ENGR 610 Applied Statistics Fall 2007 - Week 4

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# ENGR 610 Applied Statistics Fall 2007 - Week 4 - PowerPoint PPT Presentation

ENGR 610 Applied Statistics Fall 2007 - Week 4. Marshall University CITE Jack Smith. Overview for Today. Review of Ch 5 Homework problems for Ch 5 Estimation Procedures (Ch 8) Homework assignment About the 1 st exam. Chapter 5 Review. Continuous probability distributions Uniform

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### ENGR 610Applied StatisticsFall 2007 - Week 4

Marshall University

CITE

Jack Smith

Overview for Today
• Review of Ch 5
• Homework problems for Ch 5
• Estimation Procedures (Ch 8)
• Homework assignment
Chapter 5 Review
• Continuous probability distributions
• Uniform
• Normal
• Standard Normal Distribution (Z scores)
• Approximation to Binomial, Poisson distributions
• Normal probability plot
• LogNormal
• Exponential
• Sampling of the mean, proportion
• Central Limit Theorem
Continuous Probability Distributions

(Mean, expected value)

(Variance)

Normal Distribution

Gaussian with

peak at µ and

inflection points at +/- σ

FWHM = 2(2ln(2))1/2 σ

Standard Normal Distribution

68,95,99.7%

where

Is the standard normal score (“Z-score”)

With and effective mean of zero and a standard deviation of 1

Normal Approximation to Binomial Distribution
• For binomial distributionand so
• Variance, 2, should be at least 10
Normal Approximation to Poisson Distribution
• For Poisson distributionand so
• Variance, , should be at least 5
Normal Probability Plot
• Use normal probability graph paperto plot ordered cumulative percentages, Pi = (i - 0.5)/n * 100%, as Z-scores- or -
• Use Quantile-Quantile plot (see directions in text)- or -
• Use software (PHStat)!
Exponential Distribution

Poisson, with continuous rate of change, 

Only memoryless random distribution

Sampling Distribution of the Mean, Proportion
• Central Limit Theorem

Continuous data

(proportion)

Attribute data

Homework Problems (Ch 5)
• 5.66
• 5.67
• 5.68
• 5.69
Estimation Procedures
• Estimating population mean ()
• from sample mean (X-bar) and population variance (2) using Standard Normal Z distribution
• from sample mean (X-bar) and sample variance (s2) using Student’s t distribution
• Estimating population variance (2)
• from sample variance (s2) using 2 distribution
• Estimating population proportion ()
• from sample proportion (p) and binomial variance (npq)using Standard Normal Z distribution
Estimation Procedures, cont’d
• Predicting future individual values (X)
• from sample mean (X-bar) and sample variance (s2)using Student’s t distribution
• Tolerance Intervals
• One- and two-sided
• Using k-statistics
Parameter Estimation
• Statistical inference
• Conclusions about population parameters from sample statistics (mean, variation, proportion,…)
• Makes use of CLT, various sampling distributions, and degrees of freedom
• Interval estimate
• With specified level of confidence that population parameter is contained within
• When population parameters are known and distribution is Normal,
Point Estimator Properties
• Unbiased
• Average (expectation) value over all possible samples (of size n) equals population parameter
• Efficient
• Arithmetic mean most stable and precise measure of central tendency
• Consistent
• Improves with sample size n
Estimating population mean ()
• From sample mean (X-bar) and knownpopulationvariance (2)
• Using Standard Normal distribution (and CLT!)
• Where Z, the critical value, corresponds to area of (1-)/2 for a confidence level of (1-)100%
• For example, from Table A.2, Z = 1.96 corresponds to area = 0.95/2 = 0.475 for 95% confidence interval, where  = 0.05 is the sum of the upper and lower tail portions
Estimating population mean ()
• From sample mean (X-bar) and samplevariance (s2)
• Using Student’s t distribution withn-1degrees of freedom
• Where tn-1, the critical value, corresponds to area of (1-)/2 for a confidence level of (1-)100%
• For example, from Table A.4, t = 2.0639 corresponds to area = 0.95/2 = 0.475 for 95% confidence interval, where /2 = 0.025 is the area of the upper tail portion, and 24 is the number of degrees of freedom for a sample size of 25
Estimating population variance (2)
• From sample variance (s2)
• Using 2 distribution withn-1degrees of freedom
• WhereU and L, the upperandlower critical values, corresponds to areas of /2 and 1-/2 for a confidence level of (1-)100%
• For example, from Table A.6, U = 39.364 and L = 12.401 correspond to the areas of 0.975 and 0.025 for 95% confidence interval and 24 degrees of freedom
Predicting future individual values (X)
• From sample mean (X-bar) and sample variation (s2)
• Using Student’s t distribution
• Prediction interval
• Analogous to
Tolerance intervals
• An interval that includes at least a certain proportion of measurements with a stated confidence based on sample mean (X-bar) and sample variance (s2)
• Using k-statistics (Tables A.5a, A.5.b)
• WhereK1 and K2 corresponds to a confidence level of (1-)100% for p100% of measurements and a sample size of n

Two-sided

Lower Bound

Upper Bound

Estimating population proportion ()
• From binomial mean (np) and variation (npq) from sample (size n, and proportion p)
• Using Standard Normal Z distribution as approximation to binomial distribution
• Analogous towhere p = X/n
Homework
• Ch 8
• Appendix 8.1
• Problems: 8.43-44
Exam #1 (Ch 1-5,8)
• Take home
• Given out (electronically) after in-class review
• Open book, notes
• No collaboration - honor system
• Use Excel w/ PHStat where appropriate, but
• Explain, explain, explain!
• Due by beginning of class, Sept 27