2 Sample Tests

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2 Sample Tests Small Samples

• Small sample, independent groups
• Test of equality of population variances
• If variances are equal, t-test
• If variances are not equal, Wilcoxon Rank Sum
  Test
• 2. Examples

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1b. Small samples, independent groups

• We now turn to the case of comparing means for
  two independent, small samples (ns lt 30).
• There are 2 ways to do this depending upon
  whether the two population variances are equal or
  different.
• In order to know which method we should use, we
  have to test the hypothesis H0 ?12 ?22
• So for small, independent samples, there are
  always 2 steps test the variances, then test the
  means.

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1b. Small samples, independent groups

• VERY IMPORTANT POINT
• We can only use the independent groups t-test
  when the two population variances are equal.
• We must not assume that ?12 ?22.
• We must test H0 ?12 ?22.
• The test of hypothesis about the two population
  variances uses the ratio F (?12 / ?22).

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1b. Small samples, independent groups

• On an exam, you must test the hypothesis of equal
  variances before doing the independent groups
  t-test!
• If H0 ?12 ?22 is rejected, we use the Wilcoxon
  Rank Sum test instead of the t-test.
• Note before t-test only not before Z test.

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Test of hypothesis of equal variances

• Notes
• Next slide shows a formal statement of test of
  hypothesis about two population variances.
• Both one-tailed and two-tailed tests are shown.
• When you test equality of variances before doing
  small sample, independent groups t-test, always
  do a two-tailed test.
• One-tailed test of equality of variances has
  other uses.

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Test of hypothesis of equal variances

• H0 ?12 ?22 H0 ?12 ?22
• HA ?12 lt ?22 HA ?12 ? ?22
• or ?12 gt ?22
• Test statistic F S12
• S22

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Test of hypothesis of equal variances

• Rejection region
• One-tailed Two-tailed
• F gt Fa F gt Fa/2
• d.f. (n1 1), (n2 1)
• (See note on next slide.)

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a given in F table is value for upper tail. Since
the F distribution is not symmetric, we have to
compute critical F for lower tail.
a
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Computing critical F values for lower tail

• Critical F for upper tail of distribution is
  found in Table VII, using a and d.f.
• Critical F for lower tail of distribution
• 1
• Fa, n2-1, n1-1
• Note that d.f. are inverted!

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1b. Small samples, independent groups

• Now back to our t-test.
• If you do NOT reject H0 in the test of equality
  of variances, then you can pool the two sample
  variances
• Sp2 (n1-1)s12 (n2-1)s22
• n1 n2 - 2

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1b. Small samples, independent groups

• H0 ?1 ?2 D0 H0 ?1 ?2 D0
• HA ?1 ?2 gt D0 HA ?1 ?2 ? D0
• or ?1 ?2 lt D0
• Test statistic t D0

Sp2 1 1 n1 n2
(
)
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1b. Small samples, independent groups

• Rejection region
• One-tailed Two-tailed
• t lt -ta tgtta/2
• or t gt ta

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1b. Small samples, independent groups

• Wilcoxon Rank Sum Test
• first, combine the two samples and rank order
  all the observations.
• smallest number has rank 1, largest number has
  rank N ( sum of n1 and n2).
• separate samples and add up the ranks for the
  smaller sample. (If n1 n2, choose either one.)
• test statistic rank sum T for smaller sample.

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1b. Small samples, independent groups

• Wilcoxon One-tailed Hypotheses
• H0 Prob. distributions for 2 sampled populations
  are identical.
• HA Prob. distribution for Population A shifted
  to right of distribution for Population B. (Note
  could be to the left, but must be one or the
  other, not both.)

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1b. Small samples, independent groups

• Wilcoxon Two-tailed Hypotheses
• H0 Prob. distributions for 2 sampled populations
  are identical.
• HA Prob. distribution for Population A shifted
  to right or left of distribution for Population B.

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1b. Small samples, independent groups

• Wilcoxon Rejection region
• (With Sample taken from Population A being
  smaller than sample for Population B) reject H0
  if
• TA TU or TA TL

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1b. Small samples, independent groups

• Wilcoxon for n1 10 and n2 10
• Test statistic
• Z TA n1(n1 n2 1)
• 2
• n1n2(n1 n2 1)
• 12

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Wilcoxon for n1 10 and n2 10

• Rejection region
• One-tailed Two-tailed
• Z gt Za Z gt Za/2
• Note use this only when n1 10 and n2 10

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Example 1

• A retail store sales consultant is asked to
  determine whether customers at stores which
  require memberships spend more on average than
  customers at ordinary stores where you dont have
  to buy a membership to shop. She surveys as group
  of 7 randomly selected customers leaving
  no-membership-required stores and 8 different
  randomly selected customers leaving
  membersip-required stores, recording the amount
  of their purchases shown on their sales receipts.
  The data are shown below. Is there evidence that
  customers at the membership-required stores, on
  average, spend more money on a single trip to the
  store than customers at no-membership-required
  stores? (a .05)

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Example 1

• Amount spent
• No membership required Membership required
• 41 44
• 28 36
• 38 35
• 32 43
• 24 39
• 36 36
• 29 50
• 37

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Example 1

• This question involves a test of hypothesis about
  2 population means, using the means of two small,
  independent samples.
• Thus, the first step is the test of hypothesis
  about the population variances, s12 and s22.

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Example 1

• No membership required
• X 32.57
• s2 7646 7426.29 36.62
• 6
• Membership required
• X 40.0
• s2 12992 12800 27.42
• 7

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Example 1

• HO s12 s22
• HA s12 ? s22
• F critical F (6, 7, .025) 5.12 (from table)
• Fobt 36.62 1.355 lt 5.12
• 27.42
• Therefore, we can do a t-test.

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Example 1

• HO µnon2 µmem2
• HA µnon2 lt µmem2
• t critical t (n 2, a) t (13, .05) 1.771
• s2p 6 (36.62) 7 (27.42) 31.67
• 13

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Example 1

• tobt 40 32.57
• 31.67 31.67
• 7 8
• 7.43
• 4.524 3.959
• 7.43 2.55 ? reject HO.
• 2.913

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Example 2

• Many people are confused about the distinction
  between Cajun and Creole cooking. One possible
  distinction is that Cajun food contains more
  cayenne and other peppers and is thus hotter
  than Creole food. To test this hypothesis, the
  heat (measured in Scoville Units) of random
  samples of Cajun and Creole dishes is compared,
  yielding the data on the next slide. Do these
  data support the hypothesis that Cajun dishes are
  hotter than Creole dishes (a .05)?

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Example 2

• Scoville Heat Units
• Cajun Dishes Creole Dishes
• 3500 3100
• 4200 4700
• 4100 2700
• 4700 3500
• 4200 2000
• 3705 3100
• 4100 1550

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Example 2

• Cajun
• s2 890592.8572 148432.14
• 6
• Membership required
• s2 66334999.98 1055833.33
• 7

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Example 2

• HO s12 s22
• HA s12 ? s22
• F critical F (6, 6, .025) 5.82 (from table)
• Fobt 1055833.33 7.11 gt 5.82
• 148432.14
• Therefore, we cannot do a t-test. We do a
  Wilcoxon.

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Example 2

• HO Cajun distribution and Creole distribution
  are identical.
• HA Cajun distribution is shifted to the right
  relative to the Creole distribution.
• Test statistic T (rank sum total) for Cajun
  dishes
• T critical TU 66 or TL 39

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Example 2

• Scoville Heat Units
• Cajun Rank Creole Rank
• 3500 6.5 3100 4.5
• 4200 11.5 4700 13.5
• 4100 9.5 2700 3
• 4700 13.5 3500 6.5
• 4200 11.5 2000 2
• 3705 8 3100 4.5
• 4100 9.5 1550 1
• 70 35

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Example 2

• Check accuracy of rank sum total T
• TA TB 105 n(n1) 1415 105
• 2 2
• TA gt TU 70. Reject HO Cajun dishes are hotter
  than Creole dishes.

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