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ENGR 610 Applied Statistics Fall 2007 - Week 4PowerPoint Presentation

ENGR 610 Applied Statistics Fall 2007 - Week 4

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Presentation Transcript

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

- 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

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
- WhereU 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 p100% 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

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