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Chapter 11. Two-Way ANOVA for Comparing Means > 11.6 Problems
11.6 Problems
For each of the following situations, identify the appropriate type of analysis of variance that should be conducted–one-way ANOVA, two-way ANOVA without replication, or a two-way ANOVA with replication. Identify the Y-variable and the X-factor(s). Also mention any concerns that you might have about each situation fitting the assumptions of ANOVA.
A corporate ethics officer is comparing the number of sexual harassment complaints made in the past year at different plant locations. Historically, females have filed more complaints than males.
A university must determine whether there are differences in the amount of financial aid given based on veteran status and gender.
A manufacturing facility produces cell phones on two identical production lines, three shifts per day. The quality control manager is reviewing the monthly performance and believes there might be a higher number of defective cell phones produced on one of the manufacturing lines and during the evening and night shifts.
A shoe store is preparing its quarterly order and is reviewing past sales for its most popular lady's dress shoe. The store wants to determine whether there are differences in sales for the four colors stocked: black, navy blue, brown, and beige.
Military recruits choosing to specialize in communications are assigned to one of three military bases to receive training. The three bases are located in California, Kansas, and South Carolina. Upon completion of their training, recruits must pass an examination with a minimum score of 80 in order to begin working in communications. Those not achieving a passing score are assigned to another less technical job. The following table shows scores for each of the three bases and the educational level of the recruit. The military is interested in determining whether there are significant differences in the training received at the three bases, as measured by the examination scores.
High School Diploma GED Associate's Degree California 82 75 84 Kansas 83 80 92 South Carolina 90 85 96 Enter the data into a JMP data table in the format needed to carry out a two-way ANOVA.
Determine whether at least one of the bases is on average different from the others. Use the Fit Model platform to obtain an analysis of variance table. Write out the appropriate hypotheses. Find the associated F-Ratio and p-value. What is your conclusion?
From the analysis of variance options for the Military Base factor, select LSMeans Student's t. If at least one of the mean scores differs, determine specifically which means differ.
Give 95% confidence intervals for the pairwise differences in mean score.
Why was it important to take education level into account in this analysis?
A county uses four assessors to determine the values of residential properties for tax purposes. To see whether the assessors differ in their valuations, five houses, each representing a particular style, were selected. Each assessor was asked to determine the market value of these houses. The data are contained in HouseAssessments.jmp.
Write down the relevant statistical hypotheses that need to be tested.
Is there any statistically significant difference in the mean house values between assessors? If so, which assessors differ? Explain how you arrived at your answer.
Since differences between assessors were of primary importance, could you have used one-way ANOVA? If your answer is no, explain why not. If your answer is yes, run the one-way ANOVA and comment on the results.
Could you have included an interaction term in this analysis? If your answer is yes, run the model in JMP and comment on the results. If your answer is no, explain why an interaction term could not have been included.
(Optional) Based on your results, suggest actions that could be taken to achieve more consistency in assessments.
A market research firm conducts an extensive survey of customer satisfaction of bank branches writing different types of loans. Eighteen customers are asked to fill out a questionnaire in which a satisfaction score ranging from 0 (absolutely dissatisfied) to 100 (completely satisfied) is recorded. Bank branches were selected according to business volumes of small, medium and high. Customers were stratified according to type of loan, either secured or personal. Customers who had both loan types were excluded from the survey. The basic question to be addressed is whether size of branch or type of loan affects the customer satisfaction score. The results are as follows.
ID Branch Size Loan Type Satisfaction ID Branch Size Loan Type Satisfaction 1 Small Secured 83 10 Small Personal 76 2 Small Secured 81 11 Small Personal 71 3 Small Secured 78 12 Small Personal 69 4 Medium Secured 77 13 Medium Personal 73 5 Medium Secured 72 14 Medium Personal 70 6 Medium Secured 76 15 Medium Personal 67 7 High Secured 67 16 High Personal 69 8 High Secured 67 17 High Personal 67 9 High Secured 64 18 High Personal 65 Enter the data into a JMP data table in the format needed to carry out a two way ANOVA.
Based on the problem statement given, write out the appropriate hypotheses that should be tested.
Determine those factors that significantly influence customer satisfaction. Use the Fit Model platform to obtain the analysis of variance table. Find the F-Ratios and values associated with the hypotheses to be tested. What is your conclusion?
From the Fit Model platform, obtain the interaction plot. What is the meaning of interaction? Describe the nature of the interaction in this case.
List the Branch Size, Loan Type combinations that are significantly different.
Com.com, a new and growing organization, is evaluating three possible vendors for a repair contract for laptops. They would like to choose a vendor with the speediest service. Com.com's IT manager prepares seven repair tasks. Each task is completed by each vendor. The time from calling the vendor with the problem to reinstatement of the laptop to standard use is given in hours. The results of the evaluations are as follows:
Vendor Task AlCheap ComeLate DoQuick A 39 47 48 B 50 48 55 C 68 74 70 D 54 58 52 lu 44 45 54 ll 22 28 38 G 46 42 57 The data are contained in a file called ServiceVen.jmp
What advantages does Fit Model analysis using two X-factors (Vendor and Task) offer over the Fit Y by X platform using only Vendor as the single X-factor? Does it lead to the same or different recommendation of a vendor?
What does the Effect Test for Task measure and does this affect your conclusion about vendors?
Why is there no interaction term in this ANOVA?
In Fit Model, what do the LS Means plots for Task and Vendor show? How do these plots relate to the Least Squares Means in the tables?
Give your summary conclusions from this analysis.
A study needs to determine whether bed occupancy in a hospital depends on either the care level or ownership. The Y-variable is bed occupancy expressed in percentage of total beds occupied in a month. The X-factors are care level and ownership of hospitals. Hospitals are classified into one of three care levels: tertiary is the highest care level, and primary is the lowest. Hospitals can be owned by the public (typically a local, state, or federal government), a not-for-profit corporation, or an investor-owned stock corporation. There are three levels for each of the two X-factors, and so there are three times three or nine factor level combinations. The data are composed of three columns and 27 rows. This indicates that each of the nine factor combinations has been sampled three times. Data for a two-way ANOVA with replications are given in Hospital Occupancy.jmp.
Specify a two-way ANOVA model with an interaction term and run the standard ANOVA analysis.
What are the key outputs from the ANOVA table? List them and explain them.
What is the purpose of the Effect Tests table? Why is it needed in addition to the overall ANOVA table?
What does an LSMeans plot show in general?
What does the LSMeans plot for Care Level show?
What does the LSMeans plot of Ownership show in this specific application?
What does the LSMeans plot for the interaction term show?
Examine and interpret the results of the LSMeans Differences Student's t using both the Connecting Lines report and the Ordered Differences report.
Remove the interaction term from the Fit Model dialog box and rerun the analysis of variance. Write down the corresponding model. Compare the Mean Squared Errors from the two models (with and without interaction). What are the consequences of omitting the interaction term from the model?
State your final conclusions using plain English but quoting relevant evidence. List the necessary assumptions that make these valid conclusions.
A manufacturer of consumer durables wants to test the effectiveness of different advertising campaigns on sales (in 1000 units per test market). The company decides to run three different advertising programs labeled A, B, and C in 12 different test markets. Each ad program is repeated four times. The marketing VP wants to combine the ad research with the presence or absence of additional promotional discounts. The sales data (in units of 10000) for the 12 test markets with the factors AdProgram and Promotion are given in AdvertEffect.jmp.
State the hypothesis of the ANOVA and the Effect Tests tables. Interpret and explain how they help you in answering questions about advertising and promotion effects on sales.
Based on LS Means plots, what can you say about advertising and promotion effects on sales, of course subject to verification of statistical significance with p-values?
Interpret the interaction plot between Promotion and AdProgram.
Use LS Means Student's t to select significant differences between means corresponding to the interaction between AdProgram and Promotion. Check the confidence intervals as to whether they contain 0. Briefly summarize their meaning.
Stroke patients enter a rehabilitation hospital with the goal of regaining as much function as possible. This is accomplished through a variety of therapies. It is desirable to discharge patients to outpatient status as quickly as possible. The file StrokeTherapy.jmp contains data on length of stay and the amount of physical and occupational therapy received.
The Westville Meat Processing Plant has added a new product line that has required changes in the speeds at which the assembly lines are run. With rising health care costs, there is concern that an increase in line speed might lead to increases in accidents and workers' compensation costs. The number of injuries over the last year for the three shifts (day, evening, and night) and three line speeds (slow, medium, and fast) can be found in the data file Westvillelndustriallnjuries.jmp.
Perform the appropriate analysis of variance to address the issue.
Briefly summarize your findings. Include a statistical summary of the data, conclusions, and supporting statistical evidence, and the assumptions needed to make your findings valid.
