Sampling Distributions

1
Chapter 5

• Sampling Distributions

2
Introduction

• Distribution of a Sample Statistic The
  probability distribution of a sample statistic
  obtained from a random sample or a randomized
  experiment
• What values can a sample mean (or proportion)
  take on and how likely are ranges of values?
• Population Distribution Set of values for a
  variable for a population of individuals.
  Conceptually equivalent to probability
  distribution in sense of selecting an individual
  at random and observing their value of the
  variable of interest

3
Sampling Distributions for Counts and Proportions

• Binary outcomes Each individual or realization
  can be classified as a Success or Failure
  (Presence/Absence of Characteristic of interest)
• Random Variable X is the count of the number of
  successes in n trials
• Sample proportion Proportion of succeses in the
  sample
• Population proportion Proportion of successes in
  the population

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Binomial Distribution for Sample Counts

• Binomial Experiment
• Consists of n trials or observations
• Trials/observations are independent of one
  another
• Each trial/observation can end in one of two
  possible outcomes often labelled Success and
  Failure
• The probability of success, p, is constant across
  trials/observations
• Random variable, X, is the number of successes
  observed in the n trials/observations.
• Binomial Distributions Family of distributions
  for X, indexed by Success probability (p) and
  number of trials/observations (n). Notation
  XB(n,p)

5
Binomial Distributions and Sampling

• Problem when sampling from a finite sample the
  sequence of probabilities of Success is altered
  after observing earlier individuals.
• When the population is much larger than the
  sample (say at least 20 times as large), the
  effect is minimal and we say X is approximately
  binomial
• Obtaining probabilities

Table C gives probabilities for various n and p.
Note that for p gt 0.5, use 1-p and you are
obtaining P(Xn-k)
6
Example - Diagnostic Test

• Test claims to have a sensitivity of 90 (Among
  people with condition, probability of testing
  positive is .90)
• 10 people who are known to have condition are
  identified, X is the number that correctly test
  positive

• Compare with Table C, n10, p.10
• Table obtained in EXCEL with function
  BINOMDIST(k,n,p,FALSE)
• (TRUE option gives cumulative distribution
  function P(X?k)

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Binomial Mean Standard Deviation

• Let Si1 if the ith individual was a success, 0
  otherwise
• Then P(Si1) p and P(Si0) 1-p
• Then E(Si)mS 1(p) 0(1-p) p
• Note that X S1Sn and that trials are
  independent
• Then E(X)mX nmS np
• V(Si) E(Si2)-mS2 p-p2 p(1-p)
• Then V(X)sX2 np(1-p)

For the diagnostic test
8
Sample Proportions

• Counts of Successes (X) rarely reported due to
  dependency on sample size (n)
• More common is to report the sample proportion of
  successes

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Sampling Distributions for Counts Proportions

• For samples of size n, counts (and thus
  proportions) can take on only n distinct possible
  outcomes
• As the sample size n gets large, so do the number
  of possible values, and sampling distribution
  begins to approximate a normal distribution.
  Common Rule of thumb np ? 10 and n(1-p) ? 10 to
  use normal approximation

10
Sampling Distribution for XB(n1000,p0.2)
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Using Z-Table for Approximate Probabilities

• To find probabilities of certain ranges of counts
  or proportions, can make use of fact that the
  sample counts and proportions are approximately
  normally distributed for large sample sizes.
• Define range of interest
• Obtain mean of the sampling distribution
• Obtain standard deviation of sampling
  distribution
• Transform range of interest to range of Z-values
• Obtain (approximate) Probabilities from Z-table

12
Sampling Distribution of a Sample Mean

• Obtain a sample of n independent measurements of
  a quantitative variable X1,,Xn from a
  population with mean m and standard deviation s
• Averages will be less variable than the
  individual measurements
• Sampling distributions of averages will become
  more like a normal distribution as n increases
  (regardless of the shape of the population of
  individual measurements)

13
Central Limit Theorem

• When random samples of size n are selected from
  aamy population with mean m and finite standard
  deviation s, the sampling distribution of the
  sample mean will be approximately distributed for
  large n

Z-table can be used to approximate probabilities
of ranges of values for sample means, as well as
percentiles of their sampling distribution
14
Exponential Distribution

• Often used to model times survival of
  components, to complete tasks, between customer
  arrivals at a checkout line, etc. Density is
  highly skewed

Sample means of size 10 (m1, s1/100.50.32)
Individual Measurements (m1,s1)
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Miscellaneous Topics

• Normal Approximation for sample counts and
  proportions is example of CLT (XS1Sn)
• Any linear function of independent normal random
  variables is normal (use rules on means and
  variances to get parameters of distribution)
• Generalizations of CLT apply to cases where
  random variables are correlated (to an extent)
  and have different distributions (within reason)
• Variables made up of many small random influence
  will tend to be approximately normal