Biostatistics and Computer Applications

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Biostatistics and Computer Applications
Sampling mean Difference between means Students
t distribution Chi-square distribution F
distribution SAS programming 12/30/02
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Recap (Probability and variable distribution)

• Event and probability P(E)n/N (n is large)
• Additive rule and multiplication rule
• Binomial distribution
• Poisson distribution
• Normal distribution

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Binomial Distribution
Poisson Distribution
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Normal Distribution
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Standard Normal Distribution
In general, 68.26 of data within ?1? of ?,
95.45 of data within ?2? of ?, 99.73
of data within ?3? of ?, 95 ?
?1.96? 99 ? ?2.58?
0.025
0.025
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Distribution of the Sample Mean
Parameters are constant Statistics are
variable This is a critically important
concept!!!
Population N (?, )
sample of size n
sample of size n
n
n
n
Mathematical derive or Sampling
?
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Random Samples

• Infer the characteristics of a population from a
  representative sample
• Usual strategy is through a random sample
• Random sample
• Each object has a equal probability of being
  selected as a random number
• Two methods
• Exhaustive (all possible) sampling from a small
  population (Nn) (Lecture).
• Large number of sampling from a infinite
  population (SAS program).

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Sampling Distributions

• Definition
• Distribution of all possible values that can be
  assumed by some statistic, computed from samples
  of the same size randomly drawn from the same
  population, called the sampling distribution of
  that statistic
• Statistic Mean, difference between two means,
  variance etc.

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Sampling Distributions

• Construction
• From a finite population of size N, randomly draw
  all possible samples of size n (total number of
  sample Nn).
• Compute the statistic of interest for each sample
  (mean, standard deviation)
• Investigate the distribution of that statistic
  and relate it to the population parameter.

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Sampling Distribution of mean

• Example An approximately normal distributed
  population with N4, Xi2,3,3,4
• n2, independent random sampling,
• Total sample is Nn42 16.

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Sampling Distribution of mean
Mean of sampling statistics
Unbiased estimate
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Sampling Distribution of mean
Frequency distribution of sample means
If we draw n4, then total sampling is 44 256
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Distribution of Sample Mean

• 1. Mean of sampling distribution of mean has same
  mean as original population
• 2. Variance of sampling distribution of mean is
  the population variance divided by sample size
• and
• called standard error of mean. A measure of
  sampling error of x-bar, and decrease as n
  increase.

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Distribution of Sample Mean

• 3. If , then
• 4. Given a population of unknown distribution or
  nonnormal distribution with a mean and
  variance , the sampling distribution of
  
  will be approximately normally distributed
  with a mean of and variance of
  when n is large (Central limit theorem).

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Implications of Central Limit Theorem

• Assured of approximately normal sampling
  distribution
• Normally distributed population
• Nonnormal population with large sample size
• Population with unknown functional form with
  large sample size
• How large is large for sample size?
• at least 30, but as many as practical (20 may be
  ok if not much skewed)

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Application of sample distribution

• Example 1. Suppose we have a normal population
  with mean12.5 year and variance 0.78 year2.
  If we draw a random sample with n10 and, whats
  the probability the sample mean is large than
  14.7 year?

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Application of sample distribution

• Example 2. Cotton, fiber length (mm),
  . Whats the
  probability if we draw an sample with n4, the
  difference between sample mean and population
  mean is larger than 0.5 mm? What if n25?
• If n25,

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Distribution of the Sample Mean
Another example Suppose that for Oklahoma sixth
grade students the mean number of missed school
days is 5.4 days with a standard deviation of 2.8
days. What is the probability that a random
sample of size 49 will have a mean number of
missed days greater than 6 days? Find the
probability that a random sample of size 49 from
this population will have a mean greater than 6
days. ? 5.4 days ? 2.8 days n 49
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Distribution of the Sample Mean
Sampling distribution of (for samples of
size 49)
Population distribution of individual Xs
0.0668
What is the probability that a random sample
(size 49) from this population has a mean between
4 and 6 days? Check that .
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Sampling Distributions of
n2
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Sampling Distributions of
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Distribution of

• 1.
• 2.
• and
• called standard error of the difference
  between two means.

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Distribution of

• 3. If and
  , then

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Summary
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Students t-distribution
What happen if s is unknown? is normally
distributed, is not (quite)!
W.S. Gosset worked for Guinness Brewing in
Dublin, IR. He was forced to publish under the
pseudonym Student. In 1908 he derived the
distribution of which is now known as
Students t-distribution.
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Normal distribution and t distributions
When ? is unknown we replace it with the
estimate, s. The statistic has a t-distribution
with n-1 degrees of freedom.
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Students t Distribution
Cumulative function F(t) Right tail
probability Two tails probability (t
symmetrical distribution) Students
t-distribution table (two sides, selected t for
certain probabilities 0.05, 0.01)
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Students t distribution table
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Distribution of sample variance

• Same method for sample mean
• N4, Xi2,3,3,4 n2 and n4 sampling

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Distribution of sample variance

• Left skewed distribution, not z or t distribution
• Central Limit Theorem For a normal distributed
  population, when ngt100, the distribution of
  sample standard deviation s is approximately
  normal distributed with a mean and standard
  deviation of s
• Standard error of sample standard deviation

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Chi-square distribution

• Definition (population)
• Sample

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Chi-square distribution

• Probability density function
• Cumulative function
• Right side (tail) probability

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Properties of the Chi-Square Distribution

• 1.      The distribution is not symmetric,
  (as the df increases, the distribution becomes
  more symmetric) 
• 2.      All values are gt 0 (positive) 
• 3.      The distribution is different for
  each df (df n-1). As df (dfgt30) increases, the
  distribution approaches a Normal distribution.

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Example

• A normal distributed population with variance 2,
  please calculate if n5, the probability that
  and the probability that

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Chi-square table
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F distribution

• Draw two independent random samples from a normal
  distribution with variance
• Definition

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F distribution

• Probability density function
• Cumulative function
• Right tail probability

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F distribution table

• Only F values for P0.95, 0.99 etc.
• Two degree of freedom (df1, df2)
• Right side (tail) probability, arranged for
  P(FgtFi)0.05 or 0.01, denote as F0.05 and F0.01.

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F distribution table
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F distribution table
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SAS programming

• Distribution of the sample mean
• Distribution of the difference of two means
• Students t distribution
• Chi-square distribution
• F distribution

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SAS functions for sampling distribution

• Useage xfunction_name(arguments)
• Probability and density function
• PROBT(x,df lt,ncgt) probability of t
  distribution
• PROBCHI(x,df) Chi-square
• PROBF(x,ndf,ddflt,ncgt) F distribution
• Quantile function
• CINV(p,dflt,ncgt)
• FINV(p,ndf,ddflt,ncgt)
• TINV(p,dflt,ncgt)
• PROBIT(p)

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Sampling distribution of mean

• Data sample_mean
• miu100
• sigma10
• do k1 to 1000
• do i1 to 100 by .5
• zrannor(0)
• xmiuzsigma
• output
• end
• end
• proc sort
• by k

• Proc means noprint
• var x
• by k
• output outmean_sample meansample_mean
• run
• proc print datamean_sample
• run
• proc univariate datamean_sample
• var sample_mean
• run

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Sampling distribution of difference between two
means (1/3)

• / This program draw two samples from two normal
  distributions and shows the distribution of
  difference between sample means /
• Data sample_mean
• miu1100 sigma110
• miu2200 sigma210
• do k1 to 1000
• do i1 to 100 by .5
• z1rannor(0) z2rannor(0)
• x1miu1z1sigma1 x2miu2z2sigma2
• output end end

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Sampling distribution of difference between two
means (2/3)

• proc sort
• by k
• Proc means noprint
• var x1 x2 by k
• output outtwo_means meanmean1 mean2
• run
• data dmeans
• set two_means
• dmeanmean1-mean2
• run

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Sampling distribution of difference between two
means (3/3)

• proc print datadmeans
• run
• proc univariate datadmeans
• var mean1 mean2 dmean
• run

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SAS program (t distribution)

• options nodate
• data tdistribution
• df15
• do i-5 to 5 by .1
• prob_zprobnorm(i)
• prob_tprobt(i,df)
• output
• end

• proc print
• var i prob_z prob_t df
• proc plot
• plot prob_zi prob_ti
• run

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SAS program (Chi-square and F distributions)

• data distribution
• df15 df15 df220
• do i0 to 50 by .5
• prob_chi2probchi(i,df)
• prob_fprobf(i,df1,df2)
• output
• end
• proc print
• var i prob_chi2 prob_f df df1 df2
• proc plot
• plot prob_chi2i prob_fi
• run

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SAS program (distribution table)

• data sampling
• df30 df15 df210
• do p0 to 1 by 0.01
• normalprobit(p)
• ttinv(p,df)
• chi_squareCINV(p,df)
• fFINV(p,df1,df2)
• output
• end

• proc print
• var normal t chi_square f p df df1 df2
• run
• proc plot
• plot pnormal pt pchi_square pf
• run
• quit