Exam 9: Design of Experiments and Analysis of Variance

Given that the sum of squares for treatments (SST) for an ANOVA F-test is 9,000 and there are four total treatments, find the mean square for treatments (MST).

The goal of an experiment is to investigate the factors that affect visitor travel time in a complex, multilevel building on campus. Specifically, we want to determine whether the effect of directional aid (wall signs or map) on travel time depends on starting room location (interior or exterior). Three visitors were assigned to each of the combinations of directional aid and starting room location, and the travel times of each (in seconds) to reach the goal destination room were recorded. DIRECTIONAL AID STARTING Interior ROOM Exterior Wall signs 122 242 101 332 224 145 Map 126 246 106 326 236 147 Explain how to properly analyze these data.

An article in a journal examined the attitudes of three groups of professionals on the condition of the environment, and quantified their responses on a seven-point scale ( $1 =$ no problem to 7 = disaster very likely). The mean scores for the groups are: A: $4.90 , B : 4.55$ , and C: $4.51$ . Using an experimentalwise error rate of $\alpha = .05$ , Tukey's minimum significance for comparing means is $0.29$ . Use this information to conduct a multiple comparisons of the means.

Psychologists have found that people are generally reluctant to transmit bad news to their peers. This phenomenon has been named the "MUM effect." To investigate the cause of the MUM effect, undergraduates at a university participated in an experiment. Each subject was asked to administer an IQ test to another student and then provide the test taker with his or her percentile score. (Unknown to the subject, the test taker was a bogus student who was working with the researchers.) The experimenters manipulated two factors, subject visibility and success of test taker, each at two levels. Subject visibility was either visible or not visible to the test taker. Success of test taker was either top 20% or bottom 20%. Ten subjects were randomly assigned to each of the 2 x 2 $2 \times 2 = 4$ xperimental conditions. Then the time (in seconds) between the end of the test and the delivery of the percentile score from the subject to the test taker was measured. (This variable is called the latency to feedback.) Describe the experiment, including the response variable, factors, factor levels, replications, and treatments.

A certain HMO is attempting to show the benefits of managed health care to an insurance company. The HMO believes that certain types of doctors are more cost-effective than others. One theory is that both primary specialty and whether the physician is a foreign or USA medical school Graduate are an important factors in measuring the cost-effectiveness of physicians. To investigate this, the president obtained independent random samples of 40 HMO physicians, half foreign graduates and half USA graduates, from each of four primary specialties-General Practice (GP), Internal Medicine (IM), Pediatrics (PED), and Family Physician (FP)-and recorded the total per-member, per-month charges for each. Thus, information on charges were obtained for a total of $n = 160$ doctors. The sample mean charges for each of the eight categories are shown in the table Primary Specialty Foreign Grad USA Grad GP 40.10 42.50 IM 51.80 49.50 PED 23.60 25.40 FP 35.60 39.40 What type of design was used for this experiment?

An economist is investigating the impact of today's economy on workers in the manufacturing industry who have been laid off. A sample of 50 workers was randomly selected from all workers in manufacturing that have been laid off in the past year. The following variables were measured for each laid off worker: length of time jobless (number of weeks) and tax status (single, married, or married/head of household). The data for the 50 workers were entered into the computer and analyzed to determine if the mean number of weeks jobless differed for the three tax status groups. The Tukey multiple comparison printout is shown below: Tukey HSD All-Pairwise Comparisons Test of JOBLESS by STATUS STATUS Mean Tukey Groups Married 50.375 A Single 48.000 A Mar/Head 33.789 B Alpha 0.1 Critical Q Value 2.975 Alpha 0.1 Critical Q Value $2.975$ Give the population mean(s) which are in the statistically smallest group.

Given that the sum of squares for error (SSE) for an ANOVA F-test is 12,000 and there are 40 total experimental units with eight total treatments, find the mean square for error (MSE).

Given that the mean square for treatments (MST) for an ANOVA F-test is 5,000 and the mean square for error (MSE) is 3,750, find the value of the test statistic F.

A local consumer reporter wants to compare the average costs of grocery items purchased at three different supermarkets, A, B, and C. Prices (in dollars) were recorded for a sample of 60 randomly selected grocery items at each of the three supermarkets. In order to reduce item-to-item variation, the prices were recorded for each item on the same day at each supermarket. The results of the Bonferroni analysis are summarized below. Supermarket Mean Price 1.66 1.80 1.94 Fully interpret the Bonferroni analysis.

When a variable is identified as reducing variation in the response variable, but no additional knowledge concerning the variable is desired, it should be used as the blocking factor in the randomized block design.

In a study to determine the least amount of time necessary to clean an SUV while maintaining a high quality standard, the owner of a chain of car washes designed an experiment where 20 employees were divided into four groups, each with five members. Each member of each group was assigned an SUV to clean within a certain time limit. The time limits for the groups were 20 minutes, 25 minutes, 30 minutes, and 35 minutes. After the time limits for each group had expired, the owner inspected each SUV and rated the quality of the cleaning job on a scale of 1 to 10. What are the factor levels for this study?

$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\underline{\text { FACTOR B }}$ Level 1 2 3 FACTOR A 1 4.1,4.1 5.0,5.2 6.3,6.1 2 5.8,5.6 5.0,5.4 8.8,9.0 a. Calculate the mean response for each treatment b. The MINITAB ANOVA printout is shown here. Test for interaction at the $a$ =0.05 level of significance. Analysis of variance for response. Source df SS MS F A 1 0.53777 0.53777 0.11851 B 2 5.02708 2.51334 0.55391 AB 2 13.49334 6.74667 1.48678 Error 6 27.22667 4.53778 Total 11 46.28486 c. Does the result warrant tests of the two factor mean effects?

A counselor obtains SAT averages for incoming freshmen each year for a period covering 12 years, with the objective of determining the relationship between the SAT score and the year the test was given. The averages are then subjected to analysis for the purpose of drawing a conclusion regarding a trend. Determine whether the study is observational or designed.

A certain HMO is attempting to show the benefits of managed health care to an insurance company. The HMO believes that certain types of doctors are more cost-effective than others. One theory is that primary specialty is an important factor in measuring the cost-effectiveness of physicians. To investigate this, the HMO obtained independent random samples of 20 HMO physicians from each of four primary specialties-General Practice (GP), Internal Medicine (IM), pediatrics (PED), and Family Physician (FP)- and recorded the total per-member, per-month charges for each. In order to compare the mean charges for the four specialty groups, the data were be subjected to a one-way analysis of variance. The results of the Tukey analysis are summarized below. Group Sample Mean IM 55.9 GP 41.4 FP 40.00 PED 21.50 Which primary specialties have significantly lower mean charges than Internal Medicine (IM)?

A certain HMO is attempting to show the benefits of managed health care to an insurance company. The HMO believes that certain types of doctors are more cost-effective than others. One theory is that both primary specialty and whether the physician is a foreign or USA medical school graduate are an important factors in measuring the cost-effectiveness of physicians. To investigate this, the president obtained independent random samples of $40 \mathrm { HMO }$ physicians, half foreign graduates and half USA graduates, from each of four primary specialties-General Practice (GP), Internal Medicine (IM), Pediatrics (PED), and Family Physician (FP) - and recorded the total per-member, per-month charges for each. Thus, information on charges were obtained for a total of $n = 160$ doctors. The ANOVA results are summarized in the following table. Source df SS MS F Value Prob > F Specialty 3 22855 7618 60.94 .0001 Medschool 1 105 105 0.84 .6744 Interaction 3 890 297 2.38 .1348 Error 152 18950 125 Total 159 42800 Interpret the test for interaction shown in the ANOVA table. Use $\alpha = 0.025$ .

Which method generally produces wider confidence intervals?

A multiple-comparison procedure for comparing four treatment means produced the confidence intervals shown below. For each pair of means, indicate which mean is larger or indicate that there is no significant difference. - :(8,20) - :(-7,3) - :(9,21) - :(-21,-11) - :(-5,7) - :(12,22)

Psychologists have found that people are generally reluctant to transmit bad news to their peers. This phenomenon has been named the "MUM effect." To investigate the cause of the MUM effect, 40 undergraduates at a university participated in an experiment. Each subject was asked to administer an IQ test to another student and then provide the test taker with his or her percentile score. (Unknown to the subject, the test taker was a bogus student who was working with the researchers.) The experimenters manipulated two factors, subject visibility and success of test taker, each at two levels. Subject visibility was either visible or not visible to the test taker. Success of test taker was either top 20% or bottom 20%. Ten subjects were randomly assigned to each of the 2 x 2 $2 \times 2 = 4$ 4 experimental conditions, then the time (in seconds) between the end of the test and the delivery of the percentile score from the subject to the test taker was measured. (This variable is called the latency to feedback.) The data was subject to an analysis of variance, with the following results: Source df SS MS F PR >F Subject visibility 1 1380.24 1380.24 4.25 0.430 Test taker success 1 1325.16 1325.16 -1371.89 0.05 Visibility x success 1 3385.80 3385.80 10.45 .002 Error 36 11,664.00 324.00 Total 39 17,755.20 Is there evidence to indicate that subject visibility and test taker success interact? $\text { Use } \alpha = 01 .$

Four different leadership styles used by Big-Six accountants were investigated. As part of a designed study, 15 accountants were randomly selected from each of the four leadership style groups (a total of 60 accountants). Each accountant was asked to rate the degree to which their subordinates performed substandard field work on a 10-point scale-called the "substandard work scale". The objective is to compare the mean substandard work scales of the four leadership styles. The data on substandard work scales for all 60 observations were subjected to an analysis of variance. ONE-WAY ANOVA FOR SUBSTAND BY STYLE SOURCE DF SS MS F P BETWEEN 3 2728.17 909.390 5.210 0.003 WITHIN 56 9774.63 174.547 TOTAL 59 12,502.80 Interpret the results of the ANOVA $F$ -test shown on the printout for $\alpha = 0.05$ .

Suppose an experiment utilizing a random block design has 5 treatments and 8 blocks for a total of 40 observations. Assume that the total Sum of Squares for the response is SS(Total)=300 . If the Sum of Squares for Treatments (SST) is 40 % of SS(Total), and the Sum of Squares for Blocks (SSB) is 10 % of SS (Total), find the F values for this experiment.