1. Selected Statistics Examples from Lectures

2. Matlab: Histograms
>> y = [ ]’;
>> mean(y)  ans =
>> var(y)  ans =
>> std(y)  ans =
>> hist(y,9)  histogram plot with 9 bins
>> n = hist(y,9)  store result in vector n
>> x = [ ]’
>> n = hist(y,x)  create histogram with bin centers specified by vector x

3. Probability Examples
Probability that at least one coin will turn heads up from five tossed coins
Number of outcomes: 25 = 32
Probability of each outcome: 1/32
Probability of no heads: P(AC) = 1/32
Probability at least one head: P(A) = 1-P(AC) = 31/32
Probability of getting an odd number or a number less than 4 from a single dice toss
Probability of odd number: P(A) = 3/6
Probability of number less than 4: P(B) = 3/6
Probability of both:
Probability of either:
complement set

4. Permutation Examples Process for manufacturing polymer thin films
Compute the probability that the first 6 films will be too thin and the next 4 films will be too thick if the thickness is a random variable with the mean equal to the desired thickness
n1 = 6, n2 = 4
Probability
Encryption cipher
Letters arranged in five-letter words: n = 26, k = 5
Total number of different words: nk = 265 = 11,881,376
Total number of different words containing each letter no more than once:

5. Combination Examples Effect of repetitions
Three letters a, b, c taken two at a time (n = 3, k = 2)
Combinations without repetition
Combinations with repetitions
500 thin films taken 5 at a time
Combinations without repetitions

6. Matlab: Permutations and Combinations
>> perms([2 4 6])  all possible permutations of 2, 4, 6
>> randperm(6)  returns one possible permutation of 1-6
>> nchoosek(5,4)  number of combinations of 5 things taken 4 at a time without repetitions
ans = 5
>> nchoosek(2:2:10,4)  all possible combinations of 2, 4, 6, 8, 10 taken 4 at a time without repetitions

7. Continuous Distribution Example
Probability density function
Cumulative distribution function
Probability of events

8. Matlab: Normal Distribution
Normal distribution: normpdf(x,mu,sigma)
normpdf(8,10,2)  ans =
normpdf(9,10,2)  ans =
normpdf(8,10,4)  ans =
Normal cumulative distribution: normcdf(x,mu,sigma)
normcdf(8,10,2)  ans =
normcdf(12,10,2)  ans =
Inverse normal cumulative distribution: norminv(p,mu,sigma)
norminv([ ],10,2)  ans =
Random number from normal distribution: normrnd(mu,sigma,v)
normrnd(10,2,[1 5])  ans =

9. Matlab: Normal Distribution Example
The temperature of a bioreactor follows a normal distribution with an average temperature of 30oC and a standard deviation of 1oC. What percentage of the reactor operating time will the temperature be within +/-0.5oC of the average?
Calculate probability at 29.5oC and 30.5oC, then calculate the difference:
p=normcdf([ ],30,1)
p = [ ]
p(2) – p(1)
0.3829
The reactor temperature will be within +/- 0.5oC of the average ~38% of the operating time

10. Discrete Distribution Example
Matlab function: unicdf(x,n),
>> x = (0:6);
>> y = unidcdf(x,6);
>> stairs(x,y)

11. Poisson Distribution Example
Probability of a defective thin polymer film p = 0.01
What is the probability of more than 2 defects in a lot of 100 samples?
Binomial distribution: m = np = (100)(0.01) = 1
Since p <<1, can use Poisson distribution to approximate the solution.

12. Matlab: Maximum Likelihood
In a chemical vapor deposition process to make a solar cell, there is an 87% probability that a sufficient amount of silicon will be deposited in each cell. Estimate the maximum likelihood of success in 5000 cells.
s = binornd(n,p) – randomly generate the number of positive outcomes given n trials each with probability p of success
s = binornd(5000,0.87)
s=4338
phat = binofit(s,n) – returns the maximum likelihood estimate given s successful outcomes in n trials
phat = binofit(s,5000)
phat =

13. Mean Confidence Interval Example
Measurements of polymer molecular weight (scaled by 10-5)
Confidence interval

14. Variance Confidence Interval Example
Measurements of polymer molecular weight (scaled by 10-5)
Confidence interval

15. Matlab: Confidence Intervals
>> [muhat,sigmahat,muci,sigmaci] = normfit(data,alpha)
data: vector or matrix of data
alpha: confidence level = 1-alpha
muhat: estimated mean
sigmahat: estimated standard deviation
muci: confidence interval on the mean
sigmaci: confidence interval on the standard deviation
>> [muhat,sigmahat,muci,sigmaci] = normfit([ ],0.05)
muhat =
sigmahat =
muci =
1.3079
sigmaci =
0.1273

16. Mean Hypothesis Test Example
Measurements of polymer molecular weight
Hypothesis: m0 = 1.3 instead of the alternative m1 < m0
Significance level: a = 0.10
Degrees of freedom: m = 9
Critical value
Sample t
Reject hypothesis

17. Polymerization Reactor Control
Monomers
Catalysts
Solvent
Hydrogen
On-line
measurements
Lab
Polymer
Monomer Content
Viscosity
Lab measurements
Viscosity and monomer content of polymer every four hours
Three monomer content measurements: 23%, 18%, 22%
Mean content control
Expected mean and known variance: m0 = 20%, s2 = 1
Control limits:
Sample mean: 21%  process is in control

18. Goodness of Fit Example
Maximum likelihood estimates

19. Goodness of Fit Example cont.

20. Matlab: Goodness of Fit Example
Find maximum likelihood estimates for µ and σ of a normal distribution with function mle
>> data=[320 … 360];
>> phat = mle(data)
phat =
Analyze dataset for goodness of fit using function chi2gof – usage: [h,p,stats]=chi2gof(x,’edges’,edges) for data in vector x divided into intervals with the specified edges
If h = 1, reject hypothesis at 5% significance
If h = 0, accept hypothesis at 5% significance
p: probability of observing given statistic
stats: includes chi-square statistic and degrees of freedom
>> [h,p,stats]=chi2gof(data,’edges’,[-inf,325:10:405,inf]);
>> h = 0
>> p =
>> chi2stat =
>> df = 7