Sampling and Statistical Analysis for Decision Making A. A. Elimam College 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

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 PopulationRandom Sample I am 95% confident that  is between 40 & 60. Mean X = 50 Estimation Process Sample

Mean  Proportion pp s Variances 2 Population Parameters Estimated  2 X _ Point Estimate Population 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 Error = Error = Error

90% Samples 95% Samples  x _ Confidence Intervals 99% Samples 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  X _  x _ Intervals & Level of Confidence Sampling Distribution of the Mean to

Data Variation measured by  Sample Size Level of Confidence (1 -  ) Intervals Extend from Factors Affecting Interval Width X - Z  to X + Z  xx

Mean  Unknown Confidence Intervals Proportion Finite Population  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)

Mean  Unknown Confidence Intervals Proportion Finite Population  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)

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 = 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 Z t 0 t (df = 5) t (df = 13) Bell-Shaped Symmetric ‘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 X 1 = 1 (or Any Number) X 2 = 2 (or Any Number) X 3 = 3 (Cannot Vary) Mean = 2 degrees of freedom = n -1 = 3 -1 = 2 Degrees of Freedom (df)

Upper Tail Area df t 0 Assume: n = 3 df = n - 1 = 2  =.10  /2 = t 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 .   Example: Interval Estimation  Unknown

Sample of n = 30, S = Find a 99 % CI for, , the mean of each transmission system process. Therefore  =.01 and  =.005   Example: Tracway Transmission

Mean  Unknown Confidence Intervals Proportion Finite Population  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

Mean  Unknown Confidence Intervals Proportion Finite Population  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

A random sample of 1000 Voters showed 51% voted for Candidate A. Set up a 90% confidence interval estimate for p. p  Example: Estimating Proportion

Sample Size Too Big: Requires too much 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. n Z Error    .. 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 228 

Hypothesis Testing Draw inferences about two contrasting propositions (hypothesis) Determine whether two means are equal: 1.Formulate the hypothesis to test 2.Select a level of significance 3.Determine a decision rule as a base to conclusion 4.Collect data and calculate a test statistic 5.Apply the decision rule to draw conclusion

Hypothesis Formulation Null hypothesis: H 0 representing status quo Alternative hypothesis: H 1 Assumes that H 0 is true Sample evidence is obtained to determine whether H 1 is more likely to be true

Test Accept Reject Significance Level False True Type II Error 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- 

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