Sampling%20and%20Statistical%20Analysis%20for%20Decision%20Making

1
Sampling and Statistical Analysis for Decision
Making
A. A. Elimam College of Business San Francisco
State University
2
Chapter Topics

• Sampling Design and Methods
• Estimation
• Confidence Interval Estimation for the Mean
• (s Known)
• Confidence Interval Estimation for the Mean
• (s Unknown)
• Confidence Interval Estimation for the
• Proportion

3
Chapter Topics

• The Situation of Finite Populations
• Students t distribution
• Sample Size Estimation
• Hypothesis Testing
• Significance Levels
• ANOVA

4
Statistical Sampling

• Sampling Valuable tool
• Population
• Too large to deal with effectively or
  practically
• Impossible or too expensive to obtain all data
• Collect sample data to draw conclusions about
  unknown population

5
Sample design

• Representative Samples of the population
• Sampling Plan Approach to obtain samples
• Sampling Plan States
• Objectives
• Target population
• Population frame
• Method of sampling
• Data collection procedure
• Statistical analysis tools

6
Objectives

• Estimate population parameters such as a mean,
  proportion or standard deviation
• Identify if significant difference exists
  between two populations
• Population Frame
• List of all members of the target population

7
Sampling Methods

• Subjective Sampling
• Judgment select the sample (best customers)
• Convenience ease of sampling
• Probabilistic Sampling
• Simple Random Sampling
• Replacement
• Without Replacement

8
Sampling Methods

• Systematic Sampling
• Selects items periodically from population.
• First item randomly selected - may produce bias
• Example pick one sample every 7 days
• Stratified Sampling
• Populations divided into natural strata
• Allocates proper proportion of samples to each
  stratum
• Each stratum weighed by its size cost or
  significance of certain strata might suggest
  different allocation
• Example sampling of political districts - wards

9
Sampling Methods

• Cluster Sampling
• Populations divided into clusters then random
  sample each
• Items within each cluster become members of the
  sample
• Example segment customers for each geographical
  location
• Sampling Using Excel
• Population listed in spreadsheet
• Periodic
• Random

10
Sampling Methods Selection

• Systematic Sampling
• Population is large considerable effort to
  randomly select
• Stratified Sampling
• Items in each stratum homogeneous - Low
  variances
• Relatively smaller sample size than simple
  random sampling
• Cluster Sampling
• Items in each cluster are heterogeneous
• Clusters are representative of the entire
  Population
• Requires larger sample

11
Sampling Errors

• Sample does not represent target population
• (e. g. selecting inappropriate sampling method)
• Inherent errorsamples only subset of population
• Depends on size of Sample relative to population
• Accuracy of estimates
• Trade-off cost/time versus accuracy

12
Sampling From Finite Populations

• Finite without replacement (R)
• Statistical theory assumes samples selected
  with R
• When n lt .05 N difference is insignificant
• Otherwise need a correction factor
• Standard error of the mean

13
Statistical Analysis of Sample Data

• Estimation of population parameters (PP)
• Development of confidence intervals for PP
• Probability that the interval correctly
  estimates true population parameter
• Means to compare alternative decisions/process
• (comparing transmission production processes)
• Hypothesis testing validate differences among PP

14
Estimation Process
Population
Random Sample
I am 95 confident that m is between 40 60.
Mean X 50
Mean, m, is unknown
Sample
15
Population Parameters Estimated
Point Estimate
Population Parameter
_
Mean
m
X
Proportion
p
p
s
2
2
Variance
s
s
Std. Dev.
s
s
16
Confidence Interval Estimation

• Provides Range of Values
• Based on Observations from Sample
• Gives Information about Closeness to Unknown
  Population Parameter
• Stated in terms of Probability
• Never 100 Sure

17
Elements of Confidence Interval Estimation
A Probability That the Population Parameter Falls
Somewhere Within the Interval.
Sample Statistic
Confidence Interval
Confidence Limit (Lower)
Confidence Limit (Upper)
18
Example of Confidence Interval Estimation
Example 90 CI for the mean is 10 2. Point
Estimate 10 Margin of Error 2 CI 8,12
Level of Confidence 1 - ? 0.9 Probability
that true PP is not in this CI 0.1
19
Confidence Limits for Population Mean
Parameter Statistic Its Error
Error
Error
Error
Error
20
Confidence Intervals
_
X
90 Samples
95 Samples
99 Samples
21
Level of Confidence

• Probability that the unknown
• population parameter falls within the
• interval
• Denoted (1 - a) level of confidence e.g.
  90, 95, 99
• a Is Probability That the Parameter Is Not
  Within the Interval

22
Intervals Level of Confidence
Sampling Distribution of the Mean
s
_
x
a
/2
a
/2
1 -
a
_
X
Intervals Extend from
(1 - a) of Intervals Contain m. a Do Not.
to
Confidence Intervals
23
Factors Affecting Interval Width

• Data Variation
• measured by s
• Sample Size
• Level of Confidence (1 - a)

Intervals Extend from
X - Zs to X Z s
x
x
24
Confidence Interval Estimates
Confidence
Intervals

Mean
Proportion
Finite
s
s
Unknown
Known
Population
25
Confidence Intervals (s Known)

• Assumptions
• Population Standard Deviation is Known
• Population is Normally Distributed
• If Not Normal, use large samples
• Confidence Interval Estimate

26
Confidence Interval Estimates
Confidence
Intervals

Mean
Proportion
Finite
s
s
Unknown
Known
Population
27
Confidence Intervals (s Unknown)

• Assumptions
• Population Standard Deviation is Unknown
• Population Must Be Normally Distributed
• Use Students t Distribution
• Confidence Interval Estimate

28
Students t Distribution

• Shape similar to Normal Distribution
• Different t distributions based on df
• Has a larger variance than Normal
• Larger Sample size t approaches Normal
• At n 120 - virtually the same
• For any sample size true distribution of Sample
  mean is the students t
• For unknown ? and when in doubt use t

29
Students t Distribution
Standard Normal
t (df 13)
Bell-Shaped Symmetric Fatter Tails
t (df 5)
Z
t
0
30
Degrees of Freedom (df)

• Number of Observations that Are Free to Vary
  After Sample Mean Has Been Calculated
• Example
• Mean of 3 Numbers Is 2X1 1 (or Any
  Number)X2 2 (or Any Number)X3 3
  (Cannot Vary)Mean 2

degrees of freedom n -1 3 -1 2
31
Students t Table
Assume n 3 df n - 1 2 a .10
a/2 .05
Upper Tail Area
df
.25
.10
.05
1
1.000
3.078
6.314
2
0.817
1.886
2.920
.05
3
0.765
1.638
2.353
0
t
2.920
t Values
32
Example Interval Estimation s Unknown

• A random sample of n 25 has 50 and
• s 8. Set up a 95 confidence interval estimate
  for m.

.
.
m
46
69
53
30
33
Example Tracway Transmission

• Sample of n 30, S 45.4 - Find a 99 CI for,
  m , the mean of each transmission system process.
  Therefore a .01 and a/2 .005

m
266.75
312.45
34
Confidence Interval Estimates
Confidence
Intervals

Mean
Proportion
Finite
s
s
Unknown
Known
Population
35
Estimation for Finite Populations

• Assumptions
• Sample Is Large Relative to Population
• n / N gt .05
• Use Finite Population Correction Factor
• Confidence Interval (Mean, sX Unknown)

m
X
36
Confidence Interval Estimates
Confidence
Intervals

Mean
Proportion
Finite
s
s
Unknown
Known
Population
37
Confidence Interval Estimate Proportion

• Assumptions
• Two Categorical Outcomes
• Population Follows Binomial Distribution
• Normal Approximation Can Be Used
• np ³ 5 n(1 - p) ³ 5
• Confidence Interval Estimate

38
Example Estimating Proportion

• A random sample of 1000 Voters showed 51 voted
  for Candidate A. Set up a 90 confidence interval
  estimate for p.

p


.484
.536
39
Sample Size

• Too Big
• Requires too
• much resources

• Too Small
• Wont do
• the job

40
Example Sample Size for Mean

• What sample size is needed to be 90 confident of
  being correct within 5? A pilot study
  suggested that the standard deviation is 45.

2
2
2
2
Z
1
645
45
s
.

n


_at_
219
2
220
.
2
2
Error
5
Round Up
41
Example Sample Size for Proportion

• What sample size is needed to be within 5 with
  90 confidence? Out of a population of 1,000, we
  randomly selected 100 of which 30 were defective.

228
_at_
Round Up
42
Hypothesis Testing

• Draw inferences about two contrasting
  propositions (hypothesis)
• Determine whether two means are equal
• Formulate the hypothesis to test
• Select a level of significance
• Determine a decision rule as a base to conclusion
• Collect data and calculate a test statistic
• Apply the decision rule to draw conclusion

43
Hypothesis Formulation

• Null hypothesis H0 representing status quo
• Alternative hypothesis H1
• Assumes that H0 is true
• Sample evidence is obtained to determine whether
  H1 is more likely to be true

44
Significance Level
False
True
Test
Accept
Type II Error
Reject
Type I Error
Probability of making Type I error ? level of
significance Confidence Coefficient 1-
? Probability of making Type II error ? level
of significance Power of the test 1- ?
45
Decision Rules

• Sampling Distribution Normal or t distribution
• Rejection Region
• Non Rejection Region
• Two-tailed test , ?/2
• One-tailed test , ?
• P-Values

46
Hypothesis Testing Cases

• Two-Sample Means
• F-Test for Variances
• Proportions
• ANOVA Differences of several means
• Chi-square for independence

47
Chapter Summary

• Sampling Design and Methods
• Estimation
• Confidence Interval Estimation for Mean
• (s Known)
• Confidence Interval Estimation for Mean
• (s Unknown)
• Confidence Interval Estimation for Proportion

48
Chapter Summary

• Finite Populations
• Students t distribution
• Sample Size Estimation
• Hypothesis Testing
• Significance Levels Type I/II errors
• ANOVA