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An Introduction to Statistical Methods and Data Analysis

R. Lyman Ott, Michael Longnecker

Chapter 14

Analysis of Variance for Completely Randomized Designs - all with Video Answers

Educators

Chapter Questions

00:37

Problem 1

Researchers in child development are interested in developing ways to increase the spatialtemporal reasoning of preschool children. Spatial-temporal reasoning relates to the child's ability to visualize spatial patterns and mentally manipulate them over a time-ordered sequence of spatial transformations. This ability, often referred to as thinking in pictures, is important for generating and conceptualizing solutions to multistep problems and is crucial in early child development. The researchers want to design a study to evaluate which of several methods proposed to accelerate the growth in spatial-temporal reasoning yields the greatest increase in a child's development in this area. There are three methods proposed: taking piano lessons for 3 months, playing specially developed computer video games for 3 months, and playing specially designed games in small groups supervised by a trained instructor. The researchers measure the effectiveness of the three programs by assessing the children and assigning each one a reasoning score both before and after participation in the program. The difference in these two scores is the response variable. A control group is also included to measure the change in reasoning for children not given any special instruction. A pilot study with only 20 students was to be conducted prior to the complete study to determine potential problems. Demonstrate how to assign 5 of the 20 students to each of the four instructionalonditions-no instruction (control), piano lessons, computer video games, and instructor-so that the assignment is completely random.

Akhil Choudhary
Akhil Choudhary
Numerade Educator

Problem 2

Refer to Exercise 14.1. The researchers decide to use the following model, which relates the response variable $y$ to the four instructional conditions.
$$
y=\mu+\tau_i+\varepsilon_{i j} \text { for } i=1,2,3,4 \text { and } j=1,2,3,4,5
$$
a. Write an equation relating the mean reasoning score, $\mu_i$, to the parameters in the above model without any constraints on the model parameters.
b. Rewrite the equation relating the mean reasoning score, $\mu_i$, to the parameters in the above model after imposing the standard constraints placed on the model parameters.

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03:36

Problem 3

Refer to Exercise 14.1.After running the pilot study, the researchers conduct a study involving 100 students. Twenty-five students were randomly assigned to each of the four instructional conditions. The data are given here.
a. Conduct an analysis of variance, and summarize your results in an AOV table.
b. Test the research hypothesis that there is a difference in the effectiveness means of the methods of instruction. Use $\alpha=.05$.
c. Apply a multiple-comparison procedure to determine pairwise differences in the three instructional methods. Use $\alpha=.05$.
d. Was there significant evidence that all three methods of instruction produced higher mean reasoning scores than the mean reasoning score for the control?
(TABLE CAN'T COPY)

Sheryl Ezze
Sheryl Ezze
Numerade Educator

Problem 4

In order for the conclusions reached in Exercise 14.3 to be valid, the conditions of normality, equal variance, and independence must be satisfied. Use the residuals from the fitted model to assess the three conditions. (Refer to the discussion in Section 8.4.)
a. Was there significant evidence of a violation of the normality condition?
b. Was there significant evidence that the variance in reasoning scores was different for the three methods and the control?
c. What is the justification for concluding that the 100 reasoning scores are independent?
d. If the condition of normality and/or equal variance is violated, what are some alternative methods of analysis?

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Problem 5

The production manager of a large casting firm is studying different methods to increase productivity in the workforce of the company. The process engineer and personnel in the human resources department develop three new incentive plans (plans B, C, and D) and design a study to compare these incentive plans with the current plan (plan A). Twenty workers are randomly assigned to each of the four plans. The response variable is the total number of units produced by each worker during 1 month on the incentive plan. The data are given in the following table.
(TABLE CAN'T COPY)
a. State the null and alternative hypotheses being tested by the $F$ statistic in the AOV table.
b. Is there significant evidence $(\alpha=.05)$ that the mean output associated with the four incentive plans is different?
c. Use Tukey's $W$ procedure to identify the pairs of incentive plans that have different output means.

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Problem 6

In order for the conclusions reached in Exercise 14.5 to be valid, the conditions of normality, equal variance, and independence must be satisfied. Use the residuals from the fitted model to assess the three conditions. Refer to the discussion in Section 8.4.
a. Is there significant evidence of a violation of the normality condition?
b. Is there significant evidence that the variances in reasoning scores were different for the three methods and the control?
c. What is the justification for concluding that the 100 reasoning scores are independent?
d. If the condition of normality and/or equal variance is violated, what are some alternative methods of analysis?

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02:24

Problem 7

Refer to Exercise 14.5. When the normality condition is violated, an alternative to the $F$ test is the Kruskal-Wallis test (see Section 8.6).
a. Test for differences in the median outputs of the four incentive plans. Use $\alpha=.05$.
b. Why do you think the conclusions reached using the Kruskal-Wallis test differ from the conclusions reached using the $F$ test from the AOV table in Exercise 14.5?

Adriano Chikande
Adriano Chikande
Numerade Educator
03:40

Problem 8

A large advertising firm specializes in creating television commercials for children's products. The firm wants to design a study to investigate factors that may affect the lengths of time a commercial is able to hold a child's attention. A preliminary study determines that two factors that may be important are the age of the child and the type of product being advertised. The firm wants to determine whether there were large differences in the mean length of time that the commercial is able to hold the child's attention depending on these two factors. If there proves to be a difference, the firm would then attempt to determine new types of commercials depending on the product and targeted age group. Three age groups are used:
$$
A_1: 5-6 \text { years } \quad A_2: 7-8 \text { years } \quad A_3: 9-10 \text { years }
$$

The types of products selected are
$$
P_1 \text { : breakfast cereals } \quad P_2 \text { : video games }
$$

A group of 30 children is recruited in each age group, and 10 are randomly assigned to watch a 60 -second commercial for each of the two products. Researchers record their attention spans during the viewing of the commercial. The data are given here.
$$
\begin{array}{ccccccc}
\hline \text { Child } & \mathbf{A}_1-\mathbf{P}_{\mathbf{1}} & \mathbf{A}_2-\mathbf{P}_{\mathbf{1}} & \mathbf{A}_3-\mathbf{P}_{\mathbf{1}} & \mathbf{A}_{\mathbf{1}}-\mathbf{P}_{\mathbf{2}} & \mathbf{A}_{\mathbf{2}}-\mathbf{P}_{\mathbf{2}} & \mathbf{A}_3-\mathbf{P}_{\mathbf{2}} \\
\hline 1 & 19 & 19 & 37 & 39 & 30 & 51 \\
2 & 36 & 35 & 6 & 18 & 47 & 52 \\
3 & 40 & 22 & 28 & 32 & 6 & 43 \\
4 & 30 & 28 & 4 & 22 & 27 & 48 \\
5 & 4 & 1 & 32 & 16 & 44 & 39 \\
6 & 10 & 27 & 16 & 2 & 26 & 33 \\
7 & 30 & 27 & 8 & 36 & 33 & 56 \\
8 & 5 & 16 & 41 & 43 & 48 & 43 \\
9 & 34 & 3 & 29 & 7 & 23 & 40 \\
10 & 21 & 18 & 18 & 16 & 21 & 51 \\
\text { Mean } & 22.9 & 19.6 & 21.9 & 23.1 & 30.5 & 45.6 \\
\hline
\end{array}
$$
a. Identify the design.
b. Write a model for this situation, identifying all the terms in the model.
c. Estimate the parameters in the model.
d. Compute the sum of squares for the data, and summarize the information in an AOV table.

Jerelyn Nevil
Jerelyn Nevil
Numerade Educator
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Problem 9

Refer to Exercise 14.8 .
a. Draw a profile plot for the two factors, age and product type.
b. Perform appropriate $F$ tests and draw conclusions from these tests concerning the effects of age and product type on the mean attention spans of the children.

Shu Naito
Shu Naito
Numerade Educator

Problem 10

Refer to Exercise 14.8 .
Use residual plots to determine whether any of the conditions required for the validity of the $F$ tests have been violated.

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Problem 11

Commercially produced ice cream is made from a mixture of ingredients:
- A minimum of $10 \%$ milk fat
- $9-12 \%$ milk solids: this component, also known as the serum solids, contains the proteins (caseins and whey proteins) and carbohydrates (lactose) found in milk
- $12-16 \%$ sweeteners: usually a combination of sucrose and/or glucose-based corn syrup sweeteners
- $0.2-0.5 \%$ stabilizers and emulsifiers-e.g., agar or carrageenan extracted from seaweed
- $55 \%-64 \%$ water, which comes from milk solids or other ingredients

Air is incorporated with the above ingredients during the mixing process. Less-expensive ice creams contain lower-quality ingredients, and more air is incorporated during the mixing process. The finest ice creams have between $3 \%$ and $15 \%$ air. Because most ice cream is sold by volume, it is economically advantageous for producers to reduce the density of the product in order to cut costs. A food scientist is investigating how varying the amounts of the above ingredients impacts the sensory rating of the final product. The scientist decides to use three levels of milk fat: $10 \%, 12 \%, 15 \%$; three amounts of air: $5 \%, 10 \%, 15 \%$; and two levels of sweeteners: $12 \%, 16 \%$. Three replications of each of the formulations were produced and the sensory ratings ( $0-40$ ) obtained; a higher number implies a more favorable sensory rating. The data are given here.
(TABLE CAN'T COPY)
a. Identify the design and treatment structure for this study.
b. Write a model for this study, identifying all the terms in the model.
c. For each of the two levels of sweetener, draw profile plots of the effects of the percentages of air and milk fat on the sensory rating of ice cream.
d. From the profile plots, does there appear to be a three-way interaction among the effects of the percentages of sweetener, air, and milk fat on the mean sensory ratings?

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04:55

Problem 12

Refer to the study described in Exercise 14.11.
a. Perform appropriate $F$ tests and draw conclusions from these tests concerning the effects of the percentages of sweetener, air, and milk fat on the sensory ratings. Use $\alpha=.05$.
b. Are the conclusions from the $F$ tests consistent with your observations from the profile plots?

Raymond Matshanda
Raymond Matshanda
Numerade Educator

Problem 13

Refer to the study described in Exercise 14.11. Use the residuals from the fitted model to answer the following questions.
a. Is there significant evidence that the residuals have a nonnormal distribution?
b. Is there significant evidence that the residuals do not have constant variances?
c. How could we assess whether or not the residuals are independently distributed?

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01:20

Problem 14

Refer to the study described in Exercise 14.8. Use Tukey's $W$ procedure to identify significant differences in the means.
a. Use Tukey's $W$ procedure to identify significant differences in the mean attention spans of the three age groups of children.
b. Use Tukey's $W$ procedure to identify significant differences in the mean attention spans for the types of products.
c. Are your conclusions in part (a) the same for both types of products?

Manik Pulyani
Manik Pulyani
Numerade Educator
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Problem 15

Refer to the study described in Exercise 14.11.
a. Use Tukey's $W$ procedure to identify significant differences in the mean sensory ratings of the three levels of percentage of milk fat.
b. Use Tukey's $W$ procedure to identify significant differences in the mean sensory ratings of the three levels of percentage of air.
c. Which combination of percentage of milk fat, air, and sweetener appears to yield the highest mean sensory rating?

Shu Naito
Shu Naito
Numerade Educator
01:18

Problem 16

A state legislature mandates the each school district in its state must conduct an audit of the performance of the district's students on the state reading exam. The purpose is to determine if there are any extreme increases in the individual schools in the district. There are currently four software programs that are capable of conducting the audits with varying degrees of efficiency. The state board of education hires an analyst to design a study to evaluate each of the software programs. The study will involve a random sample of schools running the software on their records. One of the metrics in the evaluation will be the amount of time that the software takes to complete the audit. From the application of the software in other states, the standard deviation in the time to complete the audit was 122.5 minutes. Determine how many schools are required in the study for each software program in order to be able to detect a difference in any pair of software programs of 5 hours using a level .05 test with a power of $90 \%$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 17

A researcher seeks funding for a study from a federal agency. The study will involve the evaluation of three factors, each having two levels. From the literature, the researcher approximates the standard deviation in the responses to be approximately 9 units. How many experimental units should be included in the budget for the study so that a difference of 20 units or more in any pair of treatment means will be detected with a probability of .80 using an $\alpha=.05$ test?

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Problem 18

Refer to the study described in Exercise 14.8. Determine the number of replications needed to obtain an $\alpha=.05$ test having power of at least $80 \%$ that detects a difference of 10 in any pair of treatment means. Use the data from Exercise 14.8 to estimate the value of $\sigma_e^2$.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 19

Refer to the study described in Exercise 14.11. Suppose a new study is to be designed in which only three levels of milk fat and three levels of air will be used. Determine the number of replications needed to obtain an $\alpha=.05$ test having a power of at least $90 \%$ that detects a difference of 5 in any pair of treatment means. Use the data from Exercise 14.11 to estimate the value of $\sigma_e^2$.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 20

A study was conducted to compare the effect of four manganese rates (from $\mathrm{MnSO}_4$ ) and four copper rates (from $\mathrm{CuSO}_4 5 \mathrm{H}_2 \mathrm{O}$ ) on the yield of soybeans. A large field was subdivided into 32 separate plots. Two plots were randomly assigned to each of the 16 factor-level combinations (treatments) and the treatments were applied to the designated plots. Soybeans were then planted over the entire field in rows 3 feet apart. The yields from the 32 plots are given here (in kilograms/hectare).
(TABLE CAN'T COPY)
a. Identify the design for this experiment.
b. Write an appropriate statistical model for this experiment.
c. Construct a profile plot and describe what this plot says about the effect of Mn and Cu on soybean yield.

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Problem 21

Refer to Exercise 14.20.
a. Test for an interaction between the effects of Mn and Cu on soybean yield. Use $\alpha=.05$.
b. What level of Mn appears to produce the highest yield?
c. What level of Cu appears to produce the highest yield?
d. What combination of $\mathrm{Cu}-\mathrm{Mn}$ appears to produce the highest yield?

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Problem 22

Suppose we have a completely randomized three-factor factorial experiment with levels $3 \times 4 \times 6$, with three replications of each of the 72 treatments. Assume that the three-way interaction is not significant.
a. Write a model to describe the response $y_{i j k m}$ for this type of experiment.
b. Provide a complete AOV table for this type of experiment.
c. Sketch three profile plots to depict the following three two-way interactions: $F_1 * F_2$ significant but orderly, $F_2 * F_3$ nonsignificant, and $F_1 * F_3$ significant and disorderly.

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09:48

Problem 23

An experiment was set up to compare the effects of different soil pH and calcium additives on the increase in trunk diameters for orange trees. Elemental sulfur, gypsum, soda ash, and other ingredients were applied annually to provide pH value levels of $4,5,6$, and 7 . Three levels of a calcium supplement $(100,200$, and 300 pounds per acre) were also applied. All factor-level combinations of these two variables were used in the experiment. At the end of a 2-year period, three diameters were examined at each factor-level combination. The data appear next.
(TABLE CAN'T COPY)
a. Construct a profile plot. What do the data suggest?
b. Write an appropriate statistical model.
c. Perform an analysis of variance and identify the experimental design. Use $\alpha=.05$.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:22

Problem 24

Refer to Exercise 14.23.
a. Test for interactions and main effects. Use $\alpha=.05$.
b. What can you conclude about the effects of pH and calcium on increases in mean trunk diameter for orange trees?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:22

Problem 25

Refer to Exercise 14.23.
a. Use Tukey's $W$ procedure to determine differences in mean increases in trunk diameter among the three calcium rates. Use $\alpha=.05$.
b. Are your conclusions about the differences in mean increases in diameter among the three calcium rates the same for all four pH values?

Sheryl Ezze
Sheryl Ezze
Numerade Educator

Problem 26

Refer to Exercise 14.23.
a. Use residual analysis to determine whether any of the conditions required to conduct an appropriate $F$ test have been violated.
b. If any of the conditions have been violated, suggest ways to overcome these difficulties.

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03:52

Problem 27

Researchers conducted an experiment to compare the average oral body temperatures for persons taking one of nine different medications often prescribed for high blood pressure. The researchers were concerned that the effect of the drug may be different depending on the severity of the patient's high blood pressure disorder. Patients with high blood pressure who satisfied the study's entrance criteria were classified into one of the three levels of severity of the blood pressure disorder. The patients were then randomly assigned to receive one of the nine medications. Each patient in the study was given the assigned medication at 6:00 A.M. of the designated study day. Temperatures were taken at hourly intervals beginning at 8:00 A.m. and continuing for 10 hours. During this time, the patients were not allowed to do any physical activity and had to lie in bed. To eliminate the variability of temperature readings within a day, the average of the hourly determinations was the recorded response for each patient. These data are given in the accompanying table.
a. Identify the design for this experiment.
b. Write an appropriate statistical model and identify the parameters of the model.
(TABLE CAN'T COPY)

Joshua Argo
Joshua Argo
Numerade Educator
02:59

Problem 28

Refer to Exercise 14.27.
a. Construct an AOV table for the experiment.
b. Are the differences in mean temperatures for the nine medications the same for all three severities of the blood pressure disorders? Use $\alpha=.05$.
c. Is there a significant difference in mean temperatures for medications and severity of the disorder? Use $\alpha=.05$.
d. Use a profile plot to assist in discussing your conclusions concerning the effects of medication and severity on the mean temperatures of the patients.

James Kiss
James Kiss
Numerade Educator
01:42

Problem 29

A physician was interested in examining the relationship between the work performed by individuals in an exercise tolerance test and the excess weight (as determined by standard weight-height tables) they carried. To do this, a random sample of 28 healthy adult females, ranging in age from 25 to 40 , was selected from the community clinic during routine visits for physical examinations. The selection process was restricted so that seven persons were selected from each of the following weight classifications:
Normal weight (less than $10 \%$ underweight)
$1 \%-10 \%$ overweight
$11 \%-20 \%$ overweight
More than $20 \%$ overweight

As part of the physical examination, each person was required to exercise on a bicycle ergometer until the onset of fatigue. The time to fatigue (in minutes) was recorded for each person. The data are given next.
(TABLE CAN'T COPY)
a. Identify the experimental design and write an appropriate statistical model.
b. Use $\alpha=.05$ and perform an analysis of variance.

Adriano Chikande
Adriano Chikande
Numerade Educator

Problem 30

Refer to Exercise 14.29.
a. How would you design an experiment to investigate the effects of age, gender, and excess weight on fatigue time?
b. Suppose the physician wanted to investigate the relationship among the quantitative variables percentage overweight, age, and fatigue time. Write a possible model.

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Problem 31

An experiment was conducted to investigate the heat loss for five different designs for commercial thermal panes. The researcher, in order to obtain results that would be applicable throughout most regions of the country, decided to evaluate the panes at five temperatures: $0^{\circ} \mathrm{F}$, $20^{\circ} \mathrm{F}, 40^{\circ} \mathrm{F}, 60^{\circ} \mathrm{F}$, and $80^{\circ} \mathrm{F}$. A sample of 10 panes of each design was obtained. Two panes of each design were randomly assigned to each of the five exterior temperature settings. The interior temperature of the test was controlled at $70^{\circ} \mathrm{F}$ for all five exterior temperatures. The heat losses associated with the five pane designs are given here.
(TABLE CAN'T COPY)
a. Identify the experimental design and write an appropriate statistical model.
b. Is there a significant difference in the mean heat losses of the five pane designs? Use $\alpha=.05$.
c. Are the differences in the five designs consistent across the five temperatures? Use $\alpha=.05$ and a profile plot in reaching your conclusion.
d. Use Tukey's $W$ procedure at an $\alpha=.05$ level to compare the mean heat losses for the five pane designs.

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Problem 32

An experiment was conducted to examine the effects of different levels of reinforcement and different levels of isolation on children's ability to recall. A single analyst was to work with a random sample of 36 children selected from a relatively homogeneous group of fourth-grade students. Two levels of reinforcement (none and verbal) and three levels of isolation (20,40, and 60 minutes) were to be used. Students were randomly assigned to the six treatment groups, with a total of six students being assigned to each group.

Each student was to spend a 30 -minute session with the analyst. During this time, the student was to memorize a specific passage, with reinforcement provided as dictated by the group to which the student was assigned. Following the 30 -minute session, the student was isolated for the time specified for his or her group and then tested for recall of the memorized passage. The data appear next.
(TABLE CAN'T COPY)
a. What can you conclude about the effects of level of reinforcement and time of isolation on the average recall test score?
b. Verify that the conditions needed to validly apply your tests in part (a) are not violated.

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Problem 33

Researchers were interested in the stability of a drug product stored for four lengths of time ( $1,3,6$, and 9 months). The drug was manufactured with $30 \mathrm{mg} / \mathrm{mL}$ of the active ingredient of a drug product, and the amount of the active ingredient in the drug at the end of the storage period was to be determined. The drug was stored at a constant temperature of $30^{\circ} \mathrm{C}$. Two laboratories were used in the study, with three $2-\mathrm{mL}$ vials of the drug randomly assigned to each of the four storage times. At the end of the storage time, the amount of the active ingredient was determined for each of the vials. A measure of the pH of the drug was also recorded for each vial. The data are given here.
$$
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline \begin{array}{l}
\text { Time } \\
\text { (in months } \\
\text { at } 30^{\circ} \mathrm{C} \text { ) }
\end{array} & \text { Laboratory } & \begin{array}{l}
\mathrm{mg} / \mathrm{mL} \\
\text { of Active } \\
\text { Ingredient }
\end{array} & \mathbf{p H} & \begin{array}{l}
\text { Time } \\
\text { (in months } \\
\text { at } 30^{\circ} \mathrm{C} \text { ) }
\end{array} & \text { Laboratory } & \begin{array}{l}
\mathrm{mg} / \mathrm{mL} \\
\text { of Active } \\
\text { Ingredient }
\end{array} & \mathrm{pH} \\
\hline 1 & 1 & 30.03 & 3.61 & 1 & 2 & 30.12 & 3.87 \\
\hline 1 & 1 & 30.10 & 3.60 & 1 & 2 & 30.10 & 3.80 \\
\hline 1 & 1 & 30.14 & 3.57 & 1 & 2 & 30.02 & 3.84 \\
\hline 3 & 1 & 30.10 & 3.50 & 3 & 2 & 29.90 & 3.70 \\
\hline 3 & 1 & 30.18 & 3.45 & 3 & 2 & 29.95 & 3.80 \\
\hline 3 & 1 & 30.23 & 3.48 & 3 & 2 & 29.85 & 3.75 \\
\hline 6 & 1 & 30.03 & 3.56 & 6 & 2 & 29.75 & 3.90 \\
\hline 6 & 1 & 30.03 & 3.74 & 6 & 2 & 29.85 & 3.90 \\
\hline 6 & 1 & 29.96 & 3.81 & 6 & 2 & 29.80 & 3.90 \\
\hline 9 & 1 & 29.81 & 3.60 & 9 & 2 & 29.75 & 3.77 \\
\hline 9 & 1 & 29.79 & 3.55 & 9 & 2 & 29.85 & 3.74 \\
\hline 9 & 1 & 29.82 & 3.59 & 9 & 2 & 29.80 & 3.76 \\
\hline
\end{array}
$$
a. Write a model relating the pH measured on each vial to the factors of length of storage time and laboratory.
b. Display an analysis of variance table for the model of part (a).

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09:04

Problem 34

Refer to Exercise 14.33. Obtain an analysis of variance for both dependent variables (i.e., $y_1=\mathrm{mg} / \mathrm{mL}$ of active ingredient and $y_2=\mathrm{pH}$ ). Draw conclusions about the stability of these $2-\mathrm{mL}$ vials based on these analyses. Use $\alpha=.05$.

Raymond Matshanda
Raymond Matshanda
Numerade Educator

Problem 35

A manufacturer whose daily supply of raw materials is variable and limited can use the material to produce two different products in various proportions. The profit per unit of raw material obtained by producing each of the two products depends on the length of a product's manufacturing run and hence on the amount of raw material assigned to it. Other factors-such as worker productivity, machine breakdown, and so on-can affect the profit per unit as well, but their net effect on profit is random and uncontrollable. The manufacturer has conducted an experiment to investigate the effects of the level of supply of raw material, $S$, and the ratio of its assignment, $R$, to the two product manufacturing lines on the profit per unit of raw material. The ultimate goal is to be able to choose the best ratio, $R$, to match each day's supply of raw materials, $S$. The levels of supply of the raw material chosen for the experiment were 15,18 , and 21 tons. The levels of the ratio of allocation to the two product lines were $1 / 2,1$, and 2 . The response was the profit (in cents) per unit of raw material supply obtained from a single day's production. Three replications of each combination were conducted in a random sequence. The data for the 27 days are shown in the following table.
(TABLE CAN'T COPY)
a. Draw conclusions from an analysis of variance table. Use $\alpha=.05$.
b. Identify the two best combinations of $R$ and $S$. Are these two combinations significantly different? Use a procedure that limits the error rate of all pairwise comparisons of combinations to be no more than 0.05 .

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Problem 36

A horticulturalist at a large research institution designs a study to evaluate the effect on tomato yields of water loss due to transpiration. She decides to examine four levels of shading of the tomato plants at three stages of the tomato plant's development. The four levels of shading $(0,25 \%, 50 \%$, and $75 \%)$ were selected to reduce the solar exposure of the plants. The shading remained in place for 20 days during the early, middle, and late phases of the tomato plants' growth. There were four plots of tomatoes randomly assigned to each of the combinations of shading and growth stage. At the end of the study, the yields per plot in pounds were recorded. However, due to a problem in the harvesting of the tomatoes, a few of the plot yields were not recorded.
(TABLE CAN'T COPY)
a. Identify the design for this experiment.
b. Construct an AOV table for the experiment, and test for the main effects of shading and growth stage and an interaction between shading and growth stage.
c. Is there a linear trend in the mean yields across the levels of percent shading?
d. Which level of shading would you recommend for maximum yield?
e. During which growth stage would you apply the shading?

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Problem 37

Refer to Exercise 14.36 .
a. Are the computational formulas for obtaining the sum of squares appropriate for the data in the tomato experiment? Justify your answer.
b. Verify that there are no major violations in the conditions necessary to conduct the $F$ tests in the AOV table.
c. Write a linear model for this experiment, and estimate all the terms in your model using the data in Exercise 14.36.

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Problem 38

The following experiment is from Kuehl (2000). Sludge is a dried product remaining from processed sewage; it contains nutrients beneficial to plant growth. It can be used for fertilizer on agricultural crops provided it does not contain toxic levels of certain elements such as heavy metals (such as zinc, not rock groups). Typically, the levels of metals in sludge are assayed by growing plants in media containing different doses of the sludge.

A soil scientist hypothesized the concentration of certain heavy metals in sludge would differ among the metropolitan areas from which the sludge was obtained. The variation could result from any number of reasons, including the different industrial bases surrounding the areas and the efficiency of the various sewage treatment facilities. If this was true, then recommendations for applications on crops would have to be preceded by knowledge about the source of the sludge material. An assay was planned to determine whether there was significant variation in heavy metal concentrations among diverse metropolitan areas.

The investigator obtained sewage sludge from treatment plants located in three different metropolitan areas. Barley plants were grown in a sand medium to which sludge was added as fertilizer. The sludge was added to the sand at three different rates: $0.5,1.0$, and 1.5 metric tons/acre. Each of the nine treatment combinations was randomly assigned to four replicate containers. The containers were arranged completely at random in a growth chamber. At a certain stage of growth, the zinc contents in parts per million were determined for the barley plants grown in each of the containers. The data are given below.
(TABLE CAN'T COPY)
a. Identify the design for this experiment.
b. Write a model for this study. Identify all the terms in your model and any conditions that are placed on the terms.
c. Display estimates of all the parameters in your model.

Victor Salazar
Victor Salazar
Numerade Educator
08:10

Problem 39

Refer to Exercise 14.38.
a. Construct an AOV table for the experiment, and test for the main effects of sludge rate and source of sludge and an interaction between sludge rate and source of sludge.
b. Is there a linear trend in the mean yields across the sludge rates?
c. Which pairs of sludge rates have significant differences in their mean zinc contents?

Raymond Matshanda
Raymond Matshanda
Numerade Educator

Problem 40

Refer to Exercise 14.38. Verify that there are no major violations in the conditions necessary to conduct the tests in the AOV table.

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