Inference Concerning Populations Numeric Responses

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Chapter 7

• Inference Concerning Populations (Numeric
  Responses)

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Inference for Population Mean

• Practical Problem Sample mean has sampling
  distribution that is Normal with mean m and
  standard deviation s / ?n (when the data are
  normal, and approximately so for large samples).
  s is unknown.
• Have an estimate of s , s obtained from sample
  data. Estimated standard error of the sample mean
  is

When the sample is SRS from N(m , s) then the
t-statistic (same as z- with estimated standard
deviation) is distributed t with n-1 degrees of
freedom
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Family of t-distributions

• Symmetric, Mound-shaped, centered at 0 (like the
  standard normal (z) distribution
• Indexed by degrees of freedom (n ), the number of
  independent observations (deviations) comprising
  the estimated standard deviation. For one sample
  problems n n-1
• Have heavier tails (more probability over extreme
  ranges) than the z-distribution
• Converge to the z-distribution as n gets large
• Tables of critical values for certain upper tail
  probabilities are available (inside back cover of
  text)

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Probability
Cri t ical Values
Degrees of Freedom
Critical Values
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One-Sample Confidence Interval for m

• SRS from a population with mean m is obtained.
• Sample mean, sample standard deviation are
  obtained
• Degrees of freedom are n n-1, and confidence
  level C are selected
• Level C confidence interval of form

Procedure is theoretically derived based on
normally distributed data, but has been found to
work well regardless for large n
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1-Sample t-test (2-tailed alternative)

• 2-sided Test H0 m m0 Ha m ? m0
• Decision Rule (t obtained such that P(t(n-1)?
  t)a/2)
• Conclude m gt m0 if Test Statistic (tobs) is
  greater than t
• Conclude m lt m0 if Test Statistic (tobs) is
  less than -t
• Do not conclude Conclude m ? m0 otherwise
• P-value 2P(t(n-1)? tobs)
• Test Statistic

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P-value (2-tailed test)
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1-Sample t-test (1-tailed (upper) alternative)

• 1-sided Test H0 m m0 Ha m gt m0
• Decision Rule (t obtained such that P(t(n-1)?
  t)a)
• Conclude m gt m0 if Test Statistic (tobs) is
  greater than t
• Do not conclude m gt m0 otherwise
• P-value P(t(n-1)? tobs)
• Test Statistic

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P-value (Upper Tail Test)
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1-Sample t-test (1-tailed (lower) alternative)

• 1-sided Test H0 m m0 Ha m lt m0
• Decision Rule (t obtained such that P(t(n-1)?
  t)a)
• Conclude m lt m0 if Test Statistic (tobs) is
  less than -t
• Do not conclude m lt m0 otherwise
• P-value P(t(n-1)? tobs)
• Test Statistic

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P-value (Lower Tail Test)
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Example Mean Flight Time ATL/Honolulu

• Scheduled flight time 580 minutes
• Sample n31 flights 10/2004 (treating as SRS
  from all possible flights
• Test whether population mean flight time differs
  from scheduled time
• H0 m 580 Ha m ? 580
• Critical value (2-sided test, a 0.05, n-130
  df) t2.042
• Sample data, Test Statistic, P-value

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Paired t-test for Matched Pairs

• Goal Compare 2 Conditions on matched individuals
  (based on similarities) or the same individual
  under both conditions (e.g. before/after studies)
• Obtain the difference for each pair/individual
• Obtain the mean and standard deviation of the
  differences
• Test whether the true population means differ
  (e.g. H0mD 0)
• Test treats the differences as if they were the
  raw data

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Test Concerning mD

• Null Hypothesis H0mDD0 (almost always 0)
• Alternative Hypotheses
• 1-Sided HA mD gt D0
• 2-Sided HA mD ? D0
• Test Statistic

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Test Concerning mD
Decision Rule (Based on t-distribution with
nn-1 df) 1-sided alternative If tobs ? t
gt Conclude mD gt D0 If tobs lt t gt Do not
reject mD D0 2-sided alternative If tobs ? t
gt Conclude mD gt D0 If tobs ? -t gt
Conclude mD lt D0 If -t lt tobs lt t gt Do not
reject mD D0
Confidence Interval for mD
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Example Antiperspirant Formulations

• Subjects - 20 Volunteers armpits
• Treatments - Dry Powder vs Powder-in-Oil
• Measurements - Average Rating by Judges
• Higher scores imply more disagreeable odor
• Summary Statistics (Raw Data on next slide)

Source E. Jungermann (1974)
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Example Antiperspirant Formulations
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Example Antiperspirant Formulations
Evidence that scores are higher (more unpleasant)
for the dry powder (formulation 1)
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Comparing 2 Means - Independent Samples

• Goal Compare responses between 2 groups
  (populations, treatments, conditions)
• Observed individuals from the 2 groups are
  samples from distinct populations (identified by
  (m1,s1) and (m2,s2))
• Measurements across groups are independent
  (different individuals in the 2 groups
• Summary statistics obtained from the 2 groups

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Sampling Distribution of

• Underlying distributions normal ? sampling
  distribution is normal
• Underlying distributions nonnormal, but large
  sample sizes ? sampling distribution
  approximately normal
• Mean, variance, standard deviation

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t-test when Variances are estimated

• Case 1 Population Variances not assumed to be
  equal (s12?s22)
• Approximate degrees of freedom
• Calculated from a function of sample variances
  and sample sizes (see formula below) -
  Satterthwaites approximation
• Smaller of n1-1 and n2-1
• Estimated standard error and test statistic for
  testing H0 m1m2

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t-test when Variances are estimated

• Case 2 Population Variances assumed to be equal
  (s12s22)
• Degrees of freedom n1n2-2
• Estimated standard error and test statistic for
  testing H0 m1m2

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Example - Maze Learning (Adults/Children)

• Groups Adults (n114) / Children (n210)
• Outcome Average of Errors in Maze Learning
  Task
• Raw Data on next slide

• Conduct a 2-sided test of whether mean scores
  differ
• Construct a 95 Confidence Interval for true
  difference

Source Gould and Perrin (1916)
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Example - Maze Learning (Adults/Children)
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Example - Maze LearningCase 1 - Unequal Variances
H0 m1-m2 0 HA m1-m2 ? 0 (a
0.05)
No significant difference between 2 age
groups Note Alternative would be to use 9 df
(10-1)
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Example - Maze LearningCase 2 - Equal Variances
H0 m1-m2 0 HA m1-m2 ? 0 (a
0.05)
No significant difference between 2 age groups
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SPSS Output
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C Confidence Interval for m1-m2