Inferences Regarding Population Central Values

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Chapter 5

• Inferences Regarding Population Central Values

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Inferential Methods for Parameters

• Parameter Numeric Description of a Population
• Statistic Numeric Description of a Sample
• Statistical Inference Use of observed statistics
  to make statements regarding parameters
• Estimation Predicting the unknown parameter
  based on sample data. Can be either a single
  number (point estimate) or a range (interval
  estimate)
• Testing Using sample data to see whether we can
  rule out specific values of an unknown parameter
  with a certain level of confidence

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Estimating with Confidence

• Goal Estimate a population mean based on sample
  mean
• Unknown Parameter (m)
• Known Approximate Sampling Distribution of
  Statistic

• Recall For a random variable that is normally
  distributed, the probability that it will fall
  within 2 standard deviations of mean is
  approximately 0.95

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Estimating with Confidence

• Although the parameter is unknown, its highly
  likely that our sample mean (estimate) will lie
  within 2 standard deviations (aka standard
  errors) of the population mean (parameter)
• Margin of Error Measure of the upper bound in
  sampling error with a fixed level (we will
  typically use 95) of confidence. That will
  correspond to 2 standard errors

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Confidence Interval for a Mean m

• Confidence Coefficient (1-a) Probability (based
  on repeated samples and construction of
  intervals) that a confidence interval will
  contain the true mean m
• Common choices of 1-a and resulting intervals

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1-a
0
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1-a
m
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Philadelphia Monthly Rainfall (1825-1869)
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4 Random Samples of Size n20, 95 CIs
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Factors Effecting Confidence Interval Width

• Goal Have precise (narrow) confidence intervals
• Confidence Level (1-a) Increasing 1-a implies
  increasing probability an interval contains
  parameter implies a wider confidence interval.
  Reducing 1-a will shorten the interval (at a
  cost in confidence)
• Sample size (n) Increasing n decreases standard
  error of estimate, margin of error, and width of
  interval (Quadrupling n cuts width in half)
• Standard Deviation (s) More variable the
  individual measurements, the wider the interval.
  Potential ways to reduce s are to focus on more
  precise target population or use more precise
  measuring instrument. Often nothing can be done
  as nature determines s

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Precautions

• Data should be simple random sample from
  population (or at least can be treated as
  independent observations)
• More complex sampling designs have adjustments
  made to formulas (see Texts such as Elementary
  Survey Sampling by Scheaffer, Mendenhall, Ott)
• Biased sampling designs give meaningless results
• Small sample sizes from nonnormal distributions
  will have coverage probabilities (1-a) typically
  below the nominal level
• Typically s is unknown. Replacing it with the
  sample standard deviation s works as a good
  approximation in large samples

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Selecting the Sample Size

• Before collecting sample data, usually have a
  goal for how large the margin of error should be
  to have useful estimate of unknown parameter
  (particularly when comparing two populations)
• Let E be the desired level of the margin of error
  and s be the standard deviation of the population
  of measurements (typically will be unknown and
  must be estimated based on previous research or
  pilot study)
• The sample size giving this margin of error is

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Hypothesis Tests

• Method of using sample (observed) data to
  challenge a hypothesis regarding a state of
  nature (represented as particular parameter
  value(s))
• Begin by stating a research hypothesis that
  challenges a statement of status quo (or
  equality of 2 populations)
• State the current state or status quo as a
  statement regarding population parameter(s)
• Obtain sample data and see to what extent it
  agrees/disagrees with the status quo
• Conclude that the status quo is not true if
  observed data are highly unlikely (low
  probability) if it were true

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Elements of a Hypothesis Test (I)

• Null hypothesis (H0) Statement or theory being
  tested. Stated in terms of parameter(s) and
  contains an equality. Test is set up under the
  assumption of its truth.
• Alternative Hypothesis (Ha) Statement
  contradicting H0. Stated in terms of parameter(s)
  and contains an inequality. Will only be accepted
  if strong evidence refutes H0 based on sample
  data. May be 1-sided or 2-sided, depending on
  theory being tested.
• Test Statistic (T.S.) Quantity measuring
  discrepancy between sample statistic (estimate)
  and parameter value under H0
• Rejection Region (R.R.) Values of test statistic
  for which we reject H0 in favor of Ha
• P-value Probability (assuming H0 true) that we
  would observe sample data (test statistic) this
  extreme or more extreme in favor of the
  alternative hypothesis (Ha)

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Example Interference Effect

• Does the way items are presented effect task
  time?
• Subjects shown list of color names in 2 colors
  different/black
• yi is the difference in times to read lists for
  subject i diff-blk
• H0 No interference effect mean difference is 0
  (m 0)
• Ha Interference effect exists mean difference gt
  0 (m gt 0)
• Assume standard deviation in differences is s
  8 (unrealistic)
• Experiment to be based on n70 subjects

How likely to observe sample mean difference ?
2.39 if m 0?
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P-value
0
2.39
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Elements of a Hypothesis Test (II)

• Type I Error Test resulting in rejection of H0
  in favor of Ha when H0 is in fact true
• P(Type I error) a (typically .10, .05, or
  .01)
• Type II Error Test resulting in failure to
  reject H0 in favor of Ha when in fact Ha is true
  (H0 is false)
• P(Type II error) b (depends on true parameter
  value)
• 1-Tailed Test Test where the alternative
  hypothesis states specifically that the parameter
  is strictly above (below) the null value
• 2-Tailed Test Test where the alternative
  hypothesis is that the parameter is not equal to
  null value (simultaneously tests greater than
  and less than)

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Test Statistic

• Parameter Population mean (m ) under H0 is m0
• Statistic (Estimator) Sample mean obtained from
  sample measurements is
• Standard Error of Estimator
• Sampling Distribution of Estimator
• Normal if shape of distribution of individual
  measurements is normal
• Approximately normal regardless of shape for
  large samples
• Test Statistic (labeled simply as z in text)

Note Typically s is unknown and is replaced by
s in large samples
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Decision Rules and Rejection Regions

• Once a significance (a) level has been chosen a
  decision rule can be stated, based on a critical
  value
• 2-sided tests H0 m m0 Ha m ? m0
• If test statistic (zobs) gt za/2 Reject Ho and
  conclude m gt m0
• If test statistic (zobs) lt -za/2 Reject Ho and
  conclude m lt m0
• If -za/2 lt zobs lt za/2 Do not reject H0 m m0
• 1-sided tests (Upper Tail) H0 m ? m0 Ha m gt m0
• If test statistic (zobs) gt za Reject Ho and
  conclude m gt m0
• If zobs lt za Do not reject H0 m ? m0
• 1-sided tests (Lower Tail) H0 m ? m0 Ha m lt
  m0
• If test statistic (zobs) lt -za Reject Ho and
  conclude m lt m0
• If zobs gt -za Do not reject H0 m ? m0

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Computing the P-Value

• 2-sided Tests How likely is it to observe a
  sample mean as far of farther from the value of
  the parameter under the null hypothesis? (H0
  m m0 Ha m ? m0)

After obtaining the sample data, compute the mean
and convert it to a z-score (zobs) and find the
area above zobs and below -zobs from the
standard normal (z) table

• 1-sided Tests Obtain the area above zobs for
  upper tail tests (Ham gt m0) or below zobs for
  lower tail tests (Ham lt m0)

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Interference Effect (1-sided Test)

• Testing whether population mean time to read list
  of colors is higher when color is written in
  different color
• Data yi difference score for subject i
  (Different-Black)
• Null hypothesis (H0) No interference effect (H0
  m ? 0)
• Alternative hypothesis (Ha) Interference effect
  (Ha m gt 0)
• n 70 subjects in experiment, reasonably large
  sample

Conclude there is evidence of an interference
effect (m gt 0)
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Interference Effect (2-sided Test)

• Testing whether population mean time to read list
  of colors is effected (higher or lower) when
  color is written in different color
• Data Xi difference score for subject i
  (Different-Black)
• Null hypothesis (H0) No interference effect (H0
  m 0)
• Alternative hypothesis (Ha) Interference effect
  ( or -) (Ha m ? 0)

Again, evidence of an interference effect (m gt 0)
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Equivalence of 2-sided Tests and CIs

• For given a , a 2-sided test conducted at a
  significance level will give equivalent results
  to a (1-a) level confidence interval
• If entire interval gt m0, P-value lt a , zobs gt
  za/2 (conclude m gt m0)
• If entire interval lt m0, P-value lt a , zobs lt
  -za/2 (conclude m lt m0)
• If interval contains m0, P-value gt a , -za/2lt
  zobs lt za/2 (dont conclude m ?m0)
• Confidence interval is the set of parameter
  values that we would fail to reject the null
  hypothesis for (based on a 2-sided test)

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Power of a Test

• Power - Probability a test rejects H0 (depends on
  m)
• H0 True Power P(Type I error) a
• H0 False Power 1-P(Type II error) 1-b
• Example (Using context of interference data)
• H0 m 0 HA m gt 0
• s264 n16
• Decision Rule Reject H0 (at a0.05 significance
  level) if

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Power of a Test

• Now suppose in reality that m 3.0 (HA is true)
• Power now refers to the probability we
  (correctly) reject the null hypothesis. Note that
  the sampling distribution of the sample mean is
  approximately normal, with mean 3.0 and standard
  deviation (standard error) 2.0.
• Decision Rule (from last slide) Conclude
  population mean interference effect is positive
  (greater than 0) if the sample mean difference
  score is above 3.29
• Power for this case can be computed as

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Power of a Test

• All else being equal
• As sample size increases, power increases
• As population variance decreases, power
  increases
• As the true mean gets further from m0 , power
  increases

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Power of a Test
Distribution (H0)
Distribution (HA)
Fail to reject H0
Reject H0
.4424
.5576
.95
.05
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• Power Curves for sample sizes of 16,32,64,80 and
  varying true values m from 0 to 5 with s 8.
• For given m , power increases with sample size
• For given sample size, power increases with m

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Sample Size Calculations for Fixed Power

• Goal - Choose sample size to have a favorable
  chance of detecting important difference from m0
  in 2-sided test H0m m0 vs Ham ? m0
• Step 1 - Define an important difference to be
  detected (D)
• Case 1 s approximated from prior experience or
  pilot study - difference can be stated in units
  of the data
• Case 2 s unknown - difference must be stated in
  units of standard deviations of the data

• Step 2 - Choose the desired power to detect the
  desired important difference (1-b, typically at
  least .80). For 2-sided test

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Example - Interference Data

• 2-Sided Test H0m 0 vs Ham ? 0
• Set a P(Type I Error) 0.05
• Choose important difference of m-m0D2.0
• Choose PowerP(Reject H0D2.0) .90
• Set b P(Type II Error) 1-Power 1-.90 .10
• From study, we know s ?8

Would need 169 subjects to have a .90 probability
of detecting effect
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Potential for Abuse of Tests

• Should choose a significance (a) level in advance
  and report test conclusion (significant/nonsignifi
  cant) as well as the P-value. Significance level
  of 0.05 is widely used in the academic literature
• Very large sample sizes can detect very small
  differences for a parameter value. A clinically
  meaningful effect should be determined, and
  confidence interval reported when possible
• A nonsignificant test result does not imply no
  effect (that H0 is true).
• Many studies test many variables simultaneously.
  This can increase overall type I error rates

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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 (df), the number of
  independent observations (deviations) comprising
  the estimated standard deviation. For one sample
  problems df n-1
• Have heavier tails (more probability over extreme
  ranges) than the z-distribution
• Converge to the z-distribution as df gets large
• Tables of critical values for certain upper tail
  probabilities are available (Table 3, p. 679)

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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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Probability
Cri t ical Values
Degrees of Freedom
Critical Values
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(No Transcript)
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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 df n-1, and confidence
  level (1-a) are selected
• Level (1-a) 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 (ta/2 such that P(t(n-1)?
  ta/2)a/2)
• Conclude m gt m0 if Test Statistic (tobs) is
  greater than ta/2
• Conclude m lt m0 if Test Statistic (tobs) is
  less than -ta/2
• 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)
-tobs
tobs
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1-Sample t-test (1-tailed (upper) alternative)

• 1-sided Test H0 m m0 Ha m gt m0
• Decision Rule (ta such that P(t(n-1)? ta)a)
• Conclude m gt m0 if Test Statistic (tobs) is
  greater than ta
• 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 (ta obtained such that P(t(n-1)?
  ta)a)
• Conclude m lt m0 if Test Statistic (tobs) is
  less than -ta
• 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) t.0252.042
• Sample data, Test Statistic, P-value

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Inference on a Population Median

• Median Middle of a distribution (50th
  Percentile)
• Equal to Mean for symmetric distribution
• Below Mean for Right-skewed distribution
• Above Mean for Left-skewed dsitribution
• Confidence Interval for Population Median
• Sort observations from smallest to largest (y(1)
  ?...?y(n))
• Obtain Lower (La/2) and Upper (Ua/2) Bounds of
  Ranks
• Small Samples Obtain Ca(2),n from Table 5 (p.
  682)
• Large Samples

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Example - ATL/HNL Flight Times

• n31,
• Small-Sample C.05(2),319
• Large-Sample