Chapter 8 Fundamental Sampling Distributions and Data Distributions

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Chapter 8 Fundamental Sampling Distributions and
Data Distributions

• Wen-Hsiang Lu (???)
• Department of Computer Science and Information
  Engineering,
• National Cheng Kung University
• 2007/05/24

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8.1 Random Sampling

• Outcome of a statistical experiment
• Numerical value total value of a pair of dice
  tossed
• Descriptive representation blood types in blood
  test
• Sampling from distributions or populations
• Sample mean and sample variance
• The use of high speed computer enhance the use of
  formal statistical inference with graphical
  techniques.

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Random Sampling

• Definition 8.1 A population consists of the
  totality of the observations with which we are
  concerned.
• Finite size 600 students are classified
  according to blood type gt a population of size
  600
• Infinite size measuring the atmospheric
  pressure some infinite populations are so large
• Each observation in a population is a value of a
  random variable X having some probability
  distribution f(x).
• If one is inspecting items coming off an assembly
  line for defects, then each observation in
  population might be a value 0 or 1 of the
  binomial random variable X with probability
  distributionwhere 0 indicates a nondefective
  item and 1 indicates a defective item.

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Random Sampling

• Sometimes, it is impossible or impractical to
  observe the entire set of observations that make
  up the population.
• Definition 8.2 A sample is a subset of a
  population.
• Inference from the sample to the population are
  to be valid
• Obtain representative samples
• Bias Erroneous inferences result from selecting
  convenient sampling members
• Random sample independent and at random

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Random Sampling

• Definition 8.3 Let X1, X2 ,, Xn be n
  independent random variables, each having the
  same probability distribution f(x). We then
  define X1, X2, , Xn to be a random sample of
  size n from the population f(x) and write its
  joint probability distribution as
• If we assume the population of battery lives to
  be normal, the possible values of any random
  sample Xi, i 1, 2,, 8, will be precisely the
  same as those in the original population, and
  hence Xi has the same identical normal
  distribution as X.

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8.2 Some Important Statistics

• Definition 8.4 Any function of the random
  variables constituting a random sample is called
  a statistic.
• Definition 8.5 If X1, X2 ,, Xn represent a
  random sample of size n, then the sample mean is
  defined by the statistic
• Definition 8.6 If X1, X2 ,, Xn represent a
  random sample of size n, then the sample variance
  is defined by the statistic

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Some Important Statistics

• Example 8.1 A comparison of coffee prices at 4
  randomly selected grocery stores in San Diego
  showed increases from the previous month of 12,
  15, 17, and 20 cents for a 1-pound bag. Find the
  variance of this random sample of price
  increases.
• Solution

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Some Important Statistics

• Theorem 8.1 If S2 is the variance of a random
  sample of size n, we may write
• Proof

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Some Important Statistics

• Definition 8.7 The sample standard deviation,
  denoted by S, is the positive square root of the
  sample variance.
• Example 8.2 Find the variance of the data 3, 4,
  5, 6, 6, and 7, representing the number of trout
  caught by a random sample of 6 fishermen.
• Solution

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8.3 Data Displays and Graphical Methods

• Motivation Use creative displays to extract
  information about properties of a set.
• The stem and leaf plots provide the viewer a look
  at symmetry of the data.
• Normal probability plots and quantile plots are
  used to check normal distribution.
• Characterize statistical analysis as the process
  of drawing conclusion about system variability.
• Statistics provide single measures, whereas a
  graphical display adds additional information in
  terms of a picture.

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Box and Whisker Plot or Boxplot

• Box and whisker plot encloses the interquartile
  range of the data in a box that has median
  displayed within.
• Interquartile range between the 75th percentile
  (upper quartile) and the 25th percentile (lower
  quartile).
• Boxplot provides the viewer information about
  outliers which represent rare event.
• Example 8.3 Nicotine content was measured in a
  random sample of 40 cigarettes. The data is
  displayed right.
• Mild outliers 0.72, 0.85, and2.55

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Box and Whisker Plot or Boxplot
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Box and Whisker Plot or Boxplot

• Example 8.4 Consider the following data,
  consisting of 30 samples measuring the thickness
  of paint can ears. Figure 8.2 depicts a box and
  whisker plot for this asymmetric set of data.

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Quantile Plot

• Quantile plot
• Compare samples of data
• Draw distinctions
• Depict cumulative distribution function
• Definition 8.8 A quantile of a sample, q(f), is
  a value for which a specified fraction f of the
  data values is less than or equal to q(f).
• Sample median q(0.5) 75th percentile q(0.75)
  25th percentile q(0.25)

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Quantile Plot

• In Figure 8.3, quantile plotshows all
  observations.
• Large clusters slopes near zero
• Sparse data steeper slopes
• E.g.
• Sparse data 28-30
• High density 36-38

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Normal Quantile-Quantile Plot

• Approximation of quantileof normal distribution
• Definition 8.8 The normal quantile-quantile
  plot is a plot of

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Normal Quantile-Quantile Plot

• Construct a normal quantile-quantile plot and
  draw conclusions regarding whether or not it is
  reasonable to assume that the two samples are
  from the same N(?, ?) distribution.
• Solution
• Far from a straight line
• Station 1 reflect a few values in the lower tail
  of the distribution and several in the upper tail
• Unlikely

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8.4 Sampling Distribution

• Statistical inference is concerned with
  generalizations and predictions.
• Based on the opinions of several people
  interviewed on the street, that in a forthcoming
  election 60 of the eligible voters in the city
  of Detroit favor a certain candidate.
• Definition 8.10 The probability distribution of
  a statistic is called a sampling distribution.
• E.g., the probability distribution of is
  called the sampling distribution of the mean.
• The sampling distribution of a statistic depends
  on the size of the population, the size of the
  samples, and the method of choosing the samples.

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8.5 Sampling Distribution of Means

• Suppose that a random sample of n observations is
  taken from a normal population with mean ? and
  variance ?2.
• By the reproductive property of the normal
  distribution established in Theorem 7.11

• Theorem 7.11 If X1, X2 ,, Xn are independent
  random variables having normal distributions with
  means ?1, ?2 ,, ?n and variances ?12, ?22 ,,
  ?n2, respectively, then the random variable
  Y a1X1 a2X2 anXnhas a
  normal distribution with mean
  ?Y a1?1 a2?2 an?nand variance
  ?Y2 a12?12 a22?22 an2?n2.

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Sampling Distribution of Means

• Theorem 8.2 Central Limit Theorem If is the
  mean of a random sample of size n taken from a
  population with mean ? and finite variance ?2,
  then the limiting form of the distribution ofas
  n??, is the standard normal distribution n(z 0,
  1).
• The normal approximation for will generally
  be good if n ? 30.
• If n lt 30, the approximation is good only if the
  population is not too different from a normal
  distribution.
• If the population is known to be normal, the
  sampling distribution of will follow a
  normal distribution exactly, no matter how small
  the size of the samples.

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Sampling Distribution of Means

• Example 8.6 An electric firm manufactures light
  bulbs that have life mean equal to 800 hours and
  a standard deviation of 40 hours. Find the
  probability that a random sample of 16 bulbs will
  have an average life of less than 775 hours.
• Solution

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Sampling Distribution of Means

• Example 8.7 A engineer conjectures that the
  population mean of a certain component parts is
  5.0 millimeters. An experiment is conducted in
  which 100 parts produced by the process are
  selected randomly and the diameter measured on
  each. It is known that the population standard
  deviation ? 0.1. The experiment indicates a
  sample average diameter 5.027 millimeters.
  Does this sample information appear to support or
  refute the engineers conjecture?
• Solution

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Sampling Distribution of the Difference Between
Two Averages

• Theorem 8.3 If independent sample of size n1 and
  n2 are drawn at random from two populations,
  discrete or continuous, with means ?1 and ?2 and
  variances ?12 and ?22, respectively, then the
  sampling distribution of the differences of
  means, is approximately normally
  distributed with mean and variance given by

• Theorem 7.11 If X1, X2 ,, Xn are independent
  random variables having normal distributions with
  means ?1, ?2 ,, ?n and variances ?12, ?22 ,,
  ?n2, respectively, then the random variable
  Y a1X1 a2X2 anXnhas a normal
  distribution with mean ?Y a1?1
  a2?2 an?nand variance ?Y2
  a12?12 a22?22 an2?n2.

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Sampling Distribution of the Difference Between
Two Averages

• Example 8.8 Two independent experiments are
  being run in which two different types of paints
  are compared. Eighteen specimens are painted
  using type A and the drying time in hours is
  recorded on each. The same is done with type B.
  The population standard deviations are both known
  to be 1.0. Assuming that the mean drying time is
  equal for the two types of paint, find
• Solution

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Sampling Distribution of the Difference Between
Two Averages

• Example 8.9 The television picture tubes of
  manufacturer A have a mean lifetime of 6.5 yeas
  and a standard deviation of 0.9 year, while those
  of manufacturer B have a mean lifetime of 6.0
  years and a standard deviation of 0.8 year. What
  is the probability that a random sample of 36
  tubes from manufacturer A will have a mean
  lifetime that is at least 1 year more than the
  mean lifetime of a sample of 49 tubes from
  manufacturer B?
• Solution

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Sampling Distribution of S2

• If a random sample of size n is taken from a
  normal population with mean ? and variance ?2,
  and the sample variance S2 is computed.

Corollary If X1, X2 ,, Xn are independent
random variables having identical normal
distributions with mean ? and variances ?2
has a chi-squared
distribution with v n degrees of freedom.
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Sampling Distribution of S2

• Theorem 8.4 If S2 is the variance of a random
  sample of size n taken from a normal population
  having the variance ?2, then the statistichas
  a chi-squared distribution with v n -1 degrees
  of freedom.
• It is customary to let ??2 represent the ?2
  value above which we find an area of ?. This is
  illustrated by the shaded region in Figure 8.10.
• Table A.5

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Sampling Distribution of S2

• Example 8.10 A manufacturer of car batteries
  guarantees that his batteries will last, on the
  average, 3 years with a standard deviation of 1
  year. If five of these batteries have lifetimes
  of 1.9, 2.4, 3.0, 3.5, and 4.2 years, is the
  manufacturer still convinced that his batteries
  have a standard deviation of 1 year? Assume that
  the battery lifetime follows a normal
  distribution.
• Solution

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Degrees of Freedom As a Measure of Sample
Information

• Comparison
• Theorem 7.12 has a ?2 distribution with n
  degrees of freedom.
• Theorem 8.4 has a ?2 distribution with n -1
  degrees of freedom.(when ? is not known, a
  degree of freedom is lost in the estimation of ?,
  i.e. )

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t-Distribution

• Central Limit Theorem (Theorem 8.2)
• ? might not be known.
• Consider
• In developing the sampling distribution of T, we
  shall assume that our random sample was selected
  from a normal population.

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t-Distribution

• Theorem 8.5 Let Z be a standard normal random
  variable and V a chi-squared random variable with
  v degrees of freedom. If Z and V are independent,
  then the distribution of the random variable T,
  whereis given by the density functionThis
  is known as the t-distribution with v degrees of
  freedom.

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t-Distribution

• Corollary Let X1, X2 ,, Xn be independent
  random variables that are all normal with mean ?
  and standard deviation ?. LetThen the random
  variable has at-distribution with v
  n-1 degrees of freedom.
• Student t-distribution
• The probability distribution of T was first
  published in 1908 in a paper by W. S. Gosset.
• Employed by an Irish brewery, but disallowed
  publication.
• Published his work secretly under the name
  Student.

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t-Distribution

• T is similar to Z symmetric about ?
  0,bell-shaped.
• Difference between T and Z variance of T ? 1
  and depends on n
• T and Z are the same n ? ?

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t-Distribution

• t-value with 10 degrees of freedom leaving an
  area of 0.025 to the right is t 2.228.
• t-distribution is symmetric about 0 t1-? -t?.
• Example 8.11 The t-value with v 14 degrees of
  freedom that leaves an area of 0.025 to the left,
  and therefore an area of 0.975 to the right, is
• Example 8.12 P(-t0.025 lt T lt t0.05) 1 - 0.05
  - 0.025 0.925

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t-Distribution
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t-Distribution
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t-Distribution

• Example 8.13 Find k such that P(k lt T lt -1.761)
  0.045, for a random sample of size 15 selected
  from a normal distribution and
• Solution

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t-Distribution

• Exactly 95 of the values of a t-distribution
  with v n -1 degrees of freedom lie between
  t0.025 and t0.025.
• A t-value that falls below t0.025 or above
  t0.025 would tend to make us believe that either
  a very rare event has taken place or perhaps our
  assumption about ? is error.
• Example 8.14 A engineer claims that the
  population mean of a process is 500 grams. To
  check this claim he samples 25 batches each
  month. If the computed t-value falls between
  t0.05 and t0.05, he is satisfied with his claim.
  What conclusion should he draw from a sample that
  has a mean grams and a sample
  standard deviation s 40 grams? Assume the
  distribution of yields to be approximately
  normal.
• Solution

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t-Distribution

• The t-distribution is used extensively in
  problems that deal with
• Inference about the population mean
• Comparative samples (two sample means)
• requires that X1, X2 ,, Xn be
  normal.

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F-Distribution

• The F-distribution finds enormous application in
  comparing sample variances.
• Theorem 8.6 Let U and V be two independent
  random variables having chi-squared distribution
  with v1 and v2 degrees of freedom, respectively.
  Then the distribution of the random variable
  is given by the densityThis is
  known as the F-distribution with v1 and v2
  degrees of freedom.

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F-Distribution

• Theorem 8.7 Writing f?(v1, v2) for f? with v1
  and v2 degrees of freedom, we obtain
• E.g., f-value with 6 and 10 degrees of freedom,
  leaving an area of 0.95 to the right,

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F-Distribution
10
4.06
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F-Distribution with Two Sample Variances

• Suppose that random samples of size n1 and n2 are
  selected from two normal populations with
  variances ?12 and ?22 Let
  having chi-squared distribution
  with v1 n1 - 1 and v2 n2 1 degrees of
  freedom. Using Theorem 8.6, we obtain the
  following result
• Theorem 8.8 If S12 and S22 are the variances of
  independent random samples of size n1 and n2
  taken from normal populations with variances ?12
  and ?22, respectively, thenhas an
  F-distribution with v1 n1 - 1 and v2 n2 1
  degrees of freedom.

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F-Distribution

• If we wish to determine if the population means
  are equivalent
• The normal distribution applies nicely for
  two-sample situation.
• However, three-sample?
• F-distribution is called the variance ratio
  distribution.
• Whether sample averages could have occurred by
  chance depends on the variability within samples,
  as quantified by SA2 and SB2, and SC2.
• The notion of the important components of
  variability is best seen through some simple
  graphics

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Analysis of Variance with F-Distribution

• Two key sources of variability
• Variability within samples
• Variability between samples
• If the variability within samples is considerably
  larger than the variability between samples,
  there will be considerable overlap in the sample
  data and a signal that the data could all have
  come from a common distribution.

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Exercise

• 1, 14, 17, 29, 41, 51, 59