Simple Comparative Experiments

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Simple Comparative Experiments

• IE 4553/5553
• Fall 2004

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ProbabilityDistributions
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Discrete Random Variable

• P(Xx) denotes the probability of an event, or
  the probability that X assumes the value x

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Probability Mass Function

• For a discrete random variable X with possible
  values x1, x2, , xn, the probability mass
  function is
• f (xi) P(Xxi)
• Because f (xi) is defined as a probability, f(xi)
  ? 0 for all xi and the sum of f (xi) from i 1
  to n is equal to 1.

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Data Distribution Displays

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Cumulative Distribution

• The cumulative distribution function of a random
  discrete variable X, denoted as F(x)
• F(x) P(X ? x) Sum f (x)
• for all values less than or equal to x.
• Probability rules
• F(x) P(X ? x) Sum f (x) for all values less
  than x or equal to x
• 0 ? F(x) ? 1
• If x ? y, then F(x) ? F(y)

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Continuous Random Variable

• Probability is described by a probability density
  function f (x)
• Similar to density in other physical systems
• Probability of a value occurring within a
  specified interval is the area under f (x)
  between the end points of the interval

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Probability Density Function

• A function f (x) is a probability density
  function of the continuous random variable X if
  for any interval of real numbers a, b,
• f (x) gt 0
• Integral over the real number line 1
• P(a lt X lt b) integral evaluated over the range
  a, b

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Probability Density Function

• P(a lt Z lt b)

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Mathematical Operations

• Mean of a Probability Distribution
• Variance of a Probability Distribution
• Expected Value of a Function of R.V.
• Review Formulas in Text
• Review Elementary Relationships

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Samplingand Sampling Distributions
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Inferential Statistics
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Measures ofCentral Tendency

• Mean
• Average score in the distribution
• Use pilot study mean to identify levels
• Seriously affected by extreme scores
• Median
• Middle score in the distribution
• Use with the mean value to decide levels
• Mode
• Most frequent (typical) score in the distribution
• Not affected by extreme scores

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Measures of Variability

• Range
• Difference between the largest and smallest value
  R xmax - xmin
• Sample Variance estimates s2
• Sample Standard Deviation estimates s

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Parameter Estimation

• Statistic
• A number calculated from sample data that should
  closely estimate its corresponding parameter in
  the population
• Properties of a Good Statistic
• A statistic is an unbiased estimator of ? iff
• A statistic is a consistent estimator of ?
  iff
• The efficiency of a statistic is the measure of
  its variance relative to that of the unbiased
  estimator with the smallest variance

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Sampling Distributions

• For a sample of n independent observations from
  any distribution with a finite variance
• Distribution of the Sample Mean
• X is an unbiased estimator of the population mean
  m
• EXm
• VarX? 2/n
• Distribution of the Sample Variance
• s2 is an unbiased estimator of the population
  variance ? 2
• Es2? 2
• However, s is not an unbiased estimator of ? ,
  since
• Es??

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Central Limit Theorem

• For ANY population, the distribution of the
  sample mean will approach a Normal distribution
  for large sample sizes
• Sample Mean Distribution Properties
• Mean m
• Variance s2 / n

2.15 13.59 34.13
34.13 13.59 2.15 Std. Dev. -3s -2s
-1s m 1s 2s 3s z
score - 3 - 2 - 1 0
1 2 3
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Standard Normal

• Formed from z-scores
• Z scores indicate the deviation of raw scores
  from the sample mean in units of Std. Dev.
• Properties z N(0,1)

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Distribution Theory

• Population- target group
• Sample- subgroup of population
• 3 Distributions
• Population
• SamplE
• SamplING

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Examples of Sampling Distributions

• David Lanes Rice Virtual Lab
• http//www.ruf.rice.edu/lane/rvls.html

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More on Sampling Distributions

• Need for Sampling Distributions
• Provides a link between a statistic and our real
  interest, parameter values
• Describes degree of approximation involved with a
  statistic
• Used to calculate a statistics probability given
  a Null hypothesis about the parameter

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Hypothesis Testing
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Making Statistical Inferences

• Compare Treatment vs. Control Groups
• Sample mean of each group should estimate its
  respective population mean
• If from the same population, differences in the
  means are due only to sampling error
• If from different populations, treatment effect
  is significant (i.e., treatment caused
  differences)
• Significance Level (a)
• Criterion set by experimenter
• Statement that the probability of an observed
  mean difference is strictly due to chance
• Conventionally, difference should occur less than
  5 times in 100 if by chance alone (a .05)

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Decision Outcomes
Correct Retention
Correct Rejection
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Decision Errors

• Type I Error (a)
• Probability of rejecting H0 when it is true
• Differences are attributed to the treatment when
  sampling error was the true cause
• Type II Error (b)
• Probability of retaining H0 when it is false
• Differences due to a treatment are attributed to
  mere chance
• For fixed sample size n and (a and b) are
  inversely related
• Both can be reduced only by increasing n

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Power and Effect Size

• Power (1-b)
• Probability of a correct rejection
• Increases with larger sample sizes
• Too much power can result in the detection of
  unimportant differences
• Effect Size
• Separation distance between the means of the null
  and alternate hypotheses
• Gives some perspective on the practical
  importance of any significant differences found

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Confidence Intervals

• An interval that will contain a parameter of
  interest approximately (1- a)100 of the time
• Example
• If observations are NID(m,s2), where s2 is known,
  then
• will contain the true mean (1- a)100 of the
  time
• If observations are NID(m,s2), where s2 is
  unknown, then
• will contain the true mean (1- a)100 of the
  time
• David Lane and Rice Virtual Lab

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Hypothesis Testing

• Null Hypothesis (H0)
• States no difference exists between events
• Assumed correct until evidence to the contrary
• Always an equality (e.g., H0 m 50)
• Alternate Hypothesis (Ha or H1)
• Often the formal hypothesis of the experiment
• One-tailed m lt 50 or m gt 50
• Two-tailed m ? 50
• Critical Region
• Set of values that leads to rejecting H0

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Tests of a Single Mean

• Assumptions
• XNID(m, s2)
• Hypothesis
• H0 m m0
• Ha m ? m0
• Test Statistic
• Case 1 Large sample (n? 30) or s2 known
• Case 2 Small sample or s2 unknown

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Tests on Two Means

• Assumptions
• XNID(m, s2)
• Hypothesis
• H0 m1 m2 H0 m1 - m2 0
• Ha m1 ? m2 Ha m1 - m2 ? 0
• Test Statistic
• Case 1 Large sample (n? 30) or s12 and s22
  known

or
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Tests on Two Means

• Case 2 s12 and s22 unknown but equal (Must
  Test!)
• Test for Equal Variances
• Test Statistic

where
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Tests on Two Means

• Case 3 s12 and s22 unknown and unequal
• Test Statistic
• NOTE Round df down to the nearest whole number
  to use the t-table.
• Rejection Region

where
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Dependent Samples

• Paired Sample t-test
• Assumptions
• Dependent samples
• Take differences for each pair d x1 - x2
• Test Statistic
• Advantages and Disadvantages of Pairing
• Reducing the variability (sd) yields a larger
  calculated t value and increases the likelihood
  of rejecting the null hypothesis, BUT
• We lose degrees of freedom (cut in half) which
  makes the test less sensitive since the critical
  t value in the table will be larger

where n number of pairs
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Tests on a Single Variance

• Assumptions
• XNID (m, s2)
• Hypothesis
• H0 s2 s02
• Ha s2 ? s02 or s2 gt s02 or s2 lt s02
• Test Statistic
• Confidence Interval

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Tests on Two Variances

• Assumptions
• X1NID (m1, s12) and X2NID (m2, s22)
• Hypothesis
• H0 s12 s22
• Ha s12 ? s22
• Test Statistic
• Rejection Region