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Sampling Distributions

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Title: Sampling Distributions


1
Sampling Distributions
2
Overview
  • To frame our discussion, consider

3
Outline
4
Population
Population
Parameter the measurement of a characteristic
of an entire population
Population the complete set of objects that you
want to study
5
Sample
Sample Subset of objects that are the focus of
ones interest
  • Statistic Number calculated on
  • sample data quantifying a
  • characteristic of the sample

Population
6
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.

7
Sampling (2)
  • Convenience Sampling
  • The most convenient persons are chosen.
  • Quota Sampling
  • Subjects from various portions of the population
    are chosen.

8
Randomization
  • Statistical methods require observations from
    independent random variables. Randomization is
    used in an attempt to meet this requirement.
  • Randomization applies to the allocation of
    objects, subjects, and the order of treatments.

9
Why Randomization?
  • By random assignment you try to keep the results
    from being biased by sources of variation over
    which you have no control.

10
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.

11
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.

Demonstration
12
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

13
Using the Sample Mean
Let X1,, Xn be a random sample from a
distribution with mean value and standard
deviation Then
In addition, with To X1 Xn,
14
Normal Population Distribution
Let X1,, Xn be a random sample from a normal
distribution with mean value and standard
deviation Then for any ngt0, is
normally distributed as is To ,
15
The Central Limit Theorem
Let X1,, Xn be a random sample from a
distribution with mean value and variance
Then if n sufficiently large, has
approximately a normal distribution with
and To also has
approximately a normal distribution with
The larger the value of
n, the better the approximation.
Demonstration
16
Rule of Thumb
If n gt 30, the Central Limit Theorem can be used.
17
Central Limit Theorem (2)
  • 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.

18
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.

http//www.ruf.rice.edu/lane/stat_sim/sampling_di
st/index.html
19
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?

20
Solution - Problem 1
21
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?

22
Solution - Problem 2
23
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?

24
Solution - Problem 3
25
Sampling Distribution - Mean(s unknown)
  • If a is the mean of a random sample of size n
    is taken from a normal population having 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.

26
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.

27
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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