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of observations in a given X interval; the probability
of observing X in a given interval.
-relative f = f/n.
"Probability of observing an X above/below a particular score = the sum of the probabilities above/below the interval that contains X."
- -X = levels or values of the variable.
- -f = observed data.
-what does larger deviation mean for score location?
- The Purpose of Z scores: to identify and describe the exact location of each score in a distribution;
- note: the larger your deviation is, the closer your score tends to be within the mean. (see figure 5.1)
The mean and the standard deviation were methods to describe the ENTIRE distribution of scores, now we shift attention to the INDIVIDUAL scores WITHIN a distribution.
original, unchanged scores that are the direct result of measurement; by itself does not necessarily provide much info about it's position within a distribution. Once "standardize" and turned into a z-score; it tells us the position of the x-value. Second purpose of z-scores is to "standardize" an entire distruibution (ex IQ scores) so that it can be compared to other distributions with other z-scores
(ex: comparing an IQ of 130--with a z-score of 2 vs. an SAT score of 650--with a z-score of 1.5)
The Normal distribution & z scores are use with what kind of data?
used with either continuous data: it uses "real results" (0-100) OR Interval/Ratio Levels (uses M, SS, standard deviation)
"unit normal distribution"
=> Z-score Values to know:
-> value ranges of z-scores
-> standard deviation?
'Turns the mean into 0 and each standard deviation is "1z"; which is what you get when the z-score transformation is used with a normally distributed set of scores
- -each raw score can be transformed by using the z-score formula (Subtract x-value from the mean and divide by the standard deviation)
- -Values range from - ∞ to + ∞' mean=0 & standard deviation(σ)=1 (varies by points)
- => Z-score Values to know:
- 1. "x" raw score of interest
- 2. "μ" (population mean)
- 3. "σ" standard deviation of the raw scores (Do the first three steps)
Z-score Transformation Formula:
x= Z-score ×SD + mean
The z-score transformation is used to:
-what does positive and negative mean?
-when are z-scores no longer "normal" (what number)
turn distributions of raw scores into standardized scores or z-scores.; used to estimate probabilities of outcomes & set critical values.
-Can be positive or negative z-scores: which indicates the direction of z-score (whether it's above or below the mean of 0)
*NOTE* Below 2 and Over 2 Z-scores means the values are no longer normal! The middle composed of 95% which is average.
Size of the Standard Deviations **STUDY**
-what do smaller deviations mean to your x value location to the mean?
-the smaller the deviation, then the smaller the variance which means your x-value will be much farther from the mean (and therefore more Prestigious)
*note: Converting a frequency distribution (a) to a z-score distrubution (b) has no change the shape of the distribution; you only have a z-score distribution (OR a standard distribution) so that standardization produces a "unit normal distribution"
Z-scores & the unit normal distribution (chart meaning )
-what percent is middle?
-what z-score indicates "unlikely to be obtained from original population"
-From z-score of 0-1: how much percent?
we will use the unit normal distribution to find probabilities associated with observations and to set critical values'
-Middle 95% scores under the normal distribution are the high probability values scores near the mean.
-Below -1.96 and above 1.96 (z-scores) indicates .025 of the unit normal distribution; meaning the scores that are very unlikely to be obtained from the original population.
-Note: from 0 to 1: 34.13% and from -1 to 0: also 34.13%
Two types of statistics: Desciptive & Inferential
1. Descriptive: summarize, organize, and simplify data.
2. Inferential: Uses samples to draw conclusions about the population (surveys, z-scores)
Samples: Typical vs. "unusual or improbable"
1. If the sample is typical of what we would expect, we can conclude the sample comes from the specified population.
2. If the sample is very unusual or improbable we can conclude the sameple does NOT come from the specified population.
-This means that we must quantify two things: our expectations about the population vs. what we mean is "improbable" or unusual.
"Something unusual or Improbable" as a sample
Means they do NOT come from the population
-We need some criterion for deciding HOW different Xo (observation) has to be from the Xe (expectation) before we can conclude our observation was "unsual"
-"Inferential statistics: uses PROBABILITY to set precise criteria for deciding whether or not a sample is likely to have come from a given population.
Unusual: Expectations Vs. Observations.
-what do we need to know for our expectatons?
-> "expection": (Xe) is what we think should happen based on what we know about the population parameter (that 3 out of 6 rats will turn right because of the 50% chance-parameter)
-For our expectations, we need to know the sample size (n) and the "parameter" (turning right)
-> "Observation"(Xo): what actually happened with our sample (We saw that 5 rats turned right)
-Parameters used; and what are they based on?
-Ho: specifies? and why isn't Xe used? What is used instead and why for parameters?
Used to calculate probabilities for observations of nominal data with TWO categories.
- --> PARAMETERS: P & Q which applies to all n's (The # of events)
- -population parameters are based on prior knowledge (ex: 90% of pop is right handed)
-Ho is written in terms of P&Q rather than Xe because P&Q apply to all possible n's while Xe changes when n changes
- => Ho: specifies the proportion of the population that belongs to the event class of interest.
- ex: p(right handed) is .90 & P (right turn)=50)
Definitions & notation
Null hypothesis/ Ho:
Note that P+Q= ?
the # of events expected to display the characteristic of interest.
the # events that actually display the characteristic of interest.
a possible value of Xo.
Null hypothesis/ Ho:
specifies expectations as population parameters.
the # of events (people, coin flips, items, trials, etc…) observed.
- Probability: likelihood of observing a particular event class.
- P(A): probability of observing an outcome or characteristic that you are INTERESTED IN which belongs to event class A
probability of observing the only other possible outcome or characteristic
, belonging to event class B—the “other one.”
Note that P+Q=1.00.
Assumptions for using the binomial: Random Sampling.
-what is random sampling?
-p(selection)=? (constant probability?)
Any statistical test is only appropriate when the sample data meet "certain requirements"-Random Sampling
-"Random Sampling": Every member of the population has an EQUAL chance of being selected; Use sampling with replacement for finite populations (marbles always go back)
-p(selection) = 1/ N: Constant probability where p (of selection) ALWAYS equals 1/N.
Independence of observations:
-mutually exclusive & exhaustive?
- The probability of an element being in the sample does NOT depend on any other element's inclusion.
- -Event classes must be mutually exclusive & exhaustive
no elementary element can be a member of both event classes.
every element drawn can be categorized as one or the other event class.
Binomial Rules: "OR Rule"
To calculate the probability of selecting one event class OR the other, ADD the probabilities together:
- P(A OR B)= P(A) + P(B)-P(A & B together)
- -For binomial data, A & B can never occur together, so P(A & B together) always = 0.
Binomial Rules:"AND rule":
To calculate the probability of selecting a particular sequence of event classes, MULTIPLY the probabilities together.
P(A & B) = P(A)*P(B)
P(right turn) ⋅ n
Extra Credit : The Probability of any sequence containing X number of observations of interest
-The exact probability of a given X will depend on:
The exact probability of a given X will depend on:
- P(any sequence with a given X)
- X= # of observations of interest
- n= # of events or trials
- p= probability of observing an X on any one trial
- q= probability of observing a “not X” on any one trial
The probability of getting any sequence that contains the X & The # of different sequences that contain the X.