1. Statistika & Rancangan Percobaan
Dicky Dermawan
2- Statistics

2. Nature & Purpose of Statistics
In statistics we are concerned with method for designing and evaluating experiments to obtain information about practical problems.
In most cases the inspection of each item of population would be too expensive, time-consuming, or even impossible. Hence a few of sample are drawn at random and from this inspection conclusion about the population are inferred.

3. Parameter Estimation Population sample Size large number N
Size small number n
Mean
Average
Variance
Variance
Probability function/density f(x)
Relative frequency function
Cumulative frequency function
Distribution function F(x)

4. Processing of Sample: Frequency Table
Sample of 100 Values of the Splitting Tensile Strength (lb/in2)
320
380
340
410
360
350
370
300
420
390
440
330
400

5. Processing of Sample: Absolute Frequency Function
Sample of 100 Values of the Splitting Tensile Strength (lb/in2)

6. Processing of Sample: Relative Frequency Function
Sample of 100 Values of the Splitting Tensile Strength (lb/in2)

7. Processing of Sample: Cumulative Absolute Frequency
Sample of 100 Values of the Splitting Tensile Strength (lb/in2)

8. Processing of Sample: Cumulative Relative Frequency
Sample of 100 Values of the Splitting Tensile Strength (lb/in2)

9. Processing of Sample: Box & Whisker Plot
Min Lower Quartile Middle Quartile = Median Upper Quartile Interquartile range Max

10. Assignment: Box & Whisker Plot Portland Cement Formulation
DOX 6E Montgomery

11. Inferential Statistic
Experimental error
Hypothesis testing: null hypothesis, alternative hypothesis
Type I error α : rejecting a true hypothesis
Type II error β: accepting a false hypohesis
One-tail test vs Two-tail test
Confidence level = Significance Level
P-value
Confidence interval

12. Confidence Interval of a normal distribution if σ known: Comparing a single mean to a specified/’standar’ value
So far we have regarded the value y1, y2, ….of a sample as n observed value of a single random variable Y. We may equally well regard these n values as single observations of n random variables Y1, Y2,….that have the same distribution and are independent
If Y1, …….Yn are independent normal random variables each of which has mean  and variance σ2, then the normal random variable:
Is normal with the mean  and variance σ2/n and the random variable
Is normal with the mean 0 and variance 1
The confidence interval for  is

13. Problem: Example 2.1
A vendor submits lots of fabric to a textile manufacturer. The manufacturer wants to know if the lot average breaking strength exceeds 200 psi. If so, she wants to accept the lot. Past experience indicates that a reasonable value for the variance of breaking strength is 100 (psi)2. Four speciments are randomly selected, and the average breaking strength observed is

14. Example 2.1
The hypothesis to be tested are: This is a one-sided alternative hypothesis The value of the test statistic is: If the confidence level of 95% is chosen, i.e. type I error α = 0.05, we find Zα = Thus the difference is significant: H0 is rejected and we conclude that the lot average breaking strength exceeds 200 psi. Thus, we accept the lot. The confidence interval for  at 95% confidence level is ≤  ≤ Clearly, 200 is outside the interval. The P-value is

15. Problems

16. Problems

17. Comparing a single mean to a specified/’standar’ value Confidence Interval of a normal distribution if σ unknown
The same as previous, but we use…..
t distribution instead of normal distribution
Sample standard deviation S instead of σ
The test statistic is
The confidence interval is
At (n-1) degree of freedom

18. Comparing 2 Treatments Means
If Variance Known
The test statistic is
The confidence interval is

19. Comparing 2 Treatments Means
If Variance Unknown, but σ12 = σ22
Choose confidence level, usually 95%, then find critical t value at associated degree of freedom, i.e. t/2,
If |t0|> t /2,, we have enough reason to reject null hypothesis and conclude that the two method differ significantly
Alternatively, calculate P value, i.e. the risk of wrongly rejecting the null hypothesis
Or set confidence interval and reject null hypothesis if 0 is not included in the interval

20. Comparing 2 Treatments Means
If Variance Unknown, σ12 ≠ σ22
The test statistic is

21. Problems

22. Example: Portland Cement Formulation – Dot Diagram
Tension bond strength of portland cement mortar is an important characteristics of the product. An engineer is interested in comparing the strength of a modified formulation in which polymer latex emulsions have been added during mixing to the strength of the unmodified mortar. He collected 10 observations (Table 2.1)
Plot the dot diagram.
Plot the Box & Whisker plot
Are the two formulations really different?
Or perhaps the observed difference is the results of sampling fluctuation and the two formulations are really identical?
DOX 6E Montgomery

23. Problems

24. Problems

25. Problems

26. Inference about the difference in means Bloking: Paired Comparison Design
Bloking is a design technique used to improve the precision with which the comparisons among the factors of interest are made. Often blocking is used to reduce or eliminate the variability transmitted from nuisance factors, i.e.
factors that may influence the experimental response but in which we are not interested.
The term block refers to a relatively homogeneous experimental unit, and the block represents a restriction on complete randomization because the treatment combinations are only randomized within the block. Blocking is carried out by making comparisons within matched pairs of experimental material.
The confidence interval based on paired analysis usually much narrower than that from the independent analysis. This illustrates the noise reduction property of blocking.

27. Inference about the difference in means Bloking: Paired Comparison Design
Statistical model 4 complete randomization:
with (2ni -1) degree of freedom
Statistical model with blocking:
with only (ni pair -1) degree of freedom
The test statistic:
The confidence interval for 2-sided test:

28. Inference about the difference in means Bloking: Paired Comparison Design Example: The Story
Consider a hardness testing machine that presses a rod with a pointed tip into a metal specimen with a known force. Two different tips are available for this machine, and it is suspected that one tip produces different hardness readings than the other.
The test could be performed as follows: a number of metal specimens could randomly be selected. Half are tested by tip 1 and the other half by tip 2.
The metal specimens might be cut from different bar stock that were not exactly different in their hardness. To protect against this possibility, an alternative experimental design should be considered: divide each specimen into two part and randomly assign each tip to ½ of each specimen

29. Inference about the difference in means Bloking: Paired Comparison Design Example: Data
Speciment
Tip 1
Tip 2 1 7 6 2 3 5 4 8 9 10
- Use the paired data to determine a 95% confidence interval for the difference
- What if we use pooled or independent analysis?

30. Problems

31. Problems

32. Inference about The Variances of Normal Distributions
In some experiments it is the comparison of variability in the data that is important.
For example, in chemical laboratories, we may wish to compare the variability of two
analytical methods.
Unlike the tests on means, the procedures for tests on variances are rather sensitive to the normality assumption.
Suppose we wish to test the hypothesis weather or not the variance of a normal population equals a constant, viz. σ02 . The test statistic is:
The appropriate distribution for 02 is chi-square distribution with (n-1) degree of freedom. The confidence interval for σ02 is

33. Inference about The Variances of Normal Distributions
Suppose we wish to test equality of the variances of two normal populations.
If independent random samples of size n1 and n2 are taken from populations 1 & 2, respectively, the test statistic for:
Is the ratio of the sample variances:
The appropriate distribution for F0 is the F distribution with (n1-1) numerator degree of freedom and (n2-1) denominator degree of freedom. The null hypothesis would be rejected if F0 > Fα/2,n1-1,n2-1
The confidence interval for σ12 / σ22 is

34. Checking for Normality: Normal Probability Plot
Probability plotting is a graphical technique for determining whether sample data conform to a hypothesized distribution based on a subjective visual examination of the data.
To construct a probability plot, the observation in the sample are first rank from smallest to largest. That is, the sample y1,y2,…,yn is arranged as y(1) ,y(2) ,….,y(n) where y(1) is the smallest observation, with y(n) the largest.
The ordered observations y(j) are then plotted against their observed cumulative frequency (j-0.5)/n.
The cumulative frequency scale has been arranged so that if the hypothesized distribution adequately describes the data, the plotted points will fall approximately along a straight line. Usually, this is subjective.

35. Problems

36. Problems

37. Problems

40. Introduction to DOX An experiment is a test or a series of tests
Experiments are used widely in the engineering world
Process characterization & optimization
Evaluation of material properties
Product design & development
Component & system tolerance determination
“All experiments are designed experiments, some are poorly designed, some are well-designed”
DOX 6E Montgomery

41. The Basic Principles of DOX
Randomization
Running the trials in an experiment in random order
Notion of balancing out effects of “lurking” variables
Replication
Sample size (improving precision of effect estimation, estimation of error or background noise)
Replication versus repeat measurements? (see page 13)
Blocking
Dealing with nuisance factors
DOX 6E Montgomery

42. Strategy of Experimentation
“Best-guess” experiments
Used a lot
More successful than you might suspect, but there are disadvantages…
One-factor-at-a-time (OFAT) experiments
Sometimes associated with the “scientific” or “engineering” method
Devastated by interaction, also very inefficient
Statistically designed experiments
Based on Fisher’s factorial concept
DOX 6E Montgomery