Engineering Statistics Part 1

1
Engineering Statistics Part 1

• Random sampling
• Point estimation of parameters
• Confidence intervals

2
Random Sampling

• Random variable X
• Occur frequently in engineering applications
• Molecular weight of polymer product
• Film thickness of solar cell
• Random sampling
• Obtain samples from a population
• All outcomes must be equally likely to be sampled
• Replacement necessary for small populations
• Meaningful statistics can be obtained from
  samples
• Sample mean variance

3
Point Estimation of Parameters

• Point estimate
• Value computed from samples that approximates
  unknown parameter value of the population from
  which the samples are randomly selected
• Gaussian distribution
• Binomial distribution

4
Maximum Likelihood Estimation

• Single parameter
• Consider a discrete random variable X with
  probability density function depending on a
  single parameter f(x,q)
• Collect n random samples x1, x2,, xn
• Likelihood function probability that sample of
  size n consists of precisely these values
• Select q to maximize likelihood
• Multiple parameters f(x,q1,q2,,qr)

5
Gaussian Distribution

• Likelihood function
• Logarithm
• Estimates

6
Confidence Intervals

• Definition
• An interval in which the unknown parameter is
  contained with a certain probability g
• g confidence level
• q1, q2 confidence limits depending on g
• t-distribution (confidence interval on mean)
• m degrees of freedom
• Values tabularized (Table A9 in Appendix 5)
• Chi-square distribution (confidence interval on
  variance)
• Values tabularized (Table A10 in Appendix 5)

7
Determination of Confidence Intervals

• Consider a Gaussian distribution with unknown m
  s2
• Confidence interval on the mean
• Choose confidence level g
• Determine value c using the t-distribution with m
  n-1
• Compute the mean variance of the sample x1,
  x2,, xn
• Confidence interval
• Confidence interval on the variance
• Choose confidence level g
• Determine solutions c1 c2 using the chi-squared
  distribution with m n-1
• Compute the variance of the sample x1, x2,, xn
• Confidence interval

8
Confidence Intervals Example

• Measurements of polymer molecular weight (scaled
  by 10-5)
• Confidence interval on mean
• Confidence interval on variance

9
Matlab Confidence Intervals

• gtgt muhat,sigmahat,muci,sigmaci
  normfit(data,alpha)
• data vector or matrix of data
• alpha confidence level 100(1-alpha)
• muhat estimated mean
• sigmahat estimated standard deviation
• muci confidence interval on the mean
• sigmaci confidence interval on the standard
  deviation
• gtgt muhat,sigmahat,muci,sigmaci normfit(1.25
  1.36 1.22 1.19 1.33 1.12 1.27 1.27 1.31
  1.26,0.05)
• muhat 1.2580
• sigmahat 0.0697
• muci 1.2081
• 1.3079
• sigmaci 0.0480
• 0.1273

10
Central Limit Theorem

• Let X1,,Xn be independent random variables,
  each with same mean m variance s2
• Sum of independent variables
• Yn X1Xn has the mean nm and variance ns2
• If X1,,Xn are also normal variables, then Yn
  is a normal random variable
• Central limit theorem
• Consider following random variable Zn
• Zn is asymptotically normal with zero mean
  unity variance in the sense that its distribution
  function Fn(x) satisfies
• Can determine confidence intervals for non-normal
  distributions using previous methods if
  sufficiently large sample sizes are used

11
Matlab Test the Central Limit Theorem

• y tpdf(c,m) ? t-distribution at c for m degrees
  of freedom
• y normpdf(x,mu,sigma) ? normal distribution at
  x for mean mu and standard deviation sigma
• tpdf(-2.52.5,5)-normpdf(-2.52.5,0,1) ? ans
  0.0158 -0.0050 -0.0241 -0.0241 -0.0050
  0.0158
• tpdf(-2.52.5,25)-normpdf(-2.52.5,0,1) ? ans
  0.0042 -0.0007 -0.0050 -0.0050 -0.0007
  0.0042
• tpdf(-2.52.5,100)-normpdf(-2.52.5,0,1) ? ans
  0.0011 -0.0001 -0.0013 -0.0013 -0.0001
  0.0011