Statistical Inference

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Statistical Inference

• Statistical Inference involves estimating a
  population parameter (mean) from a sample that is
  taken from the population.
• Inference may also be concerned with the
  difference between populations on a given
  parameter (mean depression scores, etc).

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Issues In Inference

• The task of inference is to draw conclusions
  about a parameter (characteristic of the
  population) from a sample statistic.
• Because of sampling variation, we can never know
  if our inferences are exactly correct.
• The key to any problem in statistical inference
  is to discover what sample variables will occur
  in repeated sampling, and with what probability.

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Hypothesis Testing

• A hypothesis is a testable statement about a
  population parameter. A hypothesis is tested,
  and based on the outcome, is retained or rejected
  (never say proven!).
• Null Hypothesis any observed difference between
  our expected and observed values (or between
  groups) is due to chance.
• Alternative Hypothesis an observed difference
  (between expected and observed values, or between
  groups) is too drastic to be able to be explained
  by chance.

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Hypothesis Testing

• We want to test out a new therapy method on
  people with depression.
• Null Hypothesis The observed difference between
  the depression in this group and national
  depression rates can be explained by chance
  variation in the data.
• Alternative Hypothesis The observed difference
  is too drastic to be able to be explained by
  chance.
• We test the null hypothesis. We ask what type of
  results are likely, if the sample group is no
  different than the population group. If our
  sample outcome is unlikely, assuming it should
  look like the population outcome, than we reject
  the null.

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Example

• Lets say that we are interested in whether
  someone has beenusing a weighted coin while
  gambling. Our null hypothesis will be that this
  coin should not have outcomes that differ from
  what we expect of a fair coin.
• This coin comes up 5 heads in a row.It is
  possible that someone will get5 heads in 5
  tosses, but how likely is it? There is only
  about a .05chance that someone will get5 heads
  in a row.
• We reject the null hypothesis.

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Testing a Hypothesis about a Single Mean

• Remember this example from last week?
• Lets assume we have collected data from a group
  of 100 1st grade kids in schools across the
  country who have been listening to classical
  music during the school day. We know that the
  national average for IQ for 1st graders is a mean
  (?X) of 100, SD (?X) of 15.
• Does classical music influence IQ?

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Testing a Hypothesis about a Single Mean

• We are interested in testing the Null Hypothesis,
  that there is no difference between the sample
  and the population that cannot be explained by
  chance.H0 ?X 100
• The Alternative Hypothesis proposes that there is
  a difference between the sample and population
  that cannot be explained by chance.H1 ?X ? 100
• note that ? 100 indicates that this score could
  either be higher or lower than the average.

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Testing a Hypothesis about a Single Mean alpha
levels

• How do we decide whether we retain or reject the
  Null Hypothesis?
• If we get a mean sample value that falls into a
  category we dont expect to see often by chance,
  we can reject the null.
• Common practice is to reject the null if the
  sample mean is so deviant that its probability of
  occurrence is .05 (or .01) or less.
• This criterion is called the alpha level (?). We
  would note ? .05

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Testing a Hypothesis about a Single Mean
rejection regions

• In our case, we want to find out what z-score
  values will separate 95 of the general
  population distribution from the remaining 5 at
  both tails of the distribution.
• Since we are testing the null hypothesis at ?
  .05, we will only reject H0 if the obtained
  sample mean is so deviant that it falls in the
  upper or lower 2.5 of the distribution (the
  extreme 5 of cases).
• The portion of the distribution that includes
  sample mean values that lead to rejection of the
  null hypothesis are called regions of rejection.
  The z-scores that separate these areas are called
  the critical values (zcrit).

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Testing a Hypothesis about a Single Mean results

• In our case, our zcrit values are 1.96 and
  1.96 (table D.2).
• Next we compare our test statisticto these
  critical values.
• We collect data from 100 students who listened
  to classical music and have a mean IQ 105.
• This means that our sample average is 3.33 SEs
  above our expected value.
• This observed z-score (zobs 3.33) falls into
  our region of rejection. What does this mean?
  What is our conclusion?Have we proven that the
  mean is greater than 100 in these kids?

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The Five Steps of Hypothesis Testing!

• Formulate the null (H0) and alternative (H1)
  hypotheses.
• Identify the relevant test statistic (z score of
  the sample mean, or z-test) that will be used
  to discriminate between different hypotheses
  about the population parameter of interest
  (mean).
• Identify the sampling distribution of the
  statistic under consideration (for z-test -
  sampling distribution of the mean, and standard
  error).
• Determine the ? level and critical rejection
  region in which test statistics that warrant
  rejection of H0 will fall such as p lt .05).
• Conduct the experiment and collect data, then
  report the observed test statistic (zobs). Use
  this test statistic in order to make a
  statistical decision about H0, using the decision
  rule if the test statistic lies in the critical
  rejection region (or if p lt .05), then reject H0
  (support for H1). If not, retain H0 (lack of
  support for H1). Make a theoretical conclusion.

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Example using the steps of hypothesis testing

• Suppose you are a clinical psychologist and that
  you have collected data from a group of 50 people
  who have gone through your patented program to
  reduce Ophidiophobia (snake phobia). You have
  data from a national study of snake phobics that
  indicate that their pulse rates per second in the
  presence of a snake are normally distributed
  with .
• The group of people that went through the program
  had their pulse rates measured in the presence of
  a snake. Result
• Does this program influence Ophidiophobia?

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Example using the steps of hypothesis testing

• Step 1 - H0
• H1
• Step 2 - test statistic to be used?
• Step 3 - sampling distribution to be used ?
• Step 4 - a ? one-tailed or two-tailed
  test? zcrit values ?
• Step 5 - calculate (zobs) statistical
  decision? theoretical conclusion?

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But wait

• What if we had approached this analysis as a
  one-tailed test? Would our results have been
  different if we only looked at whether the
  program reduced Ophidiophobia?
• Then we would have had an a .05 at one tail,
  with a zcrit of 1.64 (see table D.2).
• What would have been
• our statistical decision and theoretical
  conclusion?

zobs -1.72
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The Null Hypothesis

• The favored null hypothesis is held innocent
  unless proved guilty, while the alternative
  hypothesis is held guilty until no other choice
  remains but to judge it innocent.
• W. W. Rozeboom, 1960

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Assumptions for inference using a single mean
(z-test)

• A random sample has been drawn from the
  population.
• The sampling has been drawn with replacement.
• The sampling distribution of X follows the normal
  curve (central limit theorem).
• The standard deviation of scores in the
  population is known.
• Problem do we always know the population
  standard deviation? Not usually! (this is dealt
  with in PSY 321!)

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Types of Errors

• When we are making statistical decisions like
  these, there are two types of error that can
  result

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Types of Errors

• The probability of a Type I error is designated
  by alpha (a) and is called the Type I error rate.
• Remember that a is also called the significance
  level, and is set by the experimenter this
  significance level is chosen in such a way to
  reduce the probability of a Type I error.
• The probability of a Type II error (the Type II
  error rate) is designated by (ß).
• A Type II error is only an error in the sense
  that an opportunity to reject the null hypothesis
  correctly was lost. It is not an error in the
  sense that an incorrect conclusion was drawn
  since no conclusion is drawn when the null
  hypothesis is not rejected.