Sampling Distributions, Hypothesis Testing and One-sample Tests

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Sampling Distributions, Hypothesis Testing and
One-sample Tests
    
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Media Violence

• Does violent content in a video affect later
  behavior?
• Bushman (1998)
• Two groups of 100 subjects saw a video
• Violent video versus nonviolent video
• Then free associated to 26 homonyms with
  aggressive nonaggressive forms.
• e.g. cuff, mug, plaster, pound, sock

Cont.
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Media Violence--cont.

• Results
• Mean number of aggressive free associates 7.10
• Assume we know that without aggressive video the
  mean would be 5.65, and the standard deviation
  4.5
• These are parameters (m and s)
• Is 7.10 enough larger than 5.65 to conclude that
  video affected results?

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Sampling Distribution of the Mean

• We need to know what kinds of sample means to
  expect if video has no effect.
• i. e. What kinds of means if m 5.65 and s
  4.5?
• This is the sampling distribution of the mean.

Cont.
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Cont.
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Sampling Distribution of the Mean--cont.

• The sampling distribution of the mean depends on
• Mean of sampled population
• Why?
• St. dev. of sampled population
• Why?
• Size of sample
• Why?

Cont.
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Sampling Distribution of the mean--cont.

• Shape of the sampling distribution
• Approaches normal
• Why?
• Rate of approach depends on sample size
• Why?
• Basic theorem
• Central limit theorem

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Central Limit Theorem

• Given a population with mean m and standard
  deviation s, the sampling distribution of the
  mean (the distribution of sample means) has a
  mean m, and a standard deviation s /?n. The
  distribution approaches normal as n, the sample
  size, increases.

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Demonstration

• Let population be very skewed
• Draw samples of 3 and calculate means
• Draw samples of 10 and calculate means
• Plot means
• Note changes in means, standard deviations, and
  shapes

Cont.
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Parent Population
Cont.
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Sampling Distribution n 3
Cont.
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Sampling Distribution n 10
Cont.
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Demonstration--cont.

• Means have stayed at 3.00 throughout--except for
  minor sampling error
• Standard deviations have decreased appropriately
• Shapes have become more normal--see superimposed
  normal distribution for reference

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Steps in Hypothesis Testing

• Define the null hypothesis.
• Decide what you would expect to find if the null
  hypothesis were true.
• Look at what you actually found.
• Reject the null if what you found is not what you
  expected.

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The Null Hypothesis

• The hypothesis that our subjects came from a
  population of normal responders.
• The hypothesis that watching a violent video does
  not change mean number of aggressive
  interpretations.
• The hypothesis we usually want to reject.

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Important Concepts

• Concepts critical to hypothesis testing
• Decision
• Type I error
• Type II error
• Critical values
• One- and two-tailed tests

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Decisions

• When we test a hypothesis we draw a conclusion
  either correct or incorrect.
• Type I error
• Reject the null hypothesis when it is actually
  correct.
• Type II error
• Retain the null hypothesis when it is actually
  false.

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Type I Errors

• Assume violent videos really have no effect on
  associations
• Assume we conclude that they do.
• This is a Type I error
• Probability set at alpha (?)
• ? usually at .05
• Therefore, probability of Type I error .05

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Type II Errors

• Assume violent videos make a difference
• Assume that we conclude they dont
• This is also an error (Type II)
• Probability denoted beta (?)
• We cant set beta easily.
• Well talk about this issue later.
• Power (1 - ?) probability of correctly
  rejecting false null hypothesis.

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Critical Values

• These represent the point at which we decide to
  reject null hypothesis.
• e.g. We might decide to reject null when (pnull)
  lt .05.
• Our test statistic has some value with p .05
• We reject when we exceed that value.
• That value is the critical value.

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One- and Two-Tailed Tests

• Two-tailed test rejects null when obtained value
  too extreme in either direction
• Decide on this before collecting data.
• One-tailed test rejects null if obtained value is
  too low (or too high)
• We only set aside one direction for rejection.

Cont.
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One- Two-Tailed Example

• One-tailed test
• Reject null if violent video group had too many
  aggressive associates
• Probably wouldnt expect too few, and therefore
  no point guarding against it.
• Two-tailed test
• Reject null if violent video group had an extreme
  number of aggressive associates either too many
  or too few.

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Testing Hypotheses s known

• H0 m 5.65
• H1 m ? 5.65 (Two-tailed)
• Calculate p (sample mean) 7.10 if m 5.65
• Use z from normal distribution
• Sampling distribution would be normal

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Using z To Test H0

• Calculate z
• If z gt 1.96, reject H0
• 3.22 gt 1.96
• The difference is significant.

Cont.
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z--cont.

• Compare computed z to histogram of sampling
  distribution
• The results should look consistent.
• Logic of test
• Calculate probability of getting this mean if
  null true.
• Reject if that probability is too small.

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Testing When s Not Known

• Assume same example, but s not known
• Cant substitute s for s because s more likely to
  be too small
• See next slide.
• Do it anyway, but call answer t
• Compare t to tabled values of t.

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Sampling Distribution of the Variance
Population variance 138.89 n 5 10,000
samples 58.94 lt 138.89
138.89
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t Test for One Mean

• Same as z except for s in place of s.
• For Bushman, s 4.40

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Degrees of Freedom

• Skewness of sampling distribution of variance
  decreases as n increases
• t will differ from z less as sample size
  increases
• Therefore need to adjust t accordingly
• df n - 1
• t based on df

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t Distribution
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Conclusions

• With n 100, t.0599 1.98
• Because t 3.30 gt 1.98, reject H0
• Conclude that viewing violent video leads to more
  aggressive free associates than normal.

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Factors Affecting t

• Difference between sample and population means
• Magnitude of sample variance
• Sample size

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Factors Affecting Decision

• Significance level a
• One-tailed versus two-tailed test

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Size of the Effect

• We know that the difference is significant.
• That doesnt mean that it is important.
• Population mean 5.65, Sample mean 7.10
• Difference is nearly 1.5 words, or 25 more
  violent words than normal.

Cont.
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Effect Size (cont.)

• Later we will express this in terms of standard
  deviations.
• 1.45 units is 1.45/4.40 1/3 of a standard
  deviation.

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Confidence Limits on Mean

• Sample mean is a point estimate
• We want interval estimate
• Probability that interval computed this way
  includes m 0.95

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For Our Data
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Confidence Interval

• The interval does not include 5.65--the
  population mean without a violent video
• Consistent with result of t test.
• Confidence interval and effect size tell us about
  the magnitude of the effect.
• What can we conclude from confidence interval?