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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

• Review of Ch 5

• Homework problems for Ch 5

• Estimation Procedures (Ch 8)

• Homework assignment

• About the 1st exam

• 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)

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

• 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)!

Poisson, with continuous rate of change, 

Only memoryless random distribution

• Central Limit Theorem

Continuous data

(proportion)

Attribute data

• 5.66

• 5.67

• 5.68

• 5.69

• 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

• 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

• 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,

• 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

• 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

• 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

• 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

• From sample mean (X-bar) and sample variation (s2)

• Using Student’s t distribution

• Prediction interval

• Analogous to

• 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

• 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

• Ch 8

• Appendix 8.1

• Problems: 8.43-44

• 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