Sampling and Statistical Analysis for Decision Making

A. A. ElimamCollege of Business

San Francisco State University

Chapter Topics• Sampling: Design and Methods• Estimation:

• Confidence Interval Estimation for the Mean(Known)

•Confidence Interval Estimation for the Mean (Unknown)

•Confidence Interval Estimation for the Proportion

Chapter Topics

• The Situation of Finite Populations• Student’s t distribution • Sample Size Estimation• Hypothesis Testing• Significance Levels• ANOVA

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

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

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

Sampling Methods

• Subjective Sampling: • Judgment: select the sample (best customers)

• Convenience: ease of sampling • Probabilistic Sampling:

• Simple Random Sampling• Replacement• Without Replacement

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

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

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

Sampling Errors

• Sample does not represent target population (e. g. selecting inappropriate sampling method)

• Inherent error:samples only subset of population• Depends on size of Sample relative to population• Accuracy of estimates• Trade-off: cost/time versus accuracy

Sampling From Finite Populations

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

1x

N nNn

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

Mean, , is unknown

Population Random SampleI am 95%

confident that is between 40 &

60.

Mean X = 50

Estimation Process

Sample

Mean

Proportion p ps

Variance s2

Population Parameters Estimated

2

X_

Point EstimatePopulation Parameter

Std. Dev. s

• 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

Confidence Interval Estimation

Confidence Interval Sample Statistic

Confidence Limit (Lower)

Confidence Limit (Upper)

A Probability That the Population Parameter Falls Somewhere Within the Interval.

Elements 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

Example of Confidence Interval Estimation

Parameter = Statistic ± Its Error

Confidence Limits for Population Mean

X Error

= Error = X

XX

XZ

xZ

XZX

Error

Error

X

90% Samples

95% Samples

x_

Confidence Intervals

xx .. 64516451

xx 96.196.1

xx .. 582582 99% Samples

nZXZX X

X_

• Probability that the unknown population parameter falls within the

interval

• Denoted (1 - ) % = level of confidence e.g. 90%, 95%, 99%

Is Probability That the Parameter Is Not Within the Interval

Level of Confidence

Confidence Intervals

Intervals Extend from (1 - ) % of

Intervals Contain . % Do Not.

1 - /2/2

X_

x_

Intervals & Level of Confidence

Sampling Distribution of

the Mean

toXZX

XZX

X

• Data Variation measured by

• Sample Size

• Level of Confidence (1 - )

Intervals Extend from

Factors Affecting Interval Width

X - Z to X + Z xx

n/XX

Mean

Unknown

ConfidenceIntervals

Proportion

FinitePopulation Known

Confidence Interval Estimates

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

• Confidence Interval Estimate

Confidence Intervals (Known)

nZX /

2

nZX /

2

Mean

Unknown

ConfidenceIntervals

Proportion

FinitePopulation Known

Confidence Interval Estimates

• Assumptions Population Standard Deviation is Unknown Population Must Be Normally Distributed

• Use Student’s t Distribution• Confidence Interval Estimate

Confidence Intervals (Unknown)

nStX n,/ 12

n

StX n,/ 12

• 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 student’s t• For unknown and when in doubt use t

Student’s t Distribution

Standard Normal

Zt0

t (df = 5)

t (df = 13)Bell-ShapedSymmetric

‘Fatter’ Tails

Student’s t Distribution

• Number of Observations that Are Free to Vary After Sample Mean Has Been Calculated

• Example Mean of 3 Numbers Is 2

X1 = 1 (or Any Number)X2 = 2 (or Any Number)X3 = 3 (Cannot Vary)Mean = 2

degrees of freedom = n -1 = 3 -1= 2

Degrees of Freedom (df)

Upper Tail Area

df .25 .10 .05

1 1.000 3.078 6.314

2 0.817 1.886 2.920

3 0.765 1.638 2.353

t0

Assume: n = 3 df = n - 1 = 2

= .10 /2 =.05

2.920t Values

.05

Student’s t Table

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

. .46 69 53 30

X

Example: Interval Estimation Unknown

nStX n,/ 12

nStX n,/ 12

2580639250 . 25

80639250 .

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

266.75 312.45

Example: Tracway Transmission

nStX n,/ 12 n

StX n,/ 12

45.4289.6 2.756430

45.4289.6 2.756430

/ 2, 1 .005,29 2.7564nt t

Mean

Unknown

ConfidenceIntervals

Proportion

FinitePopulation Known

Confidence Interval Estimates

• Assumptions Sample Is Large Relative to Population

n / N > .05• Use Finite Population Correction Factor• Confidence Interval (Mean, X Unknown)

X

Estimation for Finite Populations

nStX n,/ 12 n

StX n,/ 121

N

nN1

NnN

Mean

Unknown

ConfidenceIntervals

Proportion

FinitePopulation Known

Confidence Interval Estimates

• Assumptions Two Categorical Outcomes Population Follows Binomial Distribution Normal Approximation Can Be Used n·p 5 & n·(1 - p) 5

• Confidence Interval Estimate

Confidence Interval Estimate Proportion

n)p(pZp ss

/s

1

2 pn

)p(pZp ss/s

12

A random sample of 1000 Voters showed 51% voted for Candidate A. Set up a 90%

confidence interval estimate for p.

p .484 .536

Example: Estimating Proportion

n)p(pZp ss

/s

1

2 p

n)p(pZp ss

/s

1

2

.51(1 .51).51 1.6451000

p .51(1 .51).51 1.645

1000

Sample Size

Too Big:•Requires toomuch resources

Too Small:•Won’t do the job

What sample size is needed to be 90% confident of being correct within ± 5? A pilot study suggested that the standard

deviation is 45.

nZError

2 2

2

2 2

2

1645 45

5219 2 220

..

Example: Sample Size for Mean

Round Up

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.

Example: Sample Size for Proportion

Round Up

322705

7030645112

2

2

2

..

))(.(..error

)p(pZn

228

Hypothesis Testing

• Draw inferences about two contrasting propositions (hypothesis)

• Determine whether two means are equal:1. Formulate the hypothesis to test2. Select a level of significance3. Determine a decision rule as a base to

conclusion4. Collect data and calculate a test statistic5. Apply the decision rule to draw conclusion

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

Test

Accept Reject

Significance Level

FalseTrue

Type II ErrorType 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-

Decision Rules

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

Hypothesis Testing: Cases

• Two-Sample Means

• F-Test for Variances

• Proportions

• ANOVA: Differences of several means

• Chi-square for independence

Chapter Summary• Sampling: Design and Methods• Estimation:

• Confidence Interval Estimation for Mean(Known)

• Confidence Interval Estimation for Mean (Unknown)

• Confidence Interval Estimation for Proportion

Chapter Summary• Finite Populations• Student’s t distribution • Sample Size Estimation• Hypothesis Testing• Significance Levels: Type I/II errors • ANOVA