Test Bank For Intermediate Statistical I nvestigations 1e Nathan Tintle, Beth L. Chance, Karen Mc Gaughey, Soma Roy, Todd Swa nson, Jill VanderStoep
Chapter 1-6 Chapter 1 Intermediate Statistical Investigations Test Bank Question types:
FIB = Fill in the blank Ma = Matching MC = Multiple choice
Calc = Calculation MS = Multiple select TF = True-false
CHAPTER 1 TERMINAL LEARNING OUTCOMES TLO1-1: Apply the six-step investigative process in the context of a well-designed experiment. TLO1-2: Partitioning variation in the response variable into variation explained by the model and unexplained variation, and measuring and reporting the percentage of variation explained TLO1-3: Assess the statistical significance of the difference between two groups on a quantitative response variable using both simulation and theory-based approaches TLO1-4: Compare more than two treatments on a quantitative response using both simulation and theory-based approaches TLO1-5: Apply Post-hoc analysis after significant F-test (pairwise differences, as well as confidence and prediction intervals for single means) TLO1-6: Understand statistical power and how it is impacted by sample size, variability within groups, number of groups, and significance level
Section 1.1: Sources of Variation in an Experiment LO1.1-1: Apply the six-step investigative process. LO1.1-2: Distinguish experiments and observational studies. LO1.1-3: Review basic study design principles such as inclusion criteria and random assignment. LO1.1-4: Define terminology specific to an experimental study (e.g., treatments). LO1.1-5: Produce a Sources of Variation diagram for an experiment. Questions 1 through 3: A study published in Psychological Science in 2007 examined a possible link between mindset and health. The following is an excerpt from the abstract of the article: ―84 female room attendants working in seven different hotels were measured on physiological health variables affected by exercise. Those in the informed condition were told that the work they do (cleaning hotel rooms) is good exercise and satisfies the Surgeon General's recommendations for an active lifestyle. Examples of how their work was exercise were provided. Subjects in the control group were not given this information.‖ 1. Identify the experimental units in this study. A. The eighty-four room attendants FOR INSTRUCTOR USE ONLY
1-2 B. The seven different hotels C. The physiological health variables D. The two groups (informed and control) Ans: A; LO: 1.1-4; Difficulty: Easy; Type: MC 2. The researchers chose to include room attendants from seven different hotels (as opposed to using stricter inclusion criteria that would limit the study to room attendants at one particular hotel). Describe the consequences of this decision. Using broader inclusion criteria may ______ (increase/decrease) the amount of variation in the observed health outcomes. However, this decision also __________ (supports/limits) generalizability to a larger population of room attendants. Ans: increase, supports; LO: 1.1-3; Difficulty: Medium; Type: FIB 3. Room attendants were randomly assigned to either the informed condition or the control group. What is the most important reason for the random assignment? A. Random assignment ensures that the study is double-blind. B. Random assignment reduces the impact of outliers. C. Random assignment creates two groups of room attendants that are as similar as possible, which supports cause-and-effect conclusions. D. Random assignment makes it possible to generalize the results to the population. Ans: C; LO: 1.1-3; Difficulty: Medium; Type: MC Questions 4 through 6: An online retailer is using an experiment to decide whether to modify their website. When visitors type in the web address or click a link to the site, they are randomly re-directed to one of two versions of the website: the version that has been in use for the last year (version A) or an updated version (version B). The retailer’s goal is to maximize the amount of time (in minutes) visitors stay on the site. 4. Identify the experimental units and variables. Note: One of the answer choices will not be used. Experimental units: Explanatory variable: Response variable:
A. Version of the website (A and B) B: Online retailers C: Visitors to the website D: Time spent on the website (in minutes)
Ans: Experimental units: C, Explanatory variable: A, Response variable: D; LO: 1.1-4; Difficulty: Easy; Type: Ma 5. Consider two possible models for analyzing time spent on this retailer’s website. Single-mean model: A. Predicted time spent on site 12.33, SE of residuals 4.64 Separate-means model: FOR INSTRUCTOR USE ONLY
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9.9 forVersion A , SE of residuals 3.98 14.7 forVersion B
B. Predicted time spent on site
Does the version of the website appear to explain any of the variation in time spent on the site? Note: If more than one of these justifications is appropriate, select multiple answers. A. Yes, because the mean time spent on the site is higher for Version B than for Version A. B. Yes, because the SE of the residuals is smaller for the separate-means model than for the single-mean model. C. No, because the mean time spent on the site is not the same for Version A and for Version B. D. No, because the SE of the residuals is smaller for the separate-means model than for the single-mean model. Ans: A, B; LO: 1.1-1; Difficulty: Medium; Type: MS 6. The researcher decides that the difference between Version A and Version B in this study is meaningful. Is it reasonable to generalize these results to all customers of this retailer? A. Yes, because visitors to the website were randomly assigned to either Version A or Version B. B. Yes, because the study’s inclusion criteria would exclude potential subjects who are not customers. C. It depends whether visitors to the website knew about the research question being investigated. The study may not be double-blind. D. It depends who visited the website during the study period. The sample may not be representative. Ans: D; LO: 1.1-1; Difficulty: Medium; Type: MC Questions 7 through 8: Researchers at a university were interested in the effectiveness of a calculus workshop program for students who fail Calculus I and need to retake the course. As part of the study, students who were retaking Calculus I were allowed to enroll in a calculus workshop at their own discretion. At the end of the grading term, all students (even those with different instructors) took the same final exam. The researchers then compared the scores for those who enrolled in the workshop while re-taking calculus to those who re-took calculus without enrolling in the workshop. 7. Is this an experiment? Justify your answer. A. Yes, this is an experiment, because there was a treatment group who enrolled in the calculus workshop and a control group that did not. B. Yes, this is an experiment, because the study was double-blind (as long as the calculus teachers did not know which students enrolled in the workshop). C. No, this is an observational study, because it does not take place in a laboratory or other tightly controlled research environment. D. No, this is an observational study, because the choice of whether to participate in the workshop was made by the students not the researchers. Ans: D; LO: 1.1-2; Difficulty: Easy; Type: MC FOR INSTRUCTOR USE ONLY
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8. Which of the following are sources of unexplained variation in this study? Select all that apply. A. Whether or not students enrolled in the workshop B. Whether or not students had failed a calculus class in the past C. Student attendance in class (number of absences) D. Student motivation to study calculus E. Calculus instructor F. Difficulty of the final exam Ans: C, D, E; LO: 1.1-5; Difficulty: Easy; Type: MS 9. A study published in Athletic Training examined the effects of three different types of knee stabilizing braces on agility test speed. College football players from all different positions (running back, wide receiver, linebacker, lineman, etc.) were recruited to participate in the study. All players in the study had torn their ACL (anterior cruciate ligament) in the past, and needed to wear a knee brace to play football. Agility tests were administered in an outdoor football stadium, and the time to complete the test was recorded by a Lafayette photoelectric Cell and Light Time Unit (in seconds). Put the components of the study into the correct boxes in the Sources of Variation Diagram. Note: Some boxes will include more than one answer. Observed variation in:
Sources of explained variation
Sources of unexplained variation
Inclusion criteria: Design:
A. College football players from all different positions B. Type of knee brace C. Players’ current condition (health, mood, motivation, etc.) D. Time to complete agility test (in seconds) E. Measurement error F. History of torn ACL and need for a knee brace G. Details of the agility test H. Players’ natural speed and agility I. Environmental factors (weather, wind, etc.) Ans: Observed variation in: D; Inclusion criteria: A, F; Design: G; Sources of explained variation: B; Sources of unexplained variation: C, E, H, I; LO: 1.1-5; Difficulty: Medium; Type: Ma 10. In a separate-means model, the standard error of the residuals can be thought of as the typical deviation of an observed response from: A. The residuals (prediction errors) B. The response predicted by the model (group mean) C. The overall mean of the response variable D. The overall mean of the explanatory variable FOR INSTRUCTOR USE ONLY
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Ans: B; LO: 1.1-1; Difficulty: Easy; Type: MC 11. A study published in the Journal of Sports Science & Medicine tested the effectiveness of the Power Balance © bracelet, which has been marketed as a way to improve balance, flexibility, strength, and power through the use of hologram technology. Subjects, who were all college athletes, completed tests of their athletic performance while wearing either a Power Balance © bracelet or a plain rubber placebo bracelet. The bracelets were covered with a wristband, so the athletes and those measuring their performance could not see which bracelet was being worn. Only the researcher who analyzed the data knew which measurements corresponded to the Power Balance © bracelet and which to the placebo bracelet. Classify this study. A. This study is not blinded. B. This is a single-blind study. C. This is a double-blind study. D. There is not enough information to classify this study. Ans: C; LO: 1.1-4; Difficulty: Easy; Type: MC 12. Which of the following is an experiment? Select all that apply. A. Executives at a large department store chain selected 100 stores and randomly assigned 50 of them to reduce their hours, opening an hour later than before; hours for the other 50 stores were not changed. After six months, the executives compared revenue for the two groups of stores. B. A researcher recruited a group of American adults whose demographics were similar to the American population. The researcher measured each subject’s forced expiratory volume, an indicator of lung function. Then each subject was asked whether or not they smoke cigarettes. C. A survey was administered to a large group of high school students. The survey asked whether the students were employed outside of school (in a paying job) and how much sleep they got the night before (in hours). Ans: A; LO: 1.1-2; Difficulty: Medium; Type: MS 13. A university professor teaches two sections of introductory statistics: one section meets at 8:00 am and the other meets at 11:00 am. She wants to evaluate the effectiveness of a new method of teaching statistics compared to the standard method she has used in the past. She flips a coin to assign her sections to teaching methods and determines that she will use the standard method at 8:00 am and the new method at 11:00 am. At the end of the term, she compares the final exam scores for the two sections. Which of the following best describes the potential for confounding in this scenario? A. Different students have different levels of talent and motivation, so it is impossible to attribute differences in final exam scores to the teaching method. B. The two sections may not be exactly the same size, which would lead to inappropriate comparisons between the treatment and control groups. C. Students may find it difficult to pay attention at 8:00 am, which may negatively impact the exam scores of those taught with the standard method. D. Confounding is not a concern in this scenario, because the professor used random assignment as part of the study design. Ans: C; LO: 1.1-1; Difficulty: Medium; Type: MC FOR INSTRUCTOR USE ONLY
1-6 14. True or False: The standard error of the residuals is a way to measure the amount of variation in the response variable that remains unexplained after applying the model. Ans: True; LO: 1.1-1; Difficulty: Easy; Type: TF 15. True or False: Because of the possibility of confounding, you should always avoid using causal language (action verbs like ―affect‖ and ―lead to‖) in your conclusions. Ans: False; LO: 1.1-1; Difficulty: Easy Type: TF
Section 1.2: Quantifying Sources of Variation LO1.2-1: Partitioning variation in the response variable into variation explained by the model and unexplained variation. LO1.2-2: Measuring percentage of variation explained. LO1.2-3: Understanding effect size and practical significance. Questions 1 through 3: Dog agility is a sport where trainers guide their dogs through an obstacle course as quickly as possible. Two trainers, Abby and Lauren, have dogs participating in agility competitions. Each dog completes the same agility course, and they measure the time it takes each dog to complete the course (in seconds). The times for Abby’s three dogs were 30, 40, and 50. The times for Lauren’s three dogs were 50, 60, and 70.
1. Calculate the Sum of Squares Total (SSTotal). Solution: 30 50 40 50 50 50 50 50 60 50 70 50 1000 2
2
2
2
2
2
Ans: 1000 0; LO: 1.2-1; Difficulty: Medium; Type: Calc 2. Calculate the Sums of Squared Errors (SSError). Solution: 30 40 40 40 50 40 50 60 60 60 70 60 400 2
2
2
2
2
2
Ans: 400 0; LO: 1.2-1; Difficulty: Medium; Type: Calc 3. Calculate the Sum of Squares for the Model (SSModel). Solution: 40 50 40 50 40 50 60 50 60 50 60 50 600 2
2
2
2
2
2
Ans: 600 0; LO: 1.2-1; Difficulty: Medium; Type: Calc Questions 4 and 5: Dog agility is a sport where trainers guide their dogs through an obstacle FOR INSTRUCTOR USE ONLY
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course as quickly as possible. Two trainers, Abby and Lauren, have dogs participating in agility competitions. Each dog completes the same agility course, and they measure the time it takes each dog to complete the course in seconds. Compare the sums of squares for two possible datasets that could occur in this context.
Dataset 1: Times for Abby’s dogs: 30, 40, 50 Times for Lauren’s dogs: 50, 60, 70
Dataset 2: Times for Abby’s dogs: 20, 40, 60 Times for Lauren’s dogs: 40, 60, 80
4. SSTotal for Dataset 1 __________ (<, >, or =) SSTotal for Dataset 2 SSModel for Dataset 1 __________ (<, >, or =) SSModel for Dataset 2 SSError for Dataset 1 __________ (<, >, or =) SSError for Dataset 2 Ans: <, =, <; LO: 1.2-1; Difficulty: Medium; Type: FIB 5. The R2 value for Dataset 1 __________ (<, >, or =) the R2 value for Dataset 2 Ans: >; LO: 1.2-2; Difficulty: Medium; Type: FIB Questions 6 and 7: An office designer claims that a new ergonomic desk chair makes typing at a computer terminal faster and easier. A client company plans to test it by asking 30 employees who do a lot of typing to take part in an experiment. They will randomly assign 15 employees to use the new ergonomic chair and 15 to use a regular chair. The 30 employees will then type a selected passage for 5 minutes, recording the total number of words that are typed correctly. FOR INSTRUCTOR USE ONLY
1-8 Consider two hypothetical data sets that could result from this experiment:
6. Which of the datasets would result in a larger R2 value? A. Dataset 1 would result in a larger R2 value, because SSModel for Dataset 1 is larger than SSModel for Dataset 2. B. Dataset 1 would result in a larger R2 value, because SSError for Dataset 1 is smaller than SSError for Dataset 2. C. Dataset 2 would result in a larger R2 value, because SSModel for Dataset 2 is larger than SSModel for Dataset 1. D. Dataset 2 would result in a larger R2 value, because SSError for Dataset 2 is smaller than SSError for Dataset 1. Ans: C; LO: 1.2-2; Difficulty: Medium; Type: MC 7. Compare the value of the effects for the two datasets. A. The effects for Dataset 1 would be larger (in absolute value), because in Dataset 1 there is a smaller difference between the group means. B. The effects for Dataset 2 would be larger (in absolute value), because in Dataset 2 there is a larger difference between the group means. C. The effects for Dataset 1 would be the same size as the effects for Dataset 2, because the standard deviations are the same for both datasets. D. The effects for Dataset 1 would be the same size as the effects for Dataset 2, because the sample sizes are the same for both datasets. Ans: B; LO: 1.2-2; Difficulty: Medium; Type: MC
Questions 8 through 11: The graphs below display the outcomes of three different experiments FOR INSTRUCTOR USE ONLY
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to compare a treatment group with a control group.
8. For which of the experiments does SSModel = 0? Select one or more than one. A. Experiment A B. Experiment B C. Experiment C Ans: A, C; LO: 1.2-1; Difficulty: Easy; Type: MS 9. For which of the experiments does SSError = 0? Select one or more than one. A. Experiment A B. Experiment B C. Experiment C Ans: A, B; LO: 1.2-1; Difficulty: Easy; Type: MS 10. The R2 value for Experiment B is _______ (0, 0.5, 1), because ________ (none, half, all) FOR INSTRUCTOR USE ONLY
1-10 of the variability in outcomes is explained by the treatment group model. The R2 value for Experiment C is ________ (0, 0.5, 1), because ________ (none, half, all) of the variability in outcomes is explained by the treatment group model. Ans: 1, all, 0, none; LO: 1.2-2; Difficulty: Medium; Type: FIB 11. The value of the effect for the treatment group in Experiment B is _____ (-4, -2, 0, 2, or 4). The value of the effect for the treatment group in Experiment C is _____ (-4, -2, 0, 2, or 4). Ans: -2, 0; LO: 1.2-3; Difficulty: Easy; Type: FIB Questions 12 and 13: A statistics class conducted an experiment to investigate whether standing heart rates tend to be higher than sitting heart rates. Students were randomly assigned to either sit or stand, then the students measured their heart rates (in beats per minute). They used software to calculate the sums of squares and found that SSModel = 614.8 and SSTotal = 13232.1 12. Calculate SSError. Solution: 13232.1 – 614.8 12617.3 Ans: 12617.3 0; LO: 1.2-1; Difficulty: Easy; Type: Calc 13. Calculate the R2 value. Give your answer as a proportion. Solution: 614.8 /13232.1 0.046 Ans: 0.046 0.006; LO: 1.2-2; Difficulty: Easy; Type: Calc 14. An online retailer is using an experiment to decide whether to modify their website. When visitors type in the web address or click a link to the site, they are randomly re-directed to one of two versions of the website: the version that has been in use for the last year (version A) or an updated version (version B). The retailer’s goal is to maximize the amount of time (in minutes) visitors stay on the site. Single-mean model:
Predicted time spent on site 12.3, SE of residuals 4.64 Separate-means model:
9.9 forVersion A Predicted time spent on site , SE of residuals 3.98 14.7 forVersion B Calculate the effect for Version A. Solution: 9.9 –12.3 2.4 Ans: -2.4 0; LO: 1.2-3; Difficulty: Medium; Type: Calc Questions 15 and 16: The following output displays the amount (in dollars) that a sample of male and female college students spent on their most recent haircuts. FOR INSTRUCTOR USE ONLY
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and p-value for Variable 1 change? A. Removing Variable 2 would not affect the sum of squares or p-value for Variable 1. B. Removing Variable 2 would not affect the sum of squares for Variable 1, but the pvalue for Variable 1 may change. C. Removing Variable 2 would not affect the p-value for Variable 1, but the sum of squares for Variable 1 may change. D. Removing Variable 2 may change both the sum of squares and the p-value for Variable 1. Ans: D; LO2.3-2; Difficulty: Medium; Type: MC Questions 13 through 15: Students in a statistics class wanted to know how customers rate one of their favorite coffee shops. The students administered surveys and asked customers to rate their experience in the shop on a scale of 1-10. They also asked whether the survey respondent was a college student and how often they visit coffee shops (at least once a week or less than once a week). 13. The students decide to use a two-variable model that predicts ratings based on whether the customer is a student and the frequency of their coffee shop visits. Based on the graph below, do you expect covariation – variation in ratings that is attributed jointly to Student and Frequency?
A. Yes, because more than half of the customers in the sample are students. B. Yes, because students tend to give higher ratings than non-students. C. Yes, because students are more likely to visit coffee shops at least once per week. D. No, because the sample is evenly divided between customers who visit coffee shops at least once per week and those who visit less than once per week. Ans: C; LO2.3-3; Difficulty: Medium; Type: MC 14. Consider two possible models for predicting ratings. A one-variable model that predicts ratings based on frequency of coffee shop visits has an R2 value of 0.06. A one-variable model that predicts ratings based on whether the customer is a student has an R2 value of 0.03. What is the R2 value of a two-variable model that uses both Frequency and Student to predict ratings?
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1-50 A. R2 = 0.06 + 0.03 = 0.09 B. R2 = 0.06 - 0.03 = 0.03 C. R2 = (0.06 + 0.03) 2 = 0.045 D. There is not enough information to calculate R2 for the two-variable model. Ans: D; LO2.3-3; Difficulty: Medium; Type: MC 15. Consider two possible models to predict ratings: Model 1: {
, with
{
{
Model 2:
with Compare the adjusted sum of squares for student in each of these two models. The adjusted sum of squares for student in Model 1 ________ (>, <, =) the adjusted sum of squares for student in Model 2. Ans: >; LO2.3-2; Difficulty: Medium; Type: FIB
Chapter 3 Intermediate Statistical Investigations Test Bank Question types:
FIB = Fill in the blank Ma = Matching MC = Multiple choice
Calc = Calculation MS = Multiple select TF = True-false
CHAPTER 3 TERMINAL LEARNING OUTCOMES TLO3-1: Design and analyze a multi-factor experiment realizing the difference between blocking variables and experimental factors. TLO3-2: Understand the concept of a statistical interaction, and calculate and interpret interaction effects. TLO3-3: Explain the benefits and challenges of replication particularly as they relate to generalized block designs and within block factorial designs. TLO3-4: Run a two-variable ANOVA including an interaction term for an observational study with 2 two-level variables.
Section 3.1: Multifactor Experiments LO3.1-1: Design an experiment with more than one variable of interest. LO3.1-2: Explore the benefits of a two-variable study where the levels of both variables are asFOR INSTRUCTOR USE ONLY
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signed by the researcher. 1. A researcher is interested in the relative difficulty of two sets of mathematical tasks and how student success in completing the tasks may be affected by distractions. Forty-eight college students have been recruited to participate in a study. Which of the following is the most appropriate study design? A. Randomly assign the students into two groups of size 24. One group completes Task Set 1 with distractions, and the other completes Task Set 2 with no distractions. B. Randomly assign the students into four groups of size 12. One group completes Task Set 1 with distractions, one completes Task Set 1 without distractions, one completes Task Set 2 with distractions, and the last completes Task Set 2 without distractions. C. Divide the students into two groups of 24. Within Group 1, assign 12 students to complete Task Set 1 and 12 students to complete Task Set 2. Within Group 2, assign 12 students to work with distractions and 12 to work without distractions. D. All three study designs above are equally appropriate ways to investigate the researcher’s statistical questions. Ans: B; LO3.1-1; Difficulty: Medium; Type: MC
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1-52 Questions 2 through 5: In a balanced, full-factorial experiment, 40 volunteers were randomly assigned to receive one of two pain medications (A, B) at one of two dosages (high, low). After an hour, their response to pain was recorded on a scale of 0 to 20 (with higher numbers indicating more severe pain). The ANOVA table below corresponds to a twovariable additive model using drug and dose as predictors of pain response. Source Drug Dose Error Total
DF 1 1 37 39
SS 16.9 313.6 157.5 488.0
MS 16.9 22.22 4.26
F 3.97 73.67
p-value 0.0537 <0.0001
2. Calculate SSModel. A. SSModel = 16.9 + 313.6 = 330.5 B. SSModel = (16.9 + 313.6) / 2 = 165.25 C. SSModel = 16.9 + 313.6 – 157.5 = 173.0 D. There is not enough information to calculate SSModel, because an association between Drug and Dosage may lead to covariation. Ans: A; LO3.1-2; Difficulty: Medium; Type: MC 3. True or False: The 95% confidence interval for estimating the difference in mean pain response for low and high doses includes 0. Ans: False; LO3.1-2; Difficulty: Medium; Type: TF 4. Suppose the researchers had ignored dosage in the analysis. What values would you expect in the ANOVA table for a one-variable model using only drug as a predictor of pain response? Source Drug Error Total
DF
SS
MS
F
p-value
SSDrug in the one-variable ANOVA table would be _________ (>, <, =) 16.9 The p-value for Drug in the one-variable ANOVA table would _________ (>, <, =) 0.0537, indicating _________ (stronger/weaker/the same) evidence of a difference between Drug A and Drug B. Ans: =, >, weaker; LO3.1-2; Difficulty: Hard; Type: FIB 5. Suppose the researchers had ignored dosage in the analysis. How would this affect the width of the 95% confidence interval for estimating ? The 95% confidence interval for estimating would be ________________ (wider/narrower) than the confidence interval based on the two-variable model, because the ________________ (treatment means/residual standard error) would be different. Ans: wider, residual standard error; LO3.1-2; Difficulty: Medium; Type: FIB
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Questions 6 through 9: A large field was divided into 24 equal-sized plots to be planted with the same number of potato plants. Each plot was randomly assigned a type of fertilizer (A, B, C) and a manure level (high, low) based on a balanced, full-factorial design. The table below shows the mean yield (by weight) for each combination of fertilizer and manure. Low Manure High Manure
Fertilizer A 6 7
Fertilizer B 8 11
Fertilizer C 10 12
6. True or False: By using a balanced design, the researchers have ensured that there is no association between the type of fertilizer and the level of manure. Ans: True; LO3.1-1; Difficulty: Medium; Type: TF 7. Fill in the blanks using numerical values. In this experiment, there are ________ explanatory variables (or factors) and there are ________ treatments with ________ replications. Ans: 2, 6, 4; LO3.1-1; Difficulty: Easy; Type: FIB 8. Calculate the main effects of fertilizer and manure to fill in the additive statistical model. {
{
Ans: First column: 9; Second column: -2.5, 0.5, 2; Third column: -1, 1; LO3.1-2; Difficulty: Medium; Type: FIB 9. For the additive model that uses fertilizer type and manure level to predict yield, the standard error of the residuals is 4.5. What does the standard error of the residuals measure? A. The variability of individual plot yields around the observed mean yields given in the table B. The variability of individual plot yields around the predicted mean yields calculated based on the additive model C. The variability between the observed mean yields given in the table and the predicted mean yields calculated based on the model D. The variability between the observed mean yields given in the table, the predicted mean yields calculated based on the model, and the overall mean yield Ans: B; LO3.1-2; Difficulty: Medium; Type: MC
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1-54 Questions 10 through 13: Researchers want to reduce potato rot while potatoes are being stored for future use. In an experiment, potatoes were injected with a bacteria known to cause rot and then stored under a variety of conditions. Two of the experimental factors were temperature during storage and amount of oxygen during storage. The response variable was the diameter of the rotted area (in millimeters). A partially filled in twovariable ANOVA table is given below. Source Temperature Oxygen Error Total
DF 1 2 14 17
SS 600.89 44.44 365.77 1011.11
MS 600.89 22.22 26.14
F ?
p-value 0.0003 0.4483
10. Fill in the blanks using numerical values. Based on the ANOVA table, there were ________ levels of temperature and ________ levels of oxygen tested in this experiment. There were ________ replications for each combination of temperature and oxygen. Ans: 2, 3, 3; LO3.1-1; Difficulty: Medium; Type: FIB 11. Calculate the F-statistic for testing whether temperature has an effect on rot. Solution: Ans: 22.99 0.1; LO3.1-2; Difficulty: Medium; Type: Calc 12. What conclusions would you draw based on the p-values in this ANOVA table? After adjusting for oxygen, this study provides _______ (strong/weak) evidence that temperature has an effect on rot. After adjusting for temperature, this study provides _______ (strong/weak) evidence that oxygen level has an effect on rot. Ans: strong, weak; LO3.1-2; Difficulty: Medium; Type: FIB 13. Initially, the researchers had planned to use a one-variable model with separate means for each combination of temperature and oxygen. Which of the following are advantages of a two-variable additive model over a one-variable separate means model? Select all that apply. A. The two-variable model allows us to estimate the main effects of temperature and oxygen separately. B. The two-variable model explains more of the variability (higher SSModel) compared to the separate means model. C. The two-variable model has fewer degrees of freedom for treatment and more degrees of freedom for error. D. The two-variable model has more degrees of freedom for treatment and fewer degrees of freedom for error. Ans: A, C; LO3.1-2; Difficulty: Hard; Type: MS
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14. An experiment was conducted to compare two different applications for text messaging on cell phones, each with its own interface and software. The experiment included participants in two age groups: 15-44 years old and 45+ years old. Within each age group, half of the participants were randomly assigned to each application. The researchers then asked each participant to type the words from a given passage of text and recorded the number of words typed correctly in a fixed period of time. Describe the explanatory variables in this study. A. This study has two explanatory variables: one blocking variable and one experimental factor. B. This study has two explanatory variables, both of which are experimental factors. C. This study has four explanatory variables: two blocking variables and two experimental factors. D. This study has four explanatory variables, all of which are experimental factors. Ans: A; LO3.1-2; Difficulty: Easy; Type: MC
15. The plots below display the residuals for a two-factor model with six treatments. Match each validity condition to the description of how that condition should be checked. Two of the descriptions will not be used.
Independence:
A. Check that fitted values are spaced fairly evenly along the x-axis.
Equal variance:
B. Check that the vertical spread of the residuals at each of the fitted values is reasonably similar.
Normality:
C. Check that the histogram of the residuals is reasonably symmetric and bell-shaped. D. Check that the mean of the residuals is 0. E. Check that experimental units were randomly assigned to treatments with no repeated measures.
Ans: Independence: E; Equal variance: B; Normality: C; LO3.1-2; Difficulty: Medium; Type: Ma
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Section 3.2: Statistical Interactions LO3.2-1: Understand the concept of a statistical interaction. LO3.2-2: Interpret an interaction plot. LO3.2-3: Calculate interaction effects. LO3.2-4: Use simulation- and theory-based p-values to assess the significance of an interaction. Questions 1 through 6: A balanced, full-factorial experiment was used to investigate the effect of protein source (beef, cereal) and protein level (high, low) on weight gain (in grams) for male rats. Ten rats were assigned to each combination of source and level. A table of mean weight gains and an interaction plot are shown below. Level Source
Low
High
Beef
79.2
100
Cereal
83.9
85.9
1. Suppose the researchers used an additive model to predict weight gain, as shown below. {
{
According to the interaction plot, which treatment is expected to cause the smallest amount of weight gain?
A. Beef/Low B. Beef/High
According to the additive model, which treatment is expected to cause the smallest amount of weight gain?
C. Cereal/Low D. Cereal/High
Ans: A, C; LO3.2-1; Difficulty: Easy; Type: Ma 2. Calculate the interaction effect for the Beef/Low treatment.
{
Sol: Answer: -4.7
{
{
; 0; LO3.2-3; Difficulty: Medium; Type: Calc
3. Calculate SSInteraction. Sol: ; Answer: 883.6 0.5; LO3.2-3; Difficulty: Medium; Type: Calc 4. Which of the following statements are appropriate descriptions of the statistical interaction FOR INSTRUCTOR USE ONLY
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in this sample? Select all that apply. A. Weight gain increases when rats are fed a high protein diet instead of a low protein diet. B. When beef is the source, higher protein levels lead to higher weight gains, whereas when cereal is the source, higher protein levels do not have much impact. C. Beef has higher protein levels than cereal, and thus, beef has a positive effect on weight gain. D. Beef and cereal sources are very different in terms of weight gain when the protein level is high, but they are similar when the protein level is low. Ans: B, D; LO3.2-2; Difficulty: Medium; Type: MS 5. Use the 3S Strategy to test the significance of the interaction based on the difference in the differences statistic. The histogram to the right shows the results of simulating 10,000 trials under the assumption of no interaction between source and level.
Based on the null distribution and difference in the differences for the sample data, what do you conclude? If there were really ______ (an interaction/no interaction), we’d get a difference in the differences as or more extreme as our observed statistic about ________ (0.5, 7, or 50)% of the time. This provides ___________ (moderately/very) strong evidence of an interaction between source and level. Ans: no interaction, 7, moderately; LO3.2-4; Difficulty: Hard; Type: FIB 6. The table below shows all possible pairwise confidence intervals for the four treatments. Groups Compared Beef/High – Beef/Low Beef/High – Cereal/Low Beef/High – Cereal/High Cereal/High – Beef/Low Cereal/Low – Beef/Low Cereal/High – Cereal/Low
95% Confidence Intervals (7.23, 34.36) (2.53, 29.66) (0.54, 27.66) (-6.86, 20.26) (-8.86, 18.26) (-11.56, 15.56)
What conclusion can you draw from the table? Note that the term “significant” in these statements refers to statistical significance not practical importance. A. Beef/High is significantly different from every other treatment in this study. B. Cereal/Low is significantly different from every other treatment in this study. C. Beef/High and Beef/Low are the only two treatments that are significantly different from each other. D. Cereal/High and Cereal/Low are the only two treatments that are significantly different from each other. Ans: A; LO3.2-4; Difficulty: Medium; Type: MC FOR INSTRUCTOR USE ONLY
1-58 Questions 7 through 11: In a balanced, full-factorial experiment, 60 volunteers were randomly assigned to receive one of three pain medications (A, B, C) at one of two dosages (high, low). After an hour, their response to pain was recorded on a scale of 0 to 20 (higher numbers indicate more severe pain). 7. Suppose there were no interaction between pain medication and dosage. How would you complete the table of means below?
Low Dose High Dose
Drug A 10.4 5.3
Drug B 12.2 ?
Drug C 8.8 ?
Ans: 7.1, 3.1; LO3.2-1; Difficulty: Medium; Type: FIB 8. Which of the following statements are appropriate descriptions of the statistical interaction in this sample?
A. The mean pain response for low dosages is always higher than the mean pain response for high dosages. B. Drug A and Drug B are usually prescribed at low dosages. Drug C is prescribed at a high dosage nearly half the time. C. The dosage effect is not the same for all three drugs. The dosage effect for Drug C is much smaller than for Drug A or Drug B. D. All of the statements above are appropriate descriptions of the statistical interaction. Ans: C; LO3.2-2; Difficulty: Medium; Type: MC 9. The researchers decided to use a two-variable model with an interaction.
{
{ {
True or False: The predicted pain responses based on this model will be equal to the FOR INSTRUCTOR USE ONLY
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observed treatment means for each of the six drug-dose combinations Ans: True; LO3.2-3; Difficulty: Medium; Type: TF 10. The partially filled in ANOVA table below corresponds to the two factor model with interaction. Source Drug Dose Interaction Error Total
DF
SS 21.70 281.67 50.63 258.60 612.60
MS 10.85 281.67
F 2.27 58.82
p-value 0.1135 <0.0001 0.008
Calculate the F-statistic for the interaction. Sol: Ans: 5.29
0.05; LO3.2-4; Difficulty: Medium; Type: Calc
11. The p-value for testing the interaction between drug and dosage is 0.008. What do you conclude? You may assume . This study provides ___________ (strong/weak) evidence that there ________ (is/is not) an interaction between drug and dosage. Ans: strong, is; LO3.2-4; Difficulty: Medium; Type: FIB
12. You plan to use a theory-based p-value to test the interaction in a 2x2 full-factorial design. Does the residual plot below indicate a violation of the validity conditions?
A. Yes, because the residuals are centered at 0, so the independence condition is violated. B. Yes, because data points appear in four stacks, so the normality condition is violated. C. Yes, because the points are not evenly distributed along the x-axis, so the equal variance condition is violated. D. No, this graph does not indicate any major violation of the validity conditions. Ans: D; LO3.2-4; Difficulty: Medium; Type: MC
FOR INSTRUCTOR USE ONLY
1-60 13. Below are two interaction plots for two factors (A and B) and a quantitative response variable.
Assuming the sample sizes are the same, which graph corresponds to a smaller p-value for the interaction? A. Graph 1 would correspond to a smaller p-value for the interaction. B. Graph 2 would correspond to a smaller p-value for the interaction. C. Both samples would result in the same p-value for the interaction. D. These graphs to not provide enough information to decide which sample would correspond to a smaller p-value for the interaction. Ans: A; LO3.2-2; Difficulty: Medium; Type: MC 14. A large field was divided into 24 equal-sized plots to be planted with the same number of potato plants. Each plot was randomly assigned a type of fertilizer (A, B, C) and a manure level (high, low) based on a balanced, full-factorial design. True or False: By using a balanced design, the researchers have ensured that there is no interaction between the type of fertilizer and the level of manure. Ans: False; LO3.2-1; Difficulty: Medium; Type: TF 15. True or False: When there is a substantial interaction between Factor A and Factor B in a two-factor design, the researcher should interpret the main effects of Factor A and Factor B separately. Ans: False; LO3.2-1; Difficulty: Medium; Type: TF
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Section 3.3: Replication LO3.3-1: Explain the benefits and challenges of replication. LO3.3-2: Define and describe advantages of a generalized block design. LO3.3-3: Define and describe advantages of within-blocks factorial designs. Questions 1 through 3: Concentration is a one-person memory game in which cards are laid face down on a surface and two cards are flipped face up at a time. The object of the game is to turn over matching pairs of cards. An online version of this game includes three different sets of cards: one has images of animals on the cards, one has images of babies, and one has images of holiday scenes. Are these three variations equally difficult? To investigate, eight students tried all three versions of the game in random order. They recorded the amount of time (in seconds) it took to complete the game. 1. How would you describe this design? A. Randomized complete block design with repeated measures B. Generalized block design with replication C. Full factorial design with independent groups D. Within-block factorial design with two experimental factors Ans: A; LO3.3; Difficulty: Medium; Type: MC 2. Suppose you want to test for a person-version interaction as part of the analysis. Fill in the degrees of freedom in the ANOVA table below. Source Version Person Version Error Total
Person
DF ? ? ? ? ?
SS
MS
F
Ans: 2, 7, 14, 0, 23; LO3.3-1; Difficulty: Medium; Type: FIB 3. True or False: In this scenario, the person-version interaction is completely confounded with error, so the only way to carry out statistical tests of the version effect and the person effect is to exclude the interaction term from the model. Ans: True; LO3.3-1; Difficulty: Medium; Type: TF Questions 4 and 5: A researcher is interested in how distractions affect students as they work on mathematical tasks, so she plans to assign 150 students to work under different conditions (with or without distractions) for a fixed period of time, recording the number of math problems that each student answers correctly. The study will be carried out over the course of a week, and the researcher worries that student motivation may vary day to day, so she decides to use day as a blocking variable. Each day (Monday through Friday), she will assign 15 students to work with distractions and 15 to work without distractions. FOR INSTRUCTOR USE ONLY
1-62 4. Suppose the researcher wants to test for a condition-day interaction as part of the analysis. Fill in the degrees of freedom in the ANOVA table below. Source DF SS MS Distraction Condition ? Day ? ? Condition Day Error ? Total ? Ans: 1, 4, 4, 140, 149; LO3.3-1; Difficulty: Medium; Type: FIB
F
5. True or False: In this scenario, the condition-day interaction is completely confounded with error, so the only way to carry out statistical tests of the condition effect and the day effect is to exclude the interaction term from the model. Ans: False; LO3.3-1; Difficulty: Medium; Type: TF 6. Researchers used a paired design to investigate whether cell phone use impairs drivers’ reaction times. 64 students participated in a simulation of driving situations, pressing a brake button as soon as they saw a red light. A device recorded their reaction times (in milliseconds). Each student completed the simulation under two different conditions: once while talking on a cell phone and once while listening to music. The study design described above requires researchers to assume that the effect of the driving condition (cell phone or music) is the same for every student. How could you modify the study design in order to estimate and test for a statistically significant interaction? A. Increase the sample size by recruiting more students to participate. B. Ask each student to complete the simulation more than once under each condition. C. Add a control condition where students are not distracted by cell phones or music. D. Collect data on a potential confounding variable (how confident students feel as drivers, how much sleep they got last night, etc.). Ans: B; LO3.3-1; Difficulty: Medium; Type: MC 7. Which of the following is the best description of replication? A. Replication means there are at least 20 observational units in the study (or more if the data distribution is not symmetrical). B. Replication means the study employs random assignment and a placebo control group for comparison. C. Replication means each observational unit is measured more than once under different experimental conditions. D. Replication means each set of experimental conditions (or factor-block combination) occurs more than once in the design. Ans: D; LO3.3-1; Difficulty: Easy; Type: MC
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8. Which of the following is an advantage of replication in a study design? A. Replication reduces unexplained variation within groups, which leads to more powerful study designs. B. Replication makes more efficient use of resources, which reduces the cost and/or time required to complete a study. C. Replication makes it possible to estimate treatment and interaction effects separately from random error. D. Replication ensures that there is no association between experimental factors, which prevents confounding. Ans: C; LO3.3-1; Difficulty: Medium; Type: MC Questions 9 and 10: Ten male and ten female personnel officers were shown a front view photograph of a job applicant’s face and asked to rate the likely job success of the applicant on a scale of 0 to 20. Half of the officers in each gender were chosen at random to receive a version of the photograph in which the applicant made eye contact with the camera. 9. How would you describe this design? A. Randomized complete block design with repeated measures B. Generalized block design with replication C. Full factorial design with independent groups D. Within-block factorial design with two experimental factors Ans: B; LO3.3-2; Difficulty: Medium; Type: MC 10. The plot below shows the mean success rating for each combination of gender and eye contact. Is there a substantial interaction between gender and eye contact in this sample?
A. Yes, because the means for all four combinations of gender and eye contact are different from each other. B. Yes, because female offers tend to give higher ratings and officers who see the photo with eye contact tend to give higher ratings. C. No, because the effect of eye contact is roughly the same regardless of officer gender. D. No, because male and female officers are equally likely to see a photo with eye contact, since this is a balanced design. Ans: C; LO3.3-2; Difficulty: Medium; Type: MC
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1-64 Questions 11 through 14: A waitress works part-time at two different restaurants: one is a casual deli and the other is a more upscale Italian restaurant. She decides to conduct an experiment to investigate whether having a conversation with her customers or writing ―Thank you!‖ on the check will affect her tip percentages. At each restaurant, she will assign 32 tables of customers to one of four treatments: conversation and message on check, no conversation and message on check, conversation and no message on check, or no conversation and no message on check. 11. Identify the components of this study. Observational unit(s): Blocking variable(s): Experimental factor(s): Response variable(s):
A. Conversation B. Message on check C. Restaurant D. Tables of customers E. Tip percentage Ans: Observational units: D; Blocking variable: C; Experimental factors: A, B; Response variable: E; LO3.3-3; Difficulty: Medium; Type: Ma
12. The graphs below show the interaction between conversation and message for each of the two restaurants. Do these graphs indicate a three-way interaction between restaurant, conversation, and message in this sample?
A. Yes, because although the shapes of the graphs are similar, the tips at the Italian restaurant tend to be higher than tips at the deli. FOR INSTRUCTOR USE ONLY