1. EEM332 Design of Experiments En. Mohd Nazri Mahmud
MPhil (Cambridge, UK)
BEng (Essex, UK)
Room 2.14
Ext. 6059
EEM332 Lecture Slides

2. Agenda Statistical design of experiments Basic Statistical Concepts
Simple Comparative Experiments
Statistical Hypothesis
The t-test
EEM332 Lecture Slides

3. Experiment Is it enough to focus on designing experiments only?
How about the analysis of the data collected during the experiments?
EEM332 Lecture Slides

4. Statistical design of experiments
The process of planning the experiment so that appropriate data that can be
analysed by statistical methods will be collected for the purpose of valid and
objective conclusions to be made.
Three basic principles
Randomisation
Replication
Blocking
Randomisation
The allocation of the experimental material and the order in which the
individual runs or trials of the experiments are to be performed are randomly
determined.
Assist in “averaging out” the effect of extraneous factors that may be present
Use random number generator
EEM332 Lecture Slides

5. Statistical design of experiments
Replication
An independent repeat of each factor combination.
2 important properties
Allows experimenter to obtain an estimate of the experimental error
Allows experimenter to obtain a more precise estimate of the true
mean response.
Blocking
A design technique used to reduce or eliminate the variability due to
nuisance factors.
EEM332 Lecture Slides

6. Basic Statistical Concepts
Run – one observation of an experiment
Noise – fluctuations in the observation of the individual runs
Sampling – taking a sample from a population for a study to draw conclusions
about that population
Random samples – any of the member of the population has an equal probability
of being chosen as a sample
Sample mean – a measure of the central tendency
Suppose y1, y2,….,yn represents a sample, the sample mean,
EEM332 Lecture Slides

7. Basic Statistical Concepts
Sample variance – a measure of the dispersion of the sample
Sample standard deviation – a measure of the dispersion of the sample
EEM332 Lecture Slides

8. Example. From Montgomery p.24- Portland Cement Formulation
Refer Table 2-1 p.24
Two treatments
or two levels
of factor formulations
EEM332 Lecture Slides

9. Example. From Montgomery p.24
Refer Table 2-1 p.24
Work out the samples means and the variances
Sample 1 =16.764
Sample 2 =17.042
Sample 1 = 0.104
Sample 2 = 0.061
EEM332 Lecture Slides

10. Graphical View of the Data Dot Diagram, Fig. 2-1, pp. 24
Enables the experimenter to see quickly the general location or central tendency
of the observations and their spread.
In the example, the diagram reveals that the two formulations may differ in mean
Strength but that both formulations produce about the same variation in strength.
EEM332 Lecture Slides

11. Box Plots, Fig. 2-3, pp. 26
Displays the minimum, maximum, the lower and upper quartiles and the median.
In the example, it shows some difference in mean strength and similar spread.
EEM332 Lecture Slides

12. Simple Comparative Experiments
Experiments to compare 2 conditions ( or sometimes called treatments)
Example. P. 23 – an experiment to determine whether two different formulations
of a product give equivalent results.
Section 2.4-p. 34 discuss how the data from the simple comparative experiment
can be analysed using Hypothesis testing procedures for comparing two
treatment means.
EEM332 Lecture Slides

13. The Hypothesis Testing Framework
Statistical hypothesis testing is a useful framework for many experimental situations
Origins of the methodology date from the early 1900s
We will use a procedure known as the two-sample t-test
EEM332 Lecture Slides

14. The Hypothesis Testing Framework
Sampling from a normal distribution
Statistical hypotheses:
EEM332 Lecture Slides

15. Statistical Hypothesis
A statement either about the parameters of a probability distribution or the
parameters of a model that reflects some conjecture about the problem
situation
Hypothesis Testing (or Significance Testing)
A technique of statistical inference to assist experimenter in comparative
experiments and allows comparison to be made on objective terms, and with
knowledge of the risks associated with reaching the wrong conclusion.
(Refer example. p. 24)- it is not obvious that the two samples really are different.
One may be faced with the problem of making a definite decision with respect to
an uncertain hypothesis which is known only through its observable consequences.
EEM332 Lecture Slides

16. Estimation of Parameters
EEM332 Lecture Slides

17. The t-Test
A t test is any statistical hypothesis test for two groups in which the test statistic
has a t distribution if the null hypothesis is true.
A test of the null hypothesis that the means of two normally distributed populations
are equal.
Used for investigating the statistical significance of the difference
between two sample means, and for confidence intervals for the difference between
two population means.
Given two data sets, each characterized by its mean, standard deviation
and number of data points, we can use some kind of t test to determine whether
the means are distinct, provided that the underlying distributions can be assumed
to be normal.
The t-distribution is a probability distribution that arises in the problem of estimating
the mean of a normally distributed population when the sample size is small.
(Refer to Montgomery p. 606)
EEM332 Lecture Slides

18. The two sample t-Test
Testing for a significant difference between two means
The t statistics compares two means to see if they are significantly different from
each other (For the table of t-statistics see Table 2-3 page 47 and Table 2-4 page 48)
Used for investigating the statistical significance of the difference
between two sample means, and for confidence intervals for the difference between
two population means.
Example of two sample t-test p. 36-7
EEM332 Lecture Slides

19. Degrees of freedom (df) is a measure of the number of
independent pieces of information on which the precision of a parameter estimate
is based.
The degrees of freedom for an estimate equals the number of observations (values)
minus the number of additional parameters estimated for that calculation.
Example. P. 36
As we have to estimate more parameters, the degrees of freedom available decreases.
It can also be thought of as the number of observations (values) which are freely
available to vary given the additional parameters estimated.
It can be thought of two ways:
in terms of sample size and in terms of dimensions and parameters.
For a single sample t-test, degrees of freedom are one less than the number of
Observation (n-1).
For a two sample t-test, degrees of freedom are one less than the number of
observation in the first sample plus one less than the number of
observation in the second sample ( n1-1 + n2-1) or (n1 + n2 -2).
EEM332 Lecture Slides

20. Steps for the two sample t-test
State the null hypothesis and the alternative hypothesis
2. Set the alpha level. Example= .05, we have 5 chances in 100 of making a type I error.
3. Select an appropriate test statistics from Table 2-3p 47 or 2-4 p.48
4. Calculate the value of the appropriate statistic.
Also indicate the degrees of freedom for the statistical test. (n1+ n2 -2)
5. Write the decision rule for rejecting the null hypothesis. Refer Table 2-3 / 2-4
Example : Reject H0 if t0 is >= talpha or if t0 <= taplha
6. Write a summary statement based on the decision. Example: Reject H0
7. Write a statement of results in standard English. Example: There is a significant difference in the sample mean between
the two groups
EEM332 Lecture Slides

21. Exercise 1
A design engineer would like to compare the mean burning times of
chemical flare of two different formulations. The burning times (in minutes) are
tabulated below
Formulation A
Formulation B 65 82 64 56 81 67 71 69 57 59 83 74 66 75 70 79
State the hypothesis to be tested and using alpha = 0.05, test the hypotheses?
EEM332 Lecture Slides

22. Answer Test statistic to be used ; Degrees of freedom = n1 + n2 -2
To find Sp use formula for the estimate of
the common variance ( ie Formula 2-25.p36)
y1 mean = 70.4
Y2 mean = 70.2
Sp = 9
t0=0.050
So, do not reject H0
T0.025,18=2.101
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23. Exercise 2
A new filtering device is designed for installation in a chemical system.
Before its installation, a random sample yielded the following information
about the percentage of impurity:
After installation, a random sample yielded
Has the filtering device changed the percentage of impurity significantly?
Use alpha= 0.05.
EEM332 Lecture Slides

24. Answer Test statistic to be used ; Degrees of freedom = n1 + n2 -2
To find Sp use formula for the estimate of
the common variance ( ie Formula 2-25.p36)
Sp = 9.89
t0=0.479
Do not reject H0
T0.025,15=2.131
So, There is no evidence to indicate that the new filtering
device has affected the mean
EEM332 Lecture Slides

25. The case of different variances
If we are testing
and cannot reasonably assume that the variances of the two samples are equal,
We must use the test statistic given by equation 2-31 p 45 and degrees of freedom
given by equation 2-32 p 45.
The case when the variances for both populations are known
If we are testing
and the variances for both populations are known, use the test statistics given
by equation 2-33 p. 45.
EEM332 Lecture Slides

26. Comparing a single mean to a specified value
Some experiments involve comparing only one population mean to a specified
value
The hypotheses are
Use the test statistics given by equation 2-35 p. 46.
Example: Ex 2-1 p46
If the variance of the population is unknown use the test statistic given by
Equation 2-37 p.47.
EEM332 Lecture Slides

27. Exercise: The case when the variances for both populations are known
Two devices are used for filling containers with a net volume of
16.0 litre. The filling processes can be assumed to be normal,
with standard deviation of = and =
The quality engineering department suspects that both devices fill to the
same net volume, whether or not this volume is 16.0 litre.
An experiment is performed by taking a random sample from the output
of each device.
Device 1
Device 2
16.03
16.01
16.02
16.04
15.96
15.97
16.05
15.98
15.99
16.00
State the hypotheses that should be tested in this experiment and test these
hypotheses using = What are your conclusions?
EEM332 Lecture Slides

28. Answer Test statistic to be used = equation 2-33 z0=1.35 z0.025 =1.96
since
z0=1.35
is not greater than
Therefore, do not reject H0
EEM332 Lecture Slides

29. Confidence Interval
An interval within which the value of the parameter or parameters in question
would be expected to lie
Why ?
Hypothesis testing is useful but sometimes does not tell the entire story
It is often preferable to provide an interval within which the value of the parameter
or parameters in question would be expected to lie.
In many engineering and industrial experiments, the experimenter already knows
That the means differ, so they are more interested in a confidence interval on the
difference in means.
EEM332 Lecture Slides

30. Confidence Interval: Example. p.44
In the portland cement problem, the estimate for the difference in mean
Tension bond strength for the formulations can be expressed in the form of its 95 percent confidence interval.
General form of a confidence interval
The 100(1- α)% confidence interval on the difference in two means:
EEM332 Lecture Slides

31. Exercise: Confidence Interval
Two devices are used for filling containers with a net volume of
16.0 litre. The filling processes can be assumed to be normal,
with standard deviation of = and =
The quality engineering department suspects that both devices fill to the
same net volume, whether or not this volume is 16.0 litre.
An experiment is performed by taking a random sample from the output
of each device.
Device 1
Device 2
16.03
16.01
16.02
16.04
15.96
15.97
16.05
15.98
15.99
16.00
Find a 95 percent confidence interval on the difference in the mean fill volume
for the two machines.
EEM332 Lecture Slides

32. Answer Test statistic to be used = equation 2-33 z0=1.35 z0.025 =1.96
since
z0=1.35
is not greater than
Therefore, do not reject H0
For 95% confidence interval use equation 2-34
EEM332 Lecture Slides