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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 610 applied statistics fall 2007 week 4

ENGR 610Applied StatisticsFall 2007 - Week 4

Marshall University

CITE

Jack Smith

overview for today
Overview for Today
  • Review of Ch 5
  • Homework problems for Ch 5
  • Estimation Procedures (Ch 8)
  • Homework assignment
  • About the 1st exam
chapter 5 review
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
Continuous Probability Distributions

(Mean, expected value)

(Variance)

normal distribution
Normal Distribution

Gaussian with

peak at µ and

inflection points at +/- σ

FWHM = 2(2ln(2))1/2 σ

standard normal distribution
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
Normal Approximation to Binomial Distribution
  • For binomial distributionand so
  • Variance, 2, should be at least 10
normal approximation to poisson distribution
Normal Approximation to Poisson Distribution
  • For Poisson distributionand so
  • Variance, , should be at least 5
normal probability plot
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
Exponential Distribution

Poisson, with continuous rate of change, 

Only memoryless random distribution

sampling distribution of the mean proportion
Sampling Distribution of the Mean, Proportion
  • Central Limit Theorem

Continuous data

(proportion)

Attribute data

homework problems ch 5
Homework Problems (Ch 5)
  • 5.66
  • 5.67
  • 5.68
  • 5.69
estimation procedures
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
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
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
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
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 mean1
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
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
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
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
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
Homework
  • Ch 8
    • Appendix 8.1
    • Problems: 8.43-44
exam 1 ch 1 5 8
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
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