Experimental design practice problems (8)

(a) Identify the pieces. The experimental units are the 36 basil seedlings, since a lighting condition is assigned to each seedling. The explanatory variable is the lighting condition, which the gardener sets. The response variable is plant height in centimeters after 4 weeks.

(b) Count the treatments. The lighting variable has three levels (fluorescent, LED low, LED high), so there are 3 treatments, and $\frac{36}{3} = 12$ seedlings receive each one.

(c) Match the four elements. Comparison: three treatment groups. Random assignment: the lighting conditions are assigned at random. Replication: 12 units per treatment, well above one. Direct control: identical pot size, soil, watering schedule, and room temperature hold extraneous variables fixed from unit to unit, so the shared conditions are the direct control.

(d) Decide on causation. Because the lighting conditions were randomly assigned to the seedlings, other variables are balanced across the three groups on average, so a difference in mean height larger than chance variation would produce can be attributed to the lighting. The seedlings were not randomly selected from a larger population, so the conclusion covers seedlings like these and does not generalize further.

(a) Variables. The explanatory variable is whether a student plays a school sport, a categorical variable with two levels. The response variable is GPA.

(b) Study type. No one assigned students to play a sport or not. Students chose for themselves and the school recorded what had already happened, so this is an observational study. An observational study can establish an association but cannot establish a cause.

(c) Check the overall mean. $\frac{180(3.35) + 300(3.11)}{480} = \frac{603 + 933}{480} = \frac{1536}{480} = 3.20$, which matches the school's figure. The gap between the groups is $3.35 - 3.11 = 0.24$ GPA points.

(d) Name a confounding variable. One plausible confounder is a student's time management and academic motivation, already in place before any tryout. First link: students who organize their time well and care about school are more likely to commit to a season of practices, so the variable is tied to the explanatory variable. Second link: those same habits raise GPA on their own, whether or not the student ever joins a team. The motivated students are concentrated in the sport group, so the effect of playing a sport and the effect of the habits those students already had cannot be separated, and the 0.24 gap could belong to either one or to a mix of both. Note what does not qualify. A study hall that students attend only because they made a team is caused by the explanatory variable rather than sitting behind it, so it is part of how playing a sport might raise GPA, not a rival explanation for the gap. A confounding variable has to be a common cause of both, not a consequence of the treatment.

(e) What would fix it. Only random assignment of students to play or not play would balance motivation and every other pre-existing variable across the two groups on average, so that no one of them systematically favors the sport group. That experiment is neither practical nor ethical here, so the defensible report is that sport participation and GPA are associated at this school, not that one raises the other.

Number the units. The experimental units are the 84 volunteers. Label them 1 to 84.

Set the group sizes. Two treatments split evenly gives $\frac{84}{2} = 42$ volunteers per group.

Assign at random. Use a random number generator to produce random integers from 1 to 84, ignoring any repeat, until 42 distinct numbers have appeared. Those 42 volunteers take the tablet, and the remaining $84 - 42 = 42$ volunteers take the placebo.

Control everything else. The placebo is a pill identical in size, color, and taste to the tablet but with no active ingredient. Every volunteer takes one pill at the same time of day for the same 2 weeks and receives the same instructions.

Blind both sides. Neither the volunteers nor the nurse who records the symptom scores knows who received which pill, so the study is double-blind.

Measure and compare. After 2 weeks, record each volunteer's symptom score from 0 to 24, then compare the mean symptom score of the 42 tablet takers with the mean score of the 42 placebo takers.

Say what each control buys. The placebo gives a baseline that includes the whole experience of taking a pill, so a difference between the groups points at the active ingredient rather than at the ritual of being treated. The blinding keeps a volunteer's expectations out of the reported symptoms and the nurse's expectations out of the recorded score.

(a) Count who is blinded. The subjects do not know which cream they got, but the person measuring the response does. Only one side is hidden, so the study is single-blind, not double-blind.

(b) Name the distortion. A knee function score involves judgment. A therapist who expects the active cream to work can score the same range of motion a point or two higher for the patients he knows received it, pushing the treatment group's mean up for a reason that has nothing to do with the cream.

(c) Fix it. Have the week-6 scoring done by a therapist with no access to the assignment list, with the tubes labeled A and B by someone who measures no one. Then neither the patients nor the person recording the response knows the assignment, which makes the study double-blind.

(d) Explain the inert cream. If the control group received nothing, the two groups would differ in two ways at once: the active ingredient and the experience of applying a cream daily and expecting relief. A patient who expects relief often reports some, and the placebo effect is the difference between the mean response to a placebo and the mean response to no treatment. The inert cream makes the two experiences match on everything except the ingredient under test, so a difference in scores points at the ingredient.

(a) Why block. Experience is an extraneous variable with a large effect on finish time. A single completely randomized assignment of all 60 runners could easily leave one program with noticeably more experienced runners, and that imbalance would show up as a program difference. Blocking on experience takes the variation it causes out of the comparison, so the difference between programs is measured more precisely.

(b) Build the blocks and randomize inside each one. Block 1 is the 36 experienced runners and block 2 is the 24 novices. Within block 1, number the runners 1 to 36 and use a random number generator to draw distinct integers from 1 to 36, ignoring repeats, until 18 have appeared; those 18 follow program A and the other $36 - 18 = 18$ follow program B. Within block 2, number the runners 1 to 24 and draw distinct integers from 1 to 24 until 12 have appeared; those 12 follow program A and the other 12 follow program B. Keep coaching hours, surfaces, and the timed 10K course the same for everyone.

(c) Count the four groups. Experienced: $\frac{36}{2} = 18$ runners per program. Novice: $\frac{24}{2} = 12$ runners per program. Each program is followed by $18 + 12 = 30$ runners, and $18 + 18 + 12 + 12 = 60$, so every runner is used.

(d) Cause and effect. The treatments are still randomly assigned, now within each block, so a cause-and-effect conclusion is still supported. Compare the two programs within the experienced block and within the novice block, then combine. Blocking buys precision, not a different kind of conclusion, and because the 60 runners were not randomly selected from a population, the finding applies to runners like these.

(a) Name the design. Each volunteer supplies both measurements, so the two treatments are compared within a single person rather than across different people. That is a matched pairs design, the two-treatment case of a randomized block design with one block per volunteer. The pairing controls each volunteer's own typing skill, which varies enormously from person to person and would otherwise swamp any layout difference.

(b) Describe the randomization and count it. For each volunteer, flip a fair coin: heads means layout A is typed first, tails means layout B is typed first. Each of the 8 flips is independent with 2 outcomes, so there are $2^8 = 256$ equally likely order assignments, and the study records $8 \times 2 = 16$ typing speeds.

(c) Compute the differences. Taking B minus A volunteer by volunteer gives $65 - 62 = 3$, $57 - 58 = -1$, $76 - 71 = 5$, $54 - 49 = 5$, $68 - 66 = 2$, $58 - 54 = 4$, $63 - 60 = 3$, and $71 - 68 = 3$. Their sum is $3 + (-1) + 5 + 5 + 2 + 4 + 3 + 3 = 24$, so $\bar{d} = \frac{24}{8} = 3.0$ words per minute in favor of layout B.

(d) Explain the coin flip. If every volunteer typed the same layout first, practice on the passage would raise the second speed and fatigue would lower it, and either effect would push all the differences in one direction. Order would then be confounded with layout, and you could not tell which one produced the gap. Randomizing the order spreads any practice or fatigue effect across both layouts, and the 10-minute rest keeps fatigue small in the first place.

Note the analysis this design calls for. A matched pairs design is analyzed on the single column of 8 differences, not as two independent samples. Testing whether the true mean difference is 0 is the paired t-test worked in t-test practice.

(a) Count the treatments. A treatment is one combination of the two explanatory variables, so crossing 2 fertilizer brands with 3 watering schedules gives $2 \times 3 = 6$ treatments.

(b) Size the blocks. The blocks are the two greenhouse halves, 24 pots each, because pots within a half receive similar light. Within a block, $\frac{24}{6} = 4$ pots receive each treatment, so each treatment gets $4 \times 2 = 8$ pots overall, and the design uses $6 \times 8 = 48$ pots, which is every pot available.

(c) Randomize inside a block. Take the south half. Number its 24 pots 1 to 24 and label the 6 treatments 1 to 6. Use a random number generator to draw distinct integers from 1 to 24, ignoring repeats, until 4 have appeared; those pots receive treatment 1. Draw 4 more distinct unused integers for treatment 2, and continue until five treatments hold 4 pots each. The 4 pots whose numbers never came up receive treatment 6. Repeat the whole procedure independently in the north half.

(d) Decide whether light is confounded. It is not. Light is an extraneous variable that clearly affects growth, but every treatment appears exactly 4 times in the bright half and 4 times in the shaded half, so light does not vary with the treatments. Its variation is accounted for by the block. Light would be confounded if, for instance, all the pots on one fertilizer brand had been set in the bright half, because then a growth difference could belong to the brand or to the light and the two effects could not be separated.

(a) Count the possible assignments. Choosing which 10 of the 20 volunteers get the new routine is a combination, and every choice is equally likely: $\binom{20}{10} = 184{,}756$ different treatment groups.

(b) Find the probability of a lopsided draw. Condition on the first experienced lifter. Wherever that lifter lands, 9 slots in that group remain and 19 volunteers remain to fill them, all equally likely, so the second experienced lifter joins the first with probability $\frac{9}{19} \approx 0.474$. Counting the assignments directly agrees: both experienced lifters sit in the new-routine group for $\binom{18}{8} = 43{,}758$ of the assignments and both sit in the standard group for another $\binom{18}{10} = 43{,}758$, giving $87{,}516$ favorable assignments and a probability of $\frac{87516}{184756} \approx 0.474$.

(c) Read what the probability means. Random assignment balances extraneous variables across the groups on average, over many repetitions, not inside any single run. Here there is roughly a 47.4% chance that both experienced lifters land together, which is exactly the imbalance the researcher fears. Blocking removes it: treat the 2 experienced lifters as their own block and randomly send 1 to each routine, then randomly split the remaining 18 volunteers 9 and 9. Every group then holds exactly 1 experienced lifter by design rather than by luck.

(d) Split the claim into its two parts. Causation: the routines were randomly assigned, so a 4.2 kg gap that is larger than chance variation would produce can be attributed to the routine for these 20 volunteers. Generalization: the volunteers came from one app's mailing list and chose to sign up, so they are not a random sample of adults, and the word adults in that sentence is not supported. Rewrite it as: for volunteers like these, the new routine caused a greater mean improvement, and a randomly selected sample of adults would be needed before extending the claim to adults in general.