Intro to Statistics for the Behavioral Sciences PSYC 1900

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Intro to Statistics for the Behavioral
SciencesPSYC 1900

• Lecture 5 Probability and Hypothesis Testing

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Probability

• Relative Frequency Perspective
• Probability of some event is the limit of the
  relative frequency of occurrence as the number of
  draws (i.e., samples) approaches infinity.
• If we have 8 blue marbles and 2 red marbles, the
  probability of drawing a red 2/10 20 on any
  trial (i.e., analytic perspective).
• Across repeated trials, we would find that 20 of
  them produce a red marble.
• Note that were sampling with replacement.

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Terminology

• Sampling with replacement
• After an event, the draw or event goes back into
  the pool.
• Sampling in which an item drawn on trial N is
  replaced before the drawing of the N1 trial.
• Event
• The outcome of a trial
• Independent events
• Events where the occurrence of one has no effect
  on the probability of the occurrence of others
• Voting behavior of random citizens, marble draw
• Mutually exclusive events
• Two events are mutually exclusive when the
  occurrence of one precludes the occurrence of the
  other.
• Gender, religion, handedness

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Basic Laws of Probability

• Probabilities range from 0 to 1, where a 1 means
  the event must occur.
• Additive Rule
• Gives probs of occurrence for one or more
  mutually exclusive envents.
• 30 red marbles, 15 blue, 55 green 100 total
• p(red).30, p(blue).15, p(green) .55
• Probability of drawing a red or blue?
• Given a set of mutually exclusive events, the
  probability of one event or the other equals the
  sum of their separate probabilities.
• p(red).30 p(blue).15.45

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Basic Laws of Probability

• Multiplicative Law
• Gives the probability of the joint occurrence of
  independent events.
• 30 red marbles, 15 blue, 55 green 100 total
• p(red).30, p(blue).15, p(green) .55
• Probability of drawing a red on the first trial
  and a red on the second?
• The prob of a joint occurrence of two or more
  independent events equals the product of their
  individual probabilities.
• p(red) X p(red) .3X.3 .09

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• Sequence of coin flips
• H,H,T,H,T,T,T,H,T,T, __
• What is the probability of H on next draw?
• Prob.5 Events are independent
• What is the probability of H and H on the next
  two draws?
• Prob.5X.5.25 Events are independent
• Conditional probability of independent
  events

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Joint Probabilities

• The probability of the co-occurrence of two or
  more events
• Probability of sampling a red cube from a sample
  of red and blue marbles and cubes
• p(red,cube) p(red) x p(cube)
• If the events are independent
• If not independent (i.e., a correlation among
  events), computation of prob is more complex

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Conditional Probabilities

• The prob of one even given the occurrence of
  another event
• The prob that a person will fracture a bone given
  that he/she has osteoporosis
• p(fractureosteoporosis) Y
• If the null hypothesis is true, the probability
  of obtaining a difference between sample means of
  X size

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Bone Density No Fracture Fracture Total
Normal 153 24 177
Row 86 14 49
Column 59 24
Cell 43 7

Osteoporosis 105 76 181
Row 58 42 51
Column 41 76
Cell 29 21

Total 258 100
Column 72 28

• p(no fracture) 258/358.72
• p(norm den, no frac)153/358.43
• Why not p(norm) x p(no frac) .49x.72.35?
• p(fracosteo) .42 p(fracnorm).14
• Other conditional prob examples?

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Discrete vs. Continuous Probability Distributions

• For discrete distributions, we can calculate
  probs for specific events.
• p(Harvard, vanilla)
• 7/20.35

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Discrete vs. Continuous Probability Distributions

• For continuous distributions, case is slightly
  different.
• Prob that baby will crawl at 35 weeks?
• Almost zero at 35.00001 weeks.
• Events at a very specific point are infrequent.
• Density gives probability for specific range
• 35 weeks means from 34.5 to 35.5 weeks.
• Integrate to find area under curve which provides
  a probability as a function of proportion of
  interval area to entire area under curve (where
  total area is set to equal 1)

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Sampling Distributions Hypothesis Testing

• Until now, we have primarily focused on
  descriptive statistics.
• Although such statistics are quite useful for
  assessing the characteristics of samples, they
  cannot answer questions related to inference.
• Is the difference between two means likely to
  represent chance variation?
• To answer such questions, the remainder of this
  course will focus on the statistical process of
  inference.

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Basic Form of Inference

• The most basic question is one in which we might
  compare the means of two groups.
• If one group has a mean of 50 and the other a
  mean of 42 following some manipulation, can we
  infer that the manipulation lowered the score?

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Sampling Error

• To answer this question, we have to understand
  sampling error.
• Sampling error is the variability of a statistic
  from sample to sample due to chance.
• If I took samples from a population, the
  descriptives of the samples would cluster around,
  but not always equal the parameters of the
  population.

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Hypothesis Testing

• The basic question in hypothesis testing is
• Is the given difference large enough that it does
  not likely stem from sampling error?
• Hypothesis Testing
• A process by which decisions are made regarding
  the values of parameters.

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Sampling Distributions

• The distribution of a statistic over repeated
  sampling from a specified population.
• Both descriptive and inferential statistics
  (e.g., t, F, r) have sampling distributions.
• Tell us what values we might expect given certain
  conditions.
• A conditional probability

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Sampling Distribution of the Mean

• To determine if the difference between two means
  is likely due to sampling error, we need to know
  the sd of a distribution of means from the
  population.
• Standard Error of the Mean
• sd of a sampling distribution of means
• Sampling distritribution of the mean is the
  distribution of means collected from repeated
  sampling of the same population.

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Distribution of Sample Means
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Hypothesis Testing

• Sampling distributions allow us to test
  hypotheses.
• Sampling distributions can be derived
  mathematically.
• If the aggression mean of kids viewing a violent
  video is 6.5, and the normal population mean
  for kids is 5.65, does this difference imply that
  the such videos increase aggressive thoughts?

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Logic of Hypothesis Testing

• Set up relevant null hypothesis H0
• Sample (i.e., kids who watch violent videos)
  represents same population.
• Mean should equal population mean of 5.65
• Calculate mean of sample
• Mean 6.5
• Obtain sampling distribution and standard error
• Determine probability of obtaining a mean at
  least as large as the actual sample mean
• On that basis, decide whether to accept or reject
  the null hypothesis

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The Null Hypothesis

• At its heart, the null states that parameters are
  the same.
• For example, 2 means are equal
• The difference between the means is zero
• Any differences reflect sampling error
• Why use the null?
• Excellent starting place
• What would the alternative be?
• Wed have to specify sampling distributions for
  exact alternative parameter values?

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Test Statistics and Sampling Distributions

• The same logic applies to test statistics as well
  as means.
• ts, Fs, rs
• A sampling distribution can be calculated for
  each statistic and used to evaluate the
  corresponding null.
• For t, a sampling distribution when H0 is true
  would consist of t values from an infinite number
  of paired samples.
• Compare current t to sampling distribution to
  determine viability of null.

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Using Normal Distribution to Test Hypotheses

• The normal distribution can be used to test
  hypotheses involving individual scores or sample
  means.
• Assumes scores or sampling distributions of the
  mean are normally distributed
• Going back to our example
• Mean of kids watching violent videos 6.5
• Population parameters
• Mean 5.65, sd .45

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Using Normal Distribution to Test Hypotheses

• Convert 6.5 to a z score
• applet
• p(6.5N(5.65,0.45)).06

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Terminology

• Significance Level
• Probability with which we are willing to reject
  null when it is in fact correct
• Also called alpha level
• Rejection Region
• Set of outcomes that will lead to rejection of
  null
• Alternative Hypothesis
• Hypothesis that is adopted when null is rejected
• Usually the research hypothesis

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Type I and Type II Errors

• As weve seen, determining whether a difference
  is real or due to sampling error requires a
  choice of a critical value or significance level.
• Because we are making a choice, there is always
  the chance that the choice will be incorrect.

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Type I and Type II Errors

• If we use a significance level of .05
• 5 of the time we will reject the null hypothesis
  when it is true
• Type I Error
• p(Type I) alpha
• If we feel this amount of error is too large,
  what can we do to minimize Type I errors?

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Type I and Type II Errors

• Use a more stringent alpha level to reduce Type I
  errors
• Alpha .01 only 1 error in rejecting null
• This strategy has a trade-off
• Failing to reject the null when it is false is a
  Type II error
• p(Type II) beta

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Decision True State Of World
Null True Null False
Reject Null Type 1 Error Correct Decision
Fail to Reject Null Correct Decision Type II Error
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One-Tailed vs. Two-Tailed Tests

• Two-tailed (nondirectional) tests are most common
• Look for extremes in both tails (i.e., positive
  or negative deviations from the mean)
• Alpha .05 has .025 null rejection area in each
  tail of sampling distribution
• Used because one might never truly be sure what
  outcome to expect

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One-Tailed vs. Two-Tailed Tests

• One-tailed (directional tests) are less commonly
  used
• Look for extreme parameter values in only 1 tail
• Researcher predicts direction of difference
• Alpha.05 places total .05 null rejection area in
  a single tail
• What is the benefit in terms of power?
• Smaller differences will be viewed as significant
  due to increased null rejection area