# set global options
knitr::opts_chunk$set(
  echo = TRUE, # show code 
  warning = FALSE,  
  message = FALSE
  
) 
library(knitr)
library(kableExtra)

1.

An economist compiled data on productivity improvements last year for a sample of firms producing electronic computing equipment. The firms were classified according to the level of their average expenditures for research and development in the past three years (low, moderate, high).

  1. Is this study experimental, observational, or mixed experimental and observational? Why?
  2. Identify all factors, factor levels, and factor-level combinations.
  3. What type of study design is being implemented here?
  4. What is the basic unit of study?

Solution:

  1. Observational. It is observational because we are not intentionally altering the explanatory variable (factor) which is the level of average expenditure for the firm. We are only observing the outcome (productivity improvements) of firms in relation to their observational factor level
  2. Factor, amount of average expenditure for research and development in the past three years. The factor levels are, low, moderate, or high.
  3. Cross-sectional study
  4. A business firm

2.

In a study to investigate the effect of color of paper (blue, green, orange) on response rates for questionnaires distributed by the “windshield method” in supermarket parking lots, four supermarket parking lots were chosen in a metropolitan area and 10 questionnaires of each color were assigned at random to cars in the parking lots.

  1. Is this study experimental, observational, or mixed? Why?
  2. Identify all factors, factor levels, and factor-level combinations.
  3. What type of study design is being implemented here?
  4. What is the basic unit of study?

Solution: a. Experimental, we are intentionally choosing and altering all of our factors to assess outcomes. We are randomly choosing what parking lots to perform our experiment in, and at those parking lots we are randomly assigning cars questionnaires of varying color. Since our explanatory variables are intentionally altered by us they are experimental. b. The factors are the supermarket parking lot chosen, and the color of the questionnaire. It is crossed factor study in that cars in each supermarket parking lot selected have the possibility of receiving any color of questionnaire. c. Completely randomized experiment d. The car driver.

3.

A rehabilitation center researcher was interested in examining the relationship between physical fitness prior to surgery of persons undergoing corrective knee surgery and the time required in physical therapy until successful rehabilitation. Data on the number of days required for successful completion of physical therapy and the prior physical fitness status (below average, average, above average) were collected.

  1. Is this study experimental, observational or mixed? Why?
  2. Identify all factors, factor levels, and factor-level combinations.
  3. What type of study design is being implemented here?
  4. What is the basic unit of study?

Solution: a.) Observational. We are not intentionally altering the level of physical fitness of individuals prior to surgery (our observational factor). We are only assessing the relationship between the observational factor and the outcome (days required for successful completion of physical therapy) b.) The factor is the level of physical fitness of the individual prior to surgery. The factor has the possible levels of below average, average, and above average. It is a single-factor study. c.) Cohort. d.) Individuals who have undergone corrective knee surgery.

4.

A research laboratory was developing a new compound for the relief of severe cases of hay fever. The amounts of two active ingredients (low, medium, high) in the compound were varied at three levels each using 18 volunteers. Randomization was used in assigning volunteers to each of the treatment combinations. Data were collected on hours of relief.

  1. Is this study experimental, observational, or mixed? Why?
  2. Identify all factors. factor levels. and factor-level combinations.
  3. Describe how randomization would be performed in this study.
  4. What type of study design is being implementel here?
  5. What is the basic unit of study’?

Solution: a.) Experimental, we are intentionally altering our explanatory variables and assigning them to study units. b.) This is a two-factor study. Factor one is the active ingredient used (active ingredient one, or active ingredient two). The second factor is the amount of active ingredient used (low, medium, or high). First factor level is the active ingredient used, the second factor level is the amount of the active ingredient used (low, medium, high). Unless, individuals are assigned both active ingredients we may assume this is a nested two-factor study. c.) Completely randomized experiment d.) Individual volunteers

5. Suggest an experimental plan for each study and explain why.

An experiment involving the case hardening of lightweight shafts machined from bars of an alloy was run to study the effects of the amount of chemical agent added to the alloy in a molten state (low, high), the temperature of the hardening process (low, high), and the time duration of the hardening process (low, high). Outcome data measured the hardness of the rods tested. It will be possible to machine 32 bars in the study.

Solution: A completely randomized experiment, each bar is assigned a random combination of factor levels constituting a randomly assigned treatment. Then resulting outcomes of each randomly assigned treatment will then be compared.

6. Suggest an experimental plan for each study and explain why.

Assemblers in an electronics firm attach components to a newly developed “board” to be used in automatic-control equipment in manufacturing plants. A study was conducted to determine the effect of sequence of assembling the components (sequence 1, sequence 2 , sequence 3) on the mean time to assemble a board. Potential nuisance factors are sex of the assembler (male, female) and amount of the assembler’s prior experience (under 18 months, 18 months or more). Assume that the following assemblers are available for the study: four males with under 18 moths experience, three females with under 18 months experience, five male assemblers with 18 months or more experience, and four females with 18 months or more experience.

Solution: A randomized block design/stratified randomized experiment. The blocks could be based on prior experience (greater than 18 months, or less than 18 months) and sex, or just based on prior experience (with sex being mixed within blocks), the number of blocks would be based on sample size considerations for the number of treatment outcomes we wish to analyze. Presumably length of experience would be a greater factor than sex so regardless we should at least stratify on that. After our blocks are assembled we would then randomly assign individuals (units) within our blocks to one factor level (in this case the sequence of assembling the components) such that we end up with at least one individual to each factor level constituting a unique treatment for each block. The outcome for each treatment (the mean time to assemble a board) could then be compared within blocks and across blocks. Alternatively you could also randomly assign individuals within a block one sequence, allow them to assemble a specified number of boards, randomly assign those individuals a different sequence, allow them to assemble a specified number of boards, then finally assign those individuals the sequence that they have not previously had and allow them to assemble a specified number of boards. You could then again analyze mean time to assemble board based on each sequence number within blocks and across blocks. This second option allows us to minimize a bit more the effect individual skill level variance has in skewing our results.