EEM332 Lecture Slides1 EEM332 Design of Experiments En. Mohd Nazri Mahmud MPhil (Cambridge, UK) BEng (Essex, UK) Room 2.14 Ext. 6059

EEM332 Lecture Slides2 Agenda 1 Using P-Values in Hypothesis Testing 2 Variability in the data 3 Single factor experiment with more than two levels of factor 4 Analysis of variance 5 Demo example of ANOVA calculation using Excel 6 Assignments

EEM332 Lecture Slides3 Using P-values One way to report the results of a hypothesis testing is to state that the null Hypothesis was or was not rejected at a specified alpha-value or level of Significance. This is often inadequate because it gives the decision maker no idea about whether the value of the test statistics was just barely in the rejection region or whether it was very far into the region. To avoid this, P-value approach has been adopted widely in practice. The P-value is the probability that the test statistics will take on a value that is at least as extreme as the observed value of the statistics when the null hypothesis is true. The P-value is the smallest level of significance that would lead to rejection of the null hypothesis.

EEM332 Lecture Slides4 Using P-values-Example Minitab output The null hypothesis would be rejected at any level of significance, alpha greater And equal to 0.042

EEM332 Lecture Slides5 Comparison of the variability in the data In many experiments, we are interested in possible variability in the data because there are cases in which the variability needs to be small. Therefore, we need to examine tests of hypothesis and confidence interval for the variances using chi-square distribution and the F-distribution To test whether or not the variance is equal to a constant we use Table 2-7p53. With corresponding hypothesis, test statistics and criteria for rejection. The appropriate reference distribution is the chi-square distribution (Appendix III,p.607) With n-1 degrees of freedom To test the equality of variances, we use Table 2-7p53 with corresponding hypothesis, test statistics and criteria for rejection. The appropriate reference distribution is the F-distribution (Appendix IV,p ) With n1 -1 numerator degrees of freedom and n2-1 denominator degrees of freedom.

EEM332 Lecture Slides6 Comparison of the variability in the data Example 2.2

EEM332 Lecture Slides7 Comparison of the variability in the data Exercise

EEM332 Lecture Slides8 Single factor experiment with more than two levels of factor Single factor with 2 levels – Example 2-1p24 Single factor with > two levels – Example 3- 1p.61 –If we wish to test whether the 4 means are different or not, we do not use t-Test because it is tedious to do 6 pairs of comparison –An appropriate procedure is the Analysis of Variance (ANOVA)

EEM332 Lecture Slides9 Analysis of variance Analysis of variance (ANOVA) is based on the idea of partitioning of the total variability into its component parts. It is used for testing the equality or inequality of treatment means. The total variability in the data as measured by the Total Corrected Sum of Squares can be partitioned into a sum of squares of the differences between the treatment averages and the grand average, plus a sum of squares of the difference of observations within treatments from the treatment average If the between-treatment error is much larger than the within-treatment error, It is likely that the treatments means are different. SS T = SS Treatments + SS E

EEM332 Lecture Slides10 Analysis of variance Computing the values using Microsoft Excel Example 3-1p. 70

EEM332 Lecture Slides11 Analysis of variance – Individual Assignments using excel Question 1 The tensile strength of portland cement is being studied. Four different mixing techniques can be used economically. A completely randomised experiment was conducted and the following data collected. Mixing Technique Tensile Strength Perform ANOVA using Excel to test the hypotheses that mixing techniques affect the tensile strength

EEM332 Lecture Slides12 Analysis of variance - Assignments Q 2 A manufacturer of television sets is interested in the effect of tube conductivity of four different types of coating for color picture tubes. The following conductivity data are obtained. Coating Type Conductivity Perform ANOVA using Excel to test the hypotheses that coating types affect the conductivity.

EEM332 Lecture Slides13 Analysis of variance - Assignments Q 3 Four different designs for a digital circuits are being studied in order to compare the amount of noise present. The following data have been obtained. Circuit designs Noise observed Perform ANOVA using Excel to test the hypotheses whether the noise are the same for all the four designs or not.

EEM332 Lecture Slides14 Analysis of variance - Assignments Deadline : Friday 13 th February 12:00pm Submit hardcopies and softcopies (Excel files) Tomorrow’s class (Friday 6 th February will be in Mechatronic Lab Level 2) We will do the assignments using Excel and Minitab )