1 Doing Statistics for Business Doing Statistics for Business Data, Inference, and Decision Making Marilyn K. Pelosi Theresa M. Sandifer Chapter 14 Experimental Design and Anova

2 Doing Statistics for Business Chapter 14 Objectives Ü Motivation for Using a Designed Experiment Ü Analysis of Data From One-Way Designs Ü Assumptions of ANOVA Ü Analysis of Data from Blocked Designs

3 Doing Statistics for Business Chapter 14 Objectives (con’t) Ü Analysis of Data from Two-Way Designs Ü Other Types of Experimental Designs

4 Doing Statistics for Business Figure 14.1 A Portion of the Tissue Company Dataset

5 Doing Statistics for Business Figure 14.2 Summary Statistics for the Tissue Company

6 Doing Statistics for Business A factor is a variable that can be used to differentiate one group or population from another. It is a variable that may be related to the variable of interest. A level is one of several possible values or settings that the factor can assume.

7 Doing Statistics for Business The response variable is a quantitative variable that you are measuring or observing.

8 Doing Statistics for Business An experiment has a one-way or completely randomized design if there are several different levels of one factor being studied and the objects or people being observed/ measured are randomly assigned to one of the levels of the factor.

9 Doing Statistics for Business TRY IT NOW! One-Way Designs Designing a Simple Study Select a population that you might be interested in studying and identify a quantitative variable that you might wish to analyze. Now specify a factor that might be of interest. This should be some characteristic that you think might influence the variable you are analyzing. Indicate the various levels of the factor.

10 Doing Statistics for Business Analysis of variance (ANOVA) is the technique used to analyze the variation in the data to determine if more than two population means are equal.

11 Doing Statistics for Business A treatment is a particular setting or combination of settings of the factor(s) being studied.

12 Doing Statistics for Business TRY IT NOW! Airspace Data Getting Used to the ANOVA Notation Using the information in Figure 14.1 what is the value of x 31 ? What notation would be used to refer to the 120th observation of the data taken 2 weeks after the time of manufacturing? What is the range of values for the first subscript for group 1? What is the range of values for the second subscript?

13 Doing Statistics for Business The grand mean or the overall mean is the sample average of all the observations in the experiment. It is labeled.

14 Doing Statistics for Business The total variation or sum of squares total (SST) is a measure of the variability in the entire data set considered as a whole.

15 Doing Statistics for Business Figure 14.3 Dotplot of Airspace by Time for the Tissue Company

16 Doing Statistics for Business The treatment mean is the average of the response variable for particular treatment.

17 Doing Statistics for Business Between Groups Variation measures how different the individual treatment means are from the overall grand mean. It is often called the sum of squares between or the sum of squares among (SSA).

18 Doing Statistics for Business TRY IT NOW! Career Office Calculation of SSA The career office is interested in studying salaries for 3 different majors: Engineering, Business, and Humanities. The overall average starting salary was (X w/2 lines) = $28,200. There were 30 students in each group and the averages are shown on the following slide. Find SSA.

19 Doing Statistics for Business TRY IT NOW! Career Office Calculation of SSA (con’t)

20 Doing Statistics for Business Within groups variation measures the variability in the measurements within the groups. It is often called sum of squares within or sum of squares error (SSE).

21 Doing Statistics for Business The mean square among is labeled MSA. The mean square error is labeled MSE. The mean square total is labeled MST.

22 Doing Statistics for Business TRY IT NOW! Tissue Strengths Finding MSA and MSE Find MSA and MSE for the tissue strength data shown in Example 14.8 in your textbook. You should do the calculations using the formula and then find those values in the computer output.

23 Doing Statistics for Business TRY IT NOW! Tissue Strength Data Completing the Analysis Set up the null and the alternative hypotheses for the tissue strength example. Use the ANOVA table shown in Example 14.8 to find the F statistic and decide if you should reject the null hypothesis or fail to reject it. What do you conclude about the variable MDStrength and what is your recommendation to the company?

24 Doing Statistics for Business TRY IT NOW! Tissue Strength Data Prediction Model For the tissue strength data, write x 11 in terms of its components. What is your error for this particular observation?

25 Doing Statistics for Business Figure 14.4 Confidence Intervals for Airspace Data from Minitab

26 Doing Statistics for Business The 3 Major Assumptions of ANOVA You should always check that the three major assumptions of ANOVA are met before you use this technique: m The errors are random and independent of each other m Each population has a normal distribution m All of the populations have the same variance

27 Doing Statistics for Business Figure 14.5 Normal Probability Plot of 24-hour Airspace Data

28 Doing Statistics for Business A block is a group of objects or people that have been matched. An object or person can be matched with itself, meaning that repeated observations are taken on that object or person and these observations form a block.

29 Doing Statistics for Business An experiment has a randomized block design if several different levels of one factor are being studied and the objects or people being observed/measured have been matched. Each object or person is randomly assigned to one of the c levels of the factor.

30 Doing Statistics for Business The sum of square blocks measures the variability between the blocks. It is labeled SSBL.

31 Doing Statistics for Business TRY IT NOW! Participative Management Block Design In Chapter 10 you looked at the number of sick days used by employees in the past 12 months by employees who were traditionally managed and those who were involved in a participative management style. Management is now considering a third approach which is a mixture of the old structure with the team-based approach. The study involves 75 employees matched by age, type of work, and gender.

32 Doing Statistics for Business TRY IT NOW! Participative Management Block Design (con’t) Use the Minitab output shown below to determine if there are any differences in the mean number of sick days used by the different populations.

33 Doing Statistics for Business TRY IT NOW! Participative Management Efficiency of Block Design Calculate the relative efficiency of the block design for the sick days data. Comment on the usefulness of the blocking.

34 Doing Statistics for Business Discovery Exercise 14.1 The Benefits of Blocking In manufacturing electronics products such as loudspeakers it is important that connections are strong and hold so that they do not disconnect. The same is true for commercial heaters. One of the customers of a manufacturer of heaters complained that the pull poundage was too low. Thus, the customer did not have confidence in the terminal connection. New wires with the terminals attached were made and a pull test was done on a sample of 25. The terminals were connected and pulled apart. The pull pounds at which the terminals would disconnect from each other was recorded.

35 Doing Statistics for Business Discovery Exercise 14.1 The Benefits of Blocking (con’t) Then the same terminals were reconnected and the test was repeated 6 times. The object was to determine if the pull poundage changed over time. The minimum pull poundage required by the customer was 5 lb. The data are shown on the following slide:

36 Doing Statistics for Business

37 Doing Statistics for Business Discovery Exercise 14.1 The Benefits of Blocking (con’t) Part I. One-Way ANOVA Use a one-way ANOVA to analyze these data. A. What is the response variable and what is the factor? B. How many levels of the factor are being studied? C. Is there a difference in the average pull poundage among the trials? If so, which ones are different? D. What is your recommendation to the company and why?

38 Doing Statistics for Business Discovery Exercise 14.1 The Benefits of Blocking (con’t) Part II. Block Design Since the same terminals were connected and pulled apart 6 times, each row is actually a block. Reanalyze these data using a block design. A. Is there any difference in the average pull poundage among the trials? If so, which ones are different? B. Explain why you got a different answer when you analyzed the data as a one-way design.

39 Doing Statistics for Business Discovery Exercise 14.1 The Benefits of Blocking (con’t) Part II. Block Design C. Calculate the sample size needed to see the treatment effect using a randomized one-way design. D. Now, what is your recommendation to the company?

40 Doing Statistics for Business Figure 14.6 Schematic of a Hardroll

41 Doing Statistics for Business An experimental design is called a factorial design with two factors if there are several different levels of two factors being studied. The first factor is called factor A and there are r levels of factor A. The second factor is called factor B and there are c levels of factor B.

42 Doing Statistics for Business The design is said to have equal replication if the same number of objects or people being observed/measured are randomly selected from each population. The population is described by a specific level for each of the two factors. Each observation is called a replicate. There are n' observations or replicates observed from each population. There are n = n'rc observations in total.

43 Doing Statistics for Business The sum of squares due to factor A is labeled SSA. It measures the squared difference between the mean of each level of factor A and the grand mean.

44 Doing Statistics for Business The sum of squares due to factor B is labeled SSB. It measures the squared differences between the mean of each level of factor B and the grand mean.

45 Doing Statistics for Business The sum of squares due to the interacting effect of A and B is labeled SSAB. It measures the effect of combining factor A and factor B.

46 Doing Statistics for Business The sum of squares error is labeled SSE. It measures the variability in the measurement within the groups.

47 Doing Statistics for Business TRY IT NOW! Participative Management Two-Way Design In a previous “Try It Now” you looked at the number of sick days used in the past 12 months by employees who were traditionally managed, employees who were involved in a management style that was a mixture of the old structure with the team-based approach. Management is wondering whether the employee’s department is a factor as well.

48 Doing Statistics for Business TRY IT NOW! Participative Management Two-Way Design Use the following Minitab output show to determine if there are any differences in the mean number of sick days by management style and by department and if there is any interaction effect.

49 Doing Statistics for Business Figure 14.7 Average Airspace Based on Time on Shelf for Different Positions

50 Doing Statistics for Business Figure 14.8 Hypothetical Average Airspace if there was no Interaction Effect

51 Doing Statistics for Business ANOVA in Excel Use the ANOVA: Single Factor tool from the Tools> Data Analysis menu. 1. Enter the range that contains the data in the Input Range: textbox. If you data range contains labels, click on the textbox for Labels in First Row. 2. Since each factor is in a different column, click the radio button for Grouped By: Columns. This is the default value. 3. In the the textbox for Alpha, enter the level of significance that you want to use for the test.

52 Doing Statistics for Business ANOVA in Excel (con’t) 4. Finally, indicate where you want Excel to put the output from the analysis. 5. Click OK; the output will appear in the location specified. As with the output from the regression analysis, you might want to select Format > Column > AutoFit Selection while the output is still highlighted.

53 Doing Statistics for Business Figure ANOVA: Single Factor Dialog Box

54 Doing Statistics for Business Figure Output from ANOVA on Airspace Data

55 Doing Statistics for Business Two-Way ANOVA Designs in Excel 1. From the analysis tools list, select ANOVA: Two Factor Without Replication. 2. Enter the worksheet range for the data and fill in the other textboxes with the appropriate information. 3. Click OK; the output will be placed in the location specified.

56 Doing Statistics for Business Figure Two-way without Replication Dialog Box

57 Doing Statistics for Business Figure Output for Blocked Design

58 Doing Statistics for Business Figure Data Setup for Two-way with Replications

59 Doing Statistics for Business Figure Airspace Data for Two-way Analysis

60 Doing Statistics for Business Figure Two-way with Replications Dialog Box

61 Doing Statistics for Business Figure Output from Airspace Two- way Design

62 Doing Statistics for Business Chapter 14 Summary In this chapter you have learned: 4 ANOVA is used to draw inferences about more than 2 population means. 4 The technique of ANOVA partitions the variation in the data into components and compares the relative size of these components.

63 Doing Statistics for Business Chapter 14 Summary (con’t) 4 It is through the an analysis of variation in the data that we learn something about the means. 4 It is only by understanding variation that we can produce quality products.