Statistical Inference

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Statistical Inference
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The larger the sample size (n) the more confident
you can be that your sample mean is a good
representation of the population mean. In other
words, the "n" justifies the means.

• Ancient Kung Foole Proverb

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One- and Two Tailed Probabilities

• One-tailed
• The probability that an observation will occur at
  one end of the sampling distribution.
• Two-tailed
• The probability that an observation will occur at
  either extreme of the sampling distribution.

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Hypothesis Testing

• Conceptual (Research) Hypothesis
• A general statement about the relationship
  between the independent and dependent variables
• Statistical Hypothesis
• A statement that can be shown to be supported or
  not supported by the data.

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Statistical Significance Testing

• Indirect Proof of a Hypothesis
• Modus Tollens
• A procedure of falsification that relies on a
  single observation.
• Null Hypothesis
• A statement that specifies no relationship or
  difference on a population parameter.
• Alternative Hypothesis
• A statement that specifies some value other than
  the null hypothesis is true.

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Examples of the Null and Alternative Hypotheses
Nondirectional Test Directional Test
Ho µ 5 Ha µ ? 5 Ho µ 5 or µ 5 Ha µ lt 5 or µ gt 5
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Rejecting the Null

• Alpha Level
• The level of significant set by the experimenter.
  It is the confidence with which the researcher
  can decide to reject the null hypothesis.
• Significance Level
• The probability value used to conclude that the
  null hypothesis is an incorrect statement. Common
  significance levels are .05, .01 and .001.

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Two Types of Error

• Type I
• When a researcher rejects the null hypothesis
  when in fact it is true. The probability of a
  type I error is a.
• Type II
• An error that occurs when a researcher fails to
  reject a null hypothesis that should be rejected.
  The probability of a Type II error is ß.

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Type 1 Error Type 2 Error
Scientists Decision Reject null hypothesis
Fail to reject null hypothesis
Type 1 Error Correct Decision probability
? Probability 1- ? Correct decision Type 2
Error probability 1 - ? probability ?
Null hypothesis is true Null hypothesis is false
Type 1 Error
Type 2 Error
?
?
Cases in which you reject null hypothesis when it
is really true
Cases in which you fail to reject null hypothesis
when it is false
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The OJ Trial

• For a nice tutorial go to http//www.socialresea
  rchmethods.net/OJtrial/ojhome.htm

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One sample z-test

• Used when we know µ and s.
• Generalization of calculating the probability of
  a score.
• We are now calculating the probability of a
  sample given µ and s.

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Statistical Significance Testing
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The Problems with SST

• We misunderstand what it does tell us.
• It does not tell us what we want to know.
• We often overemphasize SST.

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Four Important Questions

• Is there a real relationship in the population?
• Statistical Significance
• How large is the relationship?
• Effect Size or Magnitude
• Is it a relationship that has important,
  powerful, useful, meaningful implications?
• Practical Significance
• Why is the relationship there?
• ??????

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SST is all about . . .

• Sampling Error
• The difference between what I see in my sample
  and what exists in the target population.
• Simply because I sampled, I could be wrong.
• This is a threat to Internal Validity

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How it works

1. Assume sampling error occurred there is no
   relationship in the population.
2. Build a statistical scenario based on this null
   hypothesis
3. How likely is it I got the sample value I got
   when the null hypothesis is true? (This is the
   fabled p-value.)

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How it works (contd)

• How unlikely does my result have to be to rule
  out sampling error? alpha (?).
• If p lt ?, then our result is statistically rare,
  is unlikely to occur when there isnt a
  relationship in the population.

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What it does tell us

• What is the probability that we would see a
  relationship in our sample when there is no
  relationship in the population?
• Can we rule out sampling error as a competing
  hypothesis for our finding?

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What it does not tell us

• Whether the null hypothesis is true.
• Whether our results will replicate.
• Whether our research hypothesis is true.
• How big the effect or relationship is.
• How important the results are.
• Why there is a relationship.

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From Z to t

• In a Z test, you compare your sample to a known
  population, with a known mean and standard
  deviation.
• In real research practice, you often compare two
  or more groups of scores to each other, without
  any direct information about populations.
• Nothing is known about the populations that the
  samples are supposed to come from.

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The t Test for a Single Sample

• The single sample t test is used to compare a
  single sample to a population with a known mean
  but an unknown variance.
• The formula for the t statistic is similar in
  structure to the Z, except that the t statistic
  uses estimated standard error.

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From Z to t
Note lowercase s.
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Degrees of Freedom

• The number you divide by (the number of scores
  minus 1) to get the estimated population variance
  is called the degrees of freedom.
• The degrees of freedom is the number of scores in
  a sample that are free to vary.

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Degrees of Freedom

• Imagine a very simple situation in which the
  individual scores that make up a distribution are
  3, 4, 5, 6, and 7.
• If you are asked to tell what the first score is
  without having seen it, the best you could do is
  a wild guess, because the first score could be
  any number.
• If you are told the first score (3) and then
  asked to give the second, it too could be any
  number.

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Degrees of Freedom

• The same is true of the third and fourth scores
  each of them has complete freedom to vary.
• But if you know those first four scores (3, 4, 5,
  and 6) and you know the mean of the distribution
  (5), then the last score can only be 7.
• If, instead of the mean and 3, 4, 5, and 6, you
  were given the mean and 3, 5, 6, and 7, the
  missing score could only be 4.

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The t Distribution

• In the Z test, you learned that when the
  population distribution follows a normal curve,
  the shape of the distribution of means will also
  be a normal curve.
• However, this changes when you do hypothesis
  testing with an estimated population variance.
• Since our estimate of ? is based on our sample
• And from sample to sample, our estimate of ? will
  change, or vary
• There is variation in our estimate of ?, and more
  variation in the t distribution.

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The t Distribution

• Just how much the t distribution differs from the
  normal curve depends on the degrees of freedom.
• The t distribution differs most from the normal
  curve when the degrees of freedom are low
  (because the estimate of the population variance
  is based on a very small sample).
• Most notably, when degrees of freedom is small,
  extremely large t ratios (either positive or
  negative) make up a larger-than-normal part of
  the distribution of samples.

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The t Distribution

• This slight difference in shape affects how
  extreme a score you need to reject the null
  hypothesis.
• As always, to reject the null hypothesis, your
  sample mean has to be in an extreme section of
  the comparison distribution of means.

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The t Distribution

• However, if the distribution has more of its
  means in the tails than a normal curve would
  have, then the point where the rejection region
  begins has to be further out on the comparison
  distribution.
• Thus, it takes a slightly more extreme sample
  mean to get a significant result when using a t
  distribution than when using a normal curve.

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The t Distribution

• For example, using the normal curve, 1.96 is the
  cut-off for a two-tailed test at the .05 level of
  significance.
• On a t distribution with 3 degrees of freedom (a
  sample size of 4), the cutoff is 3.18 for a
  two-tailed test at the .05 level of significance.
• If your estimate is based on a larger sample of
  7, the cutoff is 2.45, a critical score closer to
  that for the normal curve.

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The t Distribution

• If your sample size is infinite, the t
  distribution is the same as the normal curve.

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• Since it takes into account the changing shape of
  the distribution as n increases, there is a
  separate curve for each sample size (or degrees
  of freedom).
• However, there is not enough space in the table
  to put all of the different probabilities
  corresponding to each possible t score.
• The t table lists commonly used critical regions
  (at popular alpha levels).

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• If your study has degrees of freedom that do not
  appear on the table, use the next smallest number
  of degrees of freedom.
• Just as in the normal curve table, the table
  makes no distinction between negative and
  positive values of t because the area falling
  above a given positive value of t is the same as
  the area falling below the same negative value.

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The t Test for a Single Sample Example

• You are a chicken farmer if only you had paid
  more attention in school. Anyhow, you think that
  a new type of organic feed may lead to plumper
  chickens. As every chicken farmer knows, a fat
  chicken sells for more than a thin chicken, so
  you are excited. You know that a chicken on
  standard feed weighs, on average, 3 pounds. You
  feed a sample of 25 chickens the organic feed for
  several weeks. The average weight of a chicken
  on the new feed is 3.49 pounds with a standard
  deviation of 0.90 pounds. Should you switch to
  the organic feed? Use the .05 level of
  significance.

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Hypothesis Testing

1. State the research question.
2. State the statistical hypothesis.
3. Set decision rule.
4. Calculate the test statistic.
5. Decide if result is significant.
6. Interpret result as it relates to your research
   question.

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The t Test for a Single Sample Example

• State the research question.
• Does organic feed lead to plumper chickens?
• State the statistical hypothesis.

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• Set decision rule.

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The t Test for a Single Sample Example

• Calculate the test statistic.

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The t Test for a Single Sample Example

• Decide if result is significant.
• Reject H0, 2.72 gt 1.711
• Interpret result as it relates to your research
  question.
• The chickens on the organic feed weigh
  significantly more than the chickens on the
  standard feed.