Behavioral Statistics

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Behavioral Statistics

• Probability Theory and Sampling

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• Probability is a measure of how likely it is that
  a given event or behavior will happen.
• of Events
• Probability of an Event -----------------------
  -------
• of Possible Outcomes
• Probability is like proportions. Ranges between 0
  and 1
• A measure of randomness
• Chance behavior is unpredictable in the short run
  but has a regular and predictable pattern in the
  long run.

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• Probability of a coin flip landing on heads
• ½ or 0.5
• Probability of an Ace being selected from a deck
  of 52 cards is 4/52 or 1/13 or 0.077
• The probability of a black card being selected is
  26/52 or ½ or 0.50
• Probability that a statistics student will be
  male
• 0 of 10 0
• Mutually exclusive two events cannot occur
  simultaneously.

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• Probability can be assessed by looking at the
  sample space
• The sample space is a collection of all possible
  outcomes.
• Ex. Flip a coin two possibilities H, T
• Ex. Rolling a six sided die. The sample space is
  1, 2, 3, 4, 5, 6.
• Sample space for multiple events Flip a coin
  twice. What is the probability that H will occur
  both times?
• HH, HT, TH, TT

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• Probability of rolling a 6?

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• Probability can be calculated based on real world
  events and frequency distributions

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• Probability as relative frequency
• Probability of one of two events occurring
• What is the probability of a head or a tail?
• What is the probability of drawing a face card?
• Addition rule of probability
• p(A or B) p(A) p(B)
• Note ? This is only true for mutually exclusive
  events
• If not mutually exclusive
• p(A or B) p(A) p(B) p(A and B)

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• Examples
• Probability of rolling a 1 or a 6?

p(1 or 6) 2/6 or 1/3 ? p(1) 1/6 p(6)
1/6 p(1 or 6) 1/6 1/6
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• Probability of drawing an Ace or a diamond?
• p(Ace) 4/52
• p(diamond) 13/52
• Is the answer 17/52? No
• One card is both an Ace and a diamond
• p(Ace or diamond) p(Ace) p(diamond) p(Ace
  and diamond)
• 4/52 13/52 1/52
• 16/52

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p(A or B) p(A) p(B)
Addition Rule of Probability
p(schizophrenia or Other) p(A) p(B)
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• Probability of two events occurring together
• Multiplication rule of probability
• p(A and B) p(A) p(B)
• Examples
• What is the probability of drawing the Ace of
  Spades
• p(Ace) 4/52 .077
• p(spade) 13/52 .25
• Both .077.25 .0192
• Probability of being male and schizophrenic
• Male .5
• Schizophrenic .0108
• Both .5/0108 .0054

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• Calculating probability based on z scores
• What is the probability of scoring over 130 on an
  IQ test
• Mean100
• SD 15
• Replacement vs. no replacement
• Sampling with replacement probability is
  calculated by the formulas mentioned above.
• of Events
• Probability of an Event -----------------------
  -------
• of Possible Outcomes

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• Sampling without replacement changes the overall
  number of possible outcomes
• Example probability of you getting randomly
  selected from this class.
• Example now do it again without that person in
  it.
• Example You randomly pick 3 cards from a deck
  and they are all aces. What is the probability
  that the next one will be an ace?

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Sample problems

• A consumer survey reveals that the probability of
  a computer owner shopping on the Internet was
  0.17, while the probability of a computer owner
  downloading software was 0.33. Further, the
  probability a computer owner doing both was 0.14.
  Find the probability of the following events
• A) that a computer owner does not shop on the
  Internet
• B) that a computer owner will either shop on the
  Internet or download software
• C) that a computer owner will neither shop on the
  Internet nor download software

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• 90 students were given a survey as to which type
  of food they prefer. The following are the
  results

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• What is the probability that they prefer Mexican
  food?
• What is the probability that they prefer Mexican
  food or Chinese food?
• What is the probability that they preferred
  anything but American food?
• If half of the subjects were male, what is the
  probability that they were male and preferred
  Mexican food? (assume no sex differences)

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Common errors in calculating probabilities

• Believing that past occurrences of a repeating
  random even will influence its future occurrences
  
• Gamblers fallacy
• Example Flip a coin
• 9 straight heads
• What is the probability that the next flip will
  be a tail?
• Belief that long-term probabilities will work out
  in the short-term

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• Law of large numbers
• The larger the number of chance-determined
  repetitious events considered, the closer the
  alternatives will approach predictable ratios.

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• Error 2 looking at probabilities after the fact
• Ex. You think about a friend, the phone rings,
  and its your friend.
• Ex. You meet someone and you have the same
  birthday.
• What is the probability that 2 people in here
  have the same birthday?
• 3 people - .008
• 23 people - .5
• Sometimes unusual events just happen due to chance

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• Error 3 False belief that clusters and runs do
  not occur
• Often given a fixed probability, rare events will
  occur in clusters
• Ex. Flip a coin and get heads 10 straight times
• Ex. Cases of cancer in a community
• Ex. Tornado hits a small town twice in a week
• Discussion Is there such a thing as luck?
• Dennetts thought experiment
• If I randomly pick a card out of the deck, what
  is the probability that someone in here will
  guess it right?

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• Probability becomes especially important in
  Psychology because of sampling.
• Population total number of people
• Sample a selection of people from that
  population
• Need for representative sample
• Many types of sampling
• Random sampling randomly select individuals
  from the population to make up a sample
• All individuals must have an equally likely
  chance of being selected.
• Samples should be a reflection of the population

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• Representative sample sample must be
  representative of the population of which it
  came.
• Ex. Polls
• Ex. Ability to discriminate color
• Ex. NCAA Division I graduation rates
• Random sampling works like any random process
  individuals are selected from a group of
  potential selections.

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• Methods for random sampling
• Random number table
• Roll dice
• Random number generator http//www.random.org
• Sample must be of sufficient size to be
  representative of the population
• Bigger the better
• Very important there are many possible samples
  you could take
• Example NCAA graduation rates
• 400 schools we want to sample 50

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• Mean of our population 56
• Mean of our sample? 54
• Sample mean will be similar to the population
  mean, but not exact. Depends on the standard
  deviation of the population
• Sample 2 58
• Sample 3 51
• Sample means will be distributed around the
  population mean

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• Properties of the distribution of sample means
• Standard deviation of sample means is called the
  standard error of the mean.
• n is the size of the sample
• The greater the sample size, the smaller the
  standard error of the mean

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Central Limit Theorem
The basis for inferential statistics.
The distribution of sample means approaches a
normal distribution when n is large.
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• Why is it important that our distribution of
  sample means has a normal distribution?
• Because of the properties of the normal
  distribution
• Symmetrical
• Mean, median, mode all the same
• The further away from the mean, the less likely
  the score is to occur
• Probabilities can be calculated

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• Note the distribution of sample means is shaped
  like a normal distribution, no matter what the
  original distribution looks like
• All inferential statistics are based on the
  central limit theorem
• Especially important when considering Is the
  sample representative of the population?
• Sample means that are far from the population
  mean would be unlikely, whereas sample means that
  are close to the population mean would be more
  probable

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• Calculating probability of a sample mean the Z
  test.
• If we want to know the probability of getting a
  particular sample mean, given that we know the
  population mean, we simply need to see where the
  sample mean falls relative to the population
  mean.
• Convert the sample mean to a z-score.

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Formula for calculating z
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Examples

• Athletes performance on anxiety battery
• µ 25
• s 5
• Randomly sample 100 college athletes and give
  them the anxiety battery
• Did athletes score lower on the test than most
  college students?
• Is the difference simply due to chance?

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• If we know the population mean and standard
  deviation, we can use the z test to test for the
  probability that the sample came from the
  population

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SAT scores

• The Lebowski urban achievers (100 students) take
  the SAT and score on average 533.
• ETS data µ 500, s 110
• Did the achievers score better on the test than
  most high school students?
• Does the Lebowski sample belong to the SAT
  population?

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