Random Sampling and Sampling Distributions Chapter 6

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Random Sampling and Sampling DistributionsChapter
6

• He stuck in his thumb,
• Pulled out a plum
• and said
• what a good boy am I!
• old nursery rhyme

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Topics and Goals for Chapter 6

• Random Sampling
• Sample Statistics and Relation to Population
  Parameters
• Sampling Distribution for Sample Mean--
  The Central Limit Theorem
• Checking Normality-The Normal Probability Plot
• samples from normal distributions
• positively skewed distributions
• negatively skewed distributions
• distributions with outliers

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Populations and Samples

• A population is a large collection
  (theoretically, for the mathematician, infinite)
  of the individuals or items of interest (e.g.
  consuming public, machine line production items,
  etc.)
• To measure characteristics of the population we
  have to take a sample (smaller number).
• If we take a random sample, it is equally likely
  that any member of the population will be
  included in the sample.

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Random Sampling

• Sample represents population only if each member
  of population equally likely to be included in
  sample.
• Types of random sampling (see also Chapter 16)
• Simple Random Sampling (SRS)--
• sample whole population
• Stratified Random Sampling
• divide population into groups and sample from
  each group for example, in polls, divided
  country into four geographical regions and sample
  from each
• Cluster Sampling
• Divide population into groups and take a sample
  of a few groups from the total--e.g., looking at
  hospital performance, sample patients in few
  hospitals randomly chosen from all hospitals in
  the state.

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Sample Statistics

• Sample Mean xbar (1/N) ? xi , where xbar is
  x with a bar over it the sum is taken over all
  values of the random variable X
  measured in the sample of N units.
• xbar is an estimator of the population mean, ?.
• Sample Standard Deviation
  s 1/ (N-1) ? (xi-
  xbar)2 (1/2
• s is an unbiased estimate of the population
  standard deviation, ?.
• Note that for large samples (large N), N-1? N

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Sampling Distribution for Sample MeansThe
Central Limit Theorem--1

• In general (which means almost always), no matter
  what distribution the population follows, the
  distribution of the sample means follows a
  normal distribution with
• mean µsample means (for the population of sample
  means) equal to µ, the mean for the parent
  population, and
• standard deviation of the means
  ?sample means ? /?N. This means that the
  larger the sample size, the more accurately we
  estimate the mean.

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Sampling Distribution for Sample MeansThe
Central Limit Theorem-2

• The histogram on the left is for a sample from a
  uniform distribution (0 to 100). The sample mean
  is 50.2 and the sample standard deviation is 29.3
  (?100/?12)

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Sampling Distribution for Sample MeansThe
Central Limit Theorem-2

• The histogram on the left is for the means of 150
  samples, each size 9 (N 9). The average of
  these 150 means is 49.4 and the standard
  deviation of these 150 sample means is 9.8 which
  is about (100/?12?9), the population
  standard dev-iation of the mean.

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Normal Probability Plots (P-plots)

• The procedure to get this plot, which tests
  whether data follow a normal distribution
  procedure, is the following
• 1) order the N data
• 2) assign a rank from 1--the lowest--to N--the
  highest value
• 3) find the centile score of the mth data point
  from the relation centile score m/(N1)--e.g
  the 1st data point out of 100 has a fraction
  approximately 1/101 lower the 100th data
  point has a fraction 100/101 lower
• 4) find the z-value (standard normal variate)
  corresponding to the centile score (this would
  be the z-score or N-score).
• 5) plot the observed points versus the z-score
• If the points fall approximately on a straight
  line, the distribution is a normal distribution.
  

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Normal Probability Plots (P-plots) Examples

• Exam 2 scores were negatively skewed (range
  49-100, Q190, median92, Q3 94
• rank ordered value z-score Exam 2
• 1 0.02 -2.10 49
• 2 0.04 -1.80 72
• 3 0.05 -1.61 78
• 4 0.07 -1.47 79
• 5 0.09 -1.35 81
• 6 0.11 -1.24 85
• 7 0.13 -1.15 86
• 8 0.14 -1.07 87
• 9 0.16 -0.99 89
• etc. .

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Normal Probability Plots (P-plots) Examples
(cont.)

• This Pplot for Exam 2 scores is from the
  Statplus addin note that the axes are
  inter-changed from the previous (conventional)
  order Nscore is y-axis, actual score is x-axis
• rank ordered value z-score Exam 2
• 1 0.02 -2.10 49
• 2 0.04 -1.80 72
• 3 0.05 -1.61 78
• 4 0.07 -1.47 79
• 5 0.09 -1.35 81
• etc. .

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Qualitative Appearance of P-plots