Chapter 1: Sample and Population

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Chapter 1 Sample and Population
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Agenda

• Data summary and display
• Random Sampling
• Estimation
• Sampling distributions

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Data Summary and Display
Definition
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Example 1
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Data Summary and Display
The sample mean as a balance point for a system
of weights.
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Minitab Practice for Examples 1- 2

• Objective to Obtain Descriptive Statistics
• Data file File ? open worksheet ?.xls ?
  EX06_29
• From the menus choose
•     Stat
• Basic Statistics
•          Store Descriptive Statistics     
           Select
• Select

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Data Summary and Display
Population Mean For a finite population with N
measurements, the mean is
The is a reasonable estimate of the
.
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Data Summary and Display
Definition
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Data Summary and Display
How Does the Sample Variance Measure Variability?
How the sample variance measures variability
through the deviations .
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Example 2
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Data Summary and Display
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Data Summary and Display
Computation of s2
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Data Summary and Display
Population Variance When the population is finite
and consists of N values, we may define the
as
The is a reasonable
estimate of the .
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Data Summary and Display
Definition
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Random Sampling
Definitions
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Random Sampling
Relationship between a population and a sample.
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Random Sampling
Definitions
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Frequency Distributions and Histograms

• To construct a frequency distribution, we must
  divide the range of the data into ,
  which are usually called class ,
  or
• Constructing a Histogram (Equal Bin Widths)

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Frequency Distributions and Histograms
Histogram of compressive strength data.
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Frequency Distributions and Histograms
A histogram of the compressive strength data with
17 bins.
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Frequency Distributions and Histograms
A histogram of the compressive strength data with
nine bins.
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Frequency Distributions and Histograms
A cumulative distribution plot of the compressive
strength data.
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Frequency Distributions and Histograms
Histograms for symmetric and skewed
distributions.
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Box Plots

• The is a graphical display that
  simultaneously describes several important
  features of a data set, such as center, spread,
  departure from symmetry, and identification of
  observations that lie unusually far from the bulk
  of the data.
• Whisker
• Outlier
• Extreme outlier

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Box Plots
Description of a box plot.
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Time Sequence Plots

• A or is a data set in which the
  observations are recorded in the order in which
  they occur.
• A is a graph in which the vertical axis
  denotes the observed value of the variable (say
  x) and the horizontal axis denotes the time
  (which could be minutes, days, years, etc.).
• When measurements are plotted as a time series,
  we
• often see
• trends,
• cycles, or
• other broad features of the data

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Time Sequence Plots
Company sales by year (a) and by quarter (b).
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Probability Plots

• is a graphical method
  for determining whether sample data conform to a
  hypothesized distribution based on a subjective
  visual examination of the data.
• Probability plotting typically uses special
  graph paper, known as ,
  that has been designed for the hypothesized
  distribution. Probability paper is widely
  available for the normal, lognormal, Weibull, and
  various chi-square and gamma distributions.

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Probability Plots
Normal probability plot
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Probability Plots
Normal probability plot obtained from
standardized normal scores.
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Probability Plots
Normal probability plots indicating a non-normal
distribution. (a) Light-tailed distribution. (b)
Heavy-tailed distribution. (c ) A distribution
with positive (or right) skew.
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Minitab Practice for Data Display

• Data file EX06_29
• Histogram
• Option 1 Graph ? ? Select Graph
  variable ? Select option
• Option2 Stat ? ? descriptive
  statistics ?
• Box plots
• Option 1 Graph ? ? Select variable Y ?
  Options ? check Transpose X and Y
• Option2 Stat ? ? descriptive statistics
  ?
• Time sequence
• Graph ? ? Select variable
• Probability plot
• Graph ? ? Select distribution ? Select
  option ? Deselect include confident intervals in
  plot

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Statistical Inference for Population

• The field of statistical inference consists of
  those methods used to make decisions or to draw
  conclusions about a .
• These methods utilize the information contained
  in a from the population in
  drawing conclusions.
• Statistical inference may be divided into two
  major areas

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Statistical Inference for Population
Definition
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Statistical Inference for Population
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Statistical Inference for Population
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General Concepts of Point Estimation

• Unbiased Estimators
• Definition

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General Concepts of Point Estimation

• Variance of a Point Estimator
• Definition

The sampling distributions of two unbiased
estimators
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General Concepts of Point Estimation
Theorem 1
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General Concepts of Point Estimation

• Standard Error Reporting a Point Estimate
• Definition

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General Concepts of Point Estimation
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Example 3
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Example 3
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General Concepts of Point Estimation

• Mean Square Error of an Estimator
• Definition

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General Concepts of Point Estimation

• Mean Square Error of an Estimator

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General Concepts of Point Estimation
A biased estimator that has smaller variance
than the unbiased estimator
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Methods of Point Estimation
Definition
Definition
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Example 4
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Methods of Point Estimation

• Method of Maximum Likelihood
• Definition

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Example 5
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Example 5
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Methods of Point Estimation
Properties of the Maximum Likelihood Estimator
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Methods of Point Estimation
The Invariance Property
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Example 6
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Methods of Point Estimation

• Complications in Using Maximum Likelihood
  Estimation
• It is not always easy to maximize the
  likelihood function because the equation(s)
  obtained from dL(?)/d? 0 may be
  .
• It may be possible to use
  calculus methods directly to determine the
  .

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Example 7
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Methods of Point Estimation
The likelihood function for the uniform
distribution in Example 7.
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Sampling Distributions
is concerned with
making about a population based on
the information contained in a random sample from
that population. Definition
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Sampling Distributions of Means
Theorem 2 The Central Limit Theorem
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Sampling Distributions of Means
Distributions of average scores from throwing
dice. Adapted with permission from Box, Hunter,
and Hunter (1978).
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Example 8
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Sampling Distributions of Means
Probability for Example 8.
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Sampling Distributions of Means
Definition