ENGR 610Applied 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 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

Continuous Probability Distributions

P(aX b) f (x)dxa

b

P(X b) f (x)dx

b

E(X) xf (x)dx

2 (x )2 f (x)dx

(Mean, expected value)

(Variance)

Uniform Distribution

2

)(

)(

1 abdx

abx

b

a

f (x)

1

b aax b

0 elsewhere

12

)(

)(

1

2

)( 222 ab

dxab

abx

b

a

Normal Distribution

2]/))[(2/1(

2

1)( xxx

x

exf

Gaussian withpeak at µ andinflection points at +/- σ

FWHM = 2(2ln(2))1/2 σ

Standard Normal Distribution

f (x) 1

2e (1/ 2)Z 2

where

Z X x x

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

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

68, 95, 99.7%

Normal Approximation to Binomial Distribution

For binomial distribution

and so

Variance, 2, should be at least 10

Z X x x

X npnp(1 p)

x np

x np(1 p)

Normal Approximation to Poisson Distribution

For Poisson distribution

and so

Variance, , should be at least 5

Z X x x

X

x

x

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)!

Lognormal Distribution

f (x) 1

2 ln(x )

e (1/ 2)[(ln(X ) ln(x ) ) / ln( x ) ]2

(X ) e ln(X ) ln(X )

2 / 2

X e2 ln(X ) ln(X )2

(e ln(X )2

1)

Exponential Distribution

f (x) e x

1/

P(x X) 1 e X

Only memoryless random distribution

Poisson, with continuous rate of change,

Sampling Distribution of the Mean, Proportion

Central Limit Theorem

xx

x x / n

p (1 )

n

Continuous data

Attribute data

p (proportion)

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,

E(X) ZX X E(X) ZX

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 known population variance (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

X Zn

X Zn

Estimating population mean () From sample mean (X-bar) and sample variance (s2)

Using Student’s t distribution with n-1 degrees 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

X tn 1

s

n X tn 1

s

n

Estimating population variance (2) From sample variance (s2)

Using 2 distribution withn-1 degrees of freedom

Where U and L, the upper and lower 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

(n 1)s2

U2

2 (n 1)s2

L2

Predicting future individual values (X)

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

Using Student’s t distribution Prediction interval Analogous to

X tn 1s 11

nX X tn 1s 1

1

n

Z X Z

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) Where K1 and K2 corresponds to a confidence level of

(1-)100% for p100% of measurements and a sample size of n

X K2s

X K1s

X K1s

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

p Zp(1 p)n

p Zp(1 p)n

Z X Z

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