Hypothesis Testing

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Hypothesis Testing

• Comparing One Sample to its Population

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Hypothesis Testing w/ One Sample

• If the population mean (µ) and standard deviation
  (s) are known
• Testing if our sample mean ( ) is significantly
  different from our sampling distribution of the
  mean
• Similar to testing if how different an individual
  score is from other scores in the sample
• What is this test called?

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Hypothesis Testing w/ One Sample

• z-score formula for an individual score (x)
• z-score formula for means ( )

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Hypothesis Testing w/ One Sample

• Testing score versus standard deviation for an
  distribution of scores
• Testing mean versus standard deviation for
  distribution of sample means
• I.e. standard error

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Hypothesis Testing w/ One Sample

• Two implications of this formula
• 1. Because we are dividing by N (actually vN),
  with the same data (same sample population mean
  and s), but larger sample size, our p-value will
  be smaller (i.e. more likely to be significant)
• All statistical tests that produce p-values will
  be sensitive to sample size i.e. with enough
  people anything is significant at p lt .05

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Hypothesis Testing w/ One Sample

• Two implications of this formula
• 2. If you recall, this formula was derived from
  the formula for the normal distribution
• This means that your data must be normally
  distributed to use this test validly
• However, this test is robust to violations of
  this assumption i.e. you can violate it, if you
  have (a) a large enough sample or (b) your
  population data is normally distributed
• Why?

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Hypothesis Testing w/ One Sample

• The Central Limit Theorem
• Given a population with mean µ and variance s2,
  the sampling distribution of the mean (the
  distribution of sample means) will have a mean
  equal to µ (i.e., µ µ) and a variance (s2)
  equal to s2/N (and standard deviation, s s/vN).
  The distribution will approach the normal
  distribution as N, the sample size, increases.

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Hypothesis Testing w/ One Sample

• Example 1
• You want to test the hypothesis that the current
  crop of Kent State freshman are more depressed
  than Kent State undergraduates in general.
• What is your sample and what is your population?
• What is your Ho and your H1?
• Are you using a one- or two-tailed test?
• Assuming that for current Kent State freshman,
  their mean depression score is 15, while the mean
  for all previous Kent State undergrads (N
  100,000) is 10, and their standard deviation is 5

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Hypothesis Testing w/ One Sample

• 5/.0158 316.46
• Look up p associated with z-score in z-table
• p lt 0.0000
• Since this is less than .05 (or .025 if we were
  using a two-tailed test), we could conclude that
  the current batch of freshman are significantly
  more depressed than previous undergrads
• Also notice the effect that our large N had on
  our p-value

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Hypothesis Testing w/ One Sample

• Most often, however, we dont know the µ and s,
  because this is what were trying to estimate
  with our sample in the first place
• The formula for the t statistic accomplishes this
  by substituting s2 for s2 in the formula for the
  z statistic
• Because of this substitution, we have a different
  statistic, which requires that we use a different
  table than the z-table
• Dont worry too much about why its different

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Hypothesis Testing w/ One Sample

• Testing mean versus standard deviation for
  distribution of sample means
• I.e. standard error
• Testing mean versus standard deviation for sample

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Hypothesis Testing w/ One Sample

• After computing our t statistic, we need to
  compare it with the t-table (called the Students
  T-Table)
• Student is a pseudonym for William Gosset
• Gosset worked for the Guiness Brewing Company,
  but they wouldnt let him publish under his own
  name
• First, we will need to become familiar with the
  concept of degrees of freedom or df
• df N 1
• This represents the number of individual subjects
  data points that are free to vary, if you know
  the mean or s already

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Hypothesis Testing w/ One Sample

• For example
• If we already know that a particular set of data
  has a mean of 5, and 10 scores in total (n 10)
• Once we have nine of those scores, we can
  calculate the tenth, however, if we have eight
  scores we do not know what the other two scores
  could be
• We can solve x 5 10, but not x y 10,
  because in the latter we have more than one
  unknown (x and y)
• x and y could be 5 and 5, 8 and 2, 4 and 6, 7 and
  3, etc.
• Therefore, nine scores are free to vary, then the
  tenth is fixed

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Hypothesis Testing w/ One Sample

• Factors that influence the z and t statistics
• The difference between the sample mean and
  population mean greater differences greater t
  and z values
• The magnitude of s (or s2) since were dividing
  by s, smaller values of s result in larger values
  of t or z i.e. we want to decrease variability
  in our sample (error)
• The sample size the bigger the bigger t and z
• The significance level (a) the smaller the a,
  the higher the critical t to reject Ho although
  raising a also raises our Type I Error, so we
  probably wont want to do this without good
  reason
• Whether the test is one- or two-tailed
  two-tailed tests split a into two tails of plt
  .025, instead of one tail at p lt .05

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General Approach to Hypothesis Testing

• 1. Identify H0 and H1
• 2. Calculate df and identify the critical test
  statistic
• 3. Determine whether to use one- or two-tailed
  test, determine what value of a to use (usually
  .05), and identify the rejection region(s) that
  the critical statistic is the boundary of
• 4. Calculate your obtained test statistic
• 7. Compare your value of your test statistic to
  your rejection region to determine whether or not
  to reject H0

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Hypothesis Testing w/ One Sample

• Example 1
• Youve administered a therapy for people with
  anorexia that will supposedly assist them in
  gaining weight. The following data are amount of
  weight gained in pounds over your 16 session
  therapy for 29 participants. Does this represent
  a significantly increased degree of weight gain
  compared to the average weight gained without
  treatment (-.45 lbs.)?
• What are Ho and H1?
• Will you be using a one- or two-tailed test? Why?
• Based on this, what is your df?

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Hypothesis Testing w/ One Sample

• Example 1

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Hypothesis Testing w/ One Sample

• Example 1
• Sample Mean 87.2/29 3.0069
• s2 (1757.8 (87.2)2/29)/ 28 53.41
• s 7.3085
• t (3.0069 - -.45)/(7.3085/v29) 2.5472, p lt
  .05
• t gt Critical t and in our rejection region, which
  is above the population mean (since were only
  interested in people gaining weight), therefore
  we reject Ho and conclude that our treatment is
  more effective than no treatment at all

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Hypothesis Testing w/ One Sample

• Often, if were reporting the results of our
  experiments to the public, or the results of an
  assessment (psychological or otherwise) to a
  client, we want to emphasize to them that our
  measurements are made with error, or that our
  samples include sampling error
• We can do this by including intervals around the
  scores we report, indicating that the true
  score measured without error lies in this interval

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Hypothesis Testing w/ One Sample

• This is what is known as a Confidence Interval
• In keeping with the p lt .05 tradition, we are
  often looking for the 95 confidence interval, or
  the scores that 95 of our distribution lie, but
  we can do this for any interval
• They are calculated just like for z-scores, where
  we plug the t values into the formula and work
  backwards

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Hypothesis Testing w/ One Sample

• For a 95 CI
• Given your df (well assume df 9 for this
  example) and type of test (assume a two-tailed
  test for now), look up your critical values of t
  from a t-table (t 2.262)
• Plug into your formula with your Sample Mean and
  s (which well assume are 1.463 and .341,
  respectively), and solve for µ

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Hypothesis Testing w/ One Sample

• For a 95 CI
• 2.262 (1.463 µ)/(.341/v10)
• µ 2.262(.108) 1.463 .244 1.462
• .244 1.463 1.707
• -.244 1.463 1.219
• CI.95 1.219 µ 1.707