Sampling Distributions

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Sampling Distributions
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Overview

• To frame our discussion, consider

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Outline
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Population
Population
Parameter the measurement of a characteristic
of an entire population
Population the complete set of objects that you
want to study
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Sample
Sample Subset of subjects that are the focus
of ones study

• Statistic Number calculated on
• sample data quantifying a
• characteristic of the sample

Population
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Sampling (1)

• Random Sampling
• Subjects are chosen from the population at
  random.
• Stratified Random Sampling
• The population is divided into groups (strata)
  then random sampling is applied to the groups.

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Sampling (2)

• Convenience Sampling
• The most convenient persons are chosen.
• Quota Sampling
• Subjects from various portions of the population
  are chosen.

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Randomization

• Statistical methods require observations from
  independent random variables. Randomization is
  used to meet this requirement.
• Randomization applies to the allocation of
  objects, subjects, and the order of treatments.

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Why Randomization?

• By random assignment you try to keep the results
  from being biased by sources of variation over
  which you have no control.

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Sample Size

• The larger the variability in the population the
  larger the sample needed.
• The size of the sample impacts our ability to
  generalize since larger samples reduce error.

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Context

• Take a random sample of n observations from a
  population P. Compute the mean for the sample.
  How well does the sample mean estimate the
  population mean?
• Notice we generate statistics as estimates of
  parameters.

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Sampling Distribution - Mean(s known)

• If a random sample of size n is taken from a
  population having a mean µ and variance s2 , then
  is a random variable whose distribution has a
  mean of µ and variance

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Normal Population Distribution
Let X1,, Xn be a random sample from a normal
distribution with mean value and standard
deviation Then for any n, is
normally distributed.
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Central Limit Theorem

• If is the mean of a sample of size n taken
  from a population have mean µ and variance s2
  then
• is a random variable whose distribution function
  approaches standard normal.

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Notes

• Central Limit Theorem holds regardless of the
  population distribution.
• The sampling distribution is approximately normal
  when ngt30.
• If the population from which you are sampling has
  a normal distribution, then the sampling
  distribution is a normal distribution.

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st/index.html
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Problem 1

• Company records indicate that the time spent
  preparing for a code inspection is normally
  distributed with a mean of 55 minutes and a
  standard deviation of 15 minutes.
• What is the probability an employee spends more
  than 75 minutes preparing for a review?

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Solution - Problem 1
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Problem 2

• Company records indicate that the time spent
  preparing for a code inspection is normally
  distributed with a mean of 55 minutes and a
  standard deviation of 15 minutes.
• What is the probability that the average time for
  the review team of 6 people exceeds75 minutes?

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Solution - Problem 2
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Problem 3

• A group of women project leaders for CompuCorp is
  considering filing a sex-discrimination suit
  against the corporation. A recent report stated
  that the average salary for project leads at the
  company is 128,000 with a standard deviation of
  8,500. A random sample of 65 women taken from
  the 350 female project leads at the company had
  an average income of 125,000. If the population
  of female project managers is assumed to have
  same mean and standard deviation as project
  leads, what is the probability of observing this
  sample average?

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Solution - Problem 3
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Sampling Distribution - Mean(s unknown)

• If a is the mean of a random sample of size n
  is taken from a normal population have a mean of
  µ and variance s2 , and s2 is the variance of the
  sample, then
• is a random variable having the t distribution
  with the parameter nn-1.

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Notes

• The parameter n is referred to as the degrees of
  freedom.
• t distribution is similar to normal.
• Notice the requirement of sampling from normal
  population.
• N(0,1) is good approximation for t distribution
  when n30.

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Problem 4

• The CEO submitted a white paper indicating a few
  changes in the software development process are
  in order. His statements include a claim that
  the average effort devoted to unit testing on
  projects is 7.8 person-months. You collect
  random sample of 75 effort-logs from projects
  and determine the average effort for unit testing
  was 7.5 person-months with a standard deviation
  of 1.75 person-months. Does the data you
  collected support or refute the CEO?

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Hypothesis Testing

• For many applications, we are concerned that the
  standard deviation or variance of the population
  exceeds some specified value.

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Sampling Distribution - Variance

• If S2 is the variance of a random sample of size
  n taken from a normal population having the
  variance s2, then
• is a random variable having the chi-square
  distribution with the parameter nn-1.

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Problem 5

• You produced an algorithm for determining the
  area of polygonal regions. Your data regarding
  the area of a sample of 50 cm2 regions is
• 51.2, 47.5, 50.8, 51.5, 51.3, 49.5, 51.1, 50.7,
  46.7, 49.2, 52.1, 48.3, 51.6, 49.2, 51.5
• Assuming the area of the polygonal regions if
  normally distributed, determine s2 with 95
  confidence.

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Problem 6

• A production manager must maintain the standard
  deviation of the diameter of hard disk media to
  less than 2mm. A random sample of 26 disks
  reveal a standard deviation of 1.85mm. If the
  disks diameters are normally distributed, do the
  data indicate that the standard deviation is less
  than 2mm? Use a.05.

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Independent Samples - Variance

• The F distribution allows us to look at the ratio
  of variances from two independent random samples.
  Using the F statistic we can determine whether
  the two samples come from populations that have
  similar variances.

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• Suppose we have two normal populations. Taking a
  random sample from each population, we want to
  compare the variances of the sample. We do so
  using the F-statistic
• This random variable has the F distribution with

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• Notice the F-distribution is not symmetric.

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Problem 7

• A company is orders components from two different
  suppliers. A sample of 10 of the components from
  Supplier 1 and 16 components from Supplier 2 are
  chosen and tested. From testing we determine the
  standard deviation for Supplier 1 components
  s14.31 while the standard deviation for Supplier
  2 components is s25.01. Are the two variances
  sufficiently different at the .01 level?

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Left and Right Tails