
  • a claim about a property of a population
• statement of a relationship btwn 2 variables

Null Hypothesis (H₀)

• statement of no difference, chance, relationship
  • the value of a population parameter (eg. mean) is equal to some claimed value
• test H₀ directly: reject or not

Alternative Hypothesis (H₁)

• the statement that the parameter has a value that somehow differs from the null hypothesis
  • the symbolic form of the alternative hypothesis must use one of these symbols: ≠, <, >

Testing a Hypothesis

• Prepare: state the research question
  • identify the researh question (a goal that can be stated in terms of a hypothesis)
  • state the research question
  • identify the givens
    • variables, population parameters, sample parameters
  • identify the statistical test, e.g. single - sample means test
• Test:
  1. state H₀ and H₁ and stipulate direction
  2. describe sampling distribution
  3. state level of significance (a) and test direction and specify critical test score
  4. observe actual sample outcome
  5. make rejection decision