Problems
2.1 (drill). Code the following for sensory predicates and report counts by class (V, A, K, U), stating the boundary rule you used for any case you found ambiguous.
"It doesn't feel settled to me. On paper it looks fine — the numbers are clean, anyone can see the case — but when Dana walked me through it, something in their tone was off. I keep turning it over. It's like there's a rough edge I can't quite put my finger on, and every time I try to focus on it, it slides away."
2.2 (drill). A speaker's classified predicate counts over a session are V = 30, A = 18, K = 32, with N = 80. Compute (a) the sample proportion for each class, (b) the standard error of the kinesthetic proportion under the independence assumption, and (c) its 95% Wald confidence interval.
2.3 (drill). Using the counts in Problem 2.2, test the equiprobable null p_V = p_A = p_K = 1/3. Report χ², df, and the exact p-value using the closed form for df = 2.
2.4. For the same counts, now test against an empirical baseline of p_V = 0.48, p_A = 0.19, p_K = 0.33 estimated from comparison speakers on the same topic. Report χ² and p, and state in two sentences what the difference between your answers to 2.3 and 2.4 licenses you to conclude about the speaker.
2.5. You want to estimate a speaker's visual proportion to within ±0.04 at 95% confidence, and you have no prior information about the value. (a) How many classified predicates do you need? (b) At a density of 2.3 classified predicates per 100 words and a speech rate of 150 words per minute, how many minutes of continuous speech is that? (c) Explain why you used p = 0.5 in the sample-size formula.
2.6. Recompute the confidence interval from Problem 2.2(c) accounting for clustering, given a mean of 9 predicates per conversational turn and an intraclass correlation of ρ = 0.20. Report the design effect, the effective standard error, the corrected interval, and the corrected answer to Problem 2.5(a).
2.7. A mental-rotation experiment yields the following condition means for picture-plane (two-dimensional) rotation:
θ (deg): 0 40 80 120 160
RT (ms): 990 1580 2210 2790 3430
(a) Fit RT = a + bθ by least squares, showing x̄, ȳ, Sxx, Sxy, b, and a. (b) Convert the slope to a rotation rate in degrees per second. (c) The depth-rotation condition from Worked Example 2.2 gave b = 16.05 ms/deg and a = 1004 ms. Compare the two fits and state what the comparison implies about the mechanism.
2.8. The same participants, viewing the same stimulus pairs at the same angles with the same keypress response, are asked instead to judge whether the two figures are drawn in the same ink colour. Means:
θ (deg): 0 40 80 120 160
RT (ms): 820 835 810 845 830
(a) Fit the line. (b) Test the slope against zero (df = 3, two-tailed critical t = 3.182). (c) Explain, in terms of the chapter's central distinction, why this null result is the strongest single piece of evidence in the whole example set.
2.9. A measure X of preferred representational system has reliability r_XX = 0.20; an outcome measure Y has r_YY = 0.80. An observed correlation of r_XY = 0.09 is reported. (a) Compute the maximum correlation observable between X and Y if the underlying true-score correlation were 1.00. (b) Disattenuate the observed correlation to estimate ρ_TU. (c) State one reason the disattenuated value should be treated with suspicion rather than celebrated.
2.10. You plan a study to detect a true observable correlation of r = 0.12, at α = 0.05 two-tailed. (a) Compute the Fisher z-transform of r. (b) Find the n required for 80% power. (c) Find the power actually achieved at n = 45. (d) A colleague proposes running n = 45 and, if the result is null, concluding the effect does not exist. Write the two-sentence objection.
2.11. Treat the eye-accessing chart as a diagnostic test with sensitivity 0.65, specificity 0.60, and a prior probability of visual retrieval of 0.30. (a) Compute the posterior probability of visual retrieval given the cue, and the positive likelihood ratio. (b) Compute the mutual information in bits between cue and state, and express it as a percentage of the prior entropy. (c) The chart specifies six gaze positions crossed with four task types. Under the complete null, what is the probability of at least one result significant at α = 0.05? How many independent tests would be needed for that probability to reach 0.90?
2.12 (extension). You are asked to design the study that should have been run in 1980, and to state in advance what each outcome would mean.
Consider a fully crossed within-participant design after Brooks (1968): task modality (spatial-imagery vs. verbal) × response modality (manual pointing vs. spoken response), with each participant completing many trials in each of the four cells. The dependent measure is a per-trial performance score.
(a) State the pattern of results predicted by the task-modality hypothesis (thought runs in the channel the task recruits) and the pattern predicted by the trait-PRS hypothesis (each person has a dominant channel). Be explicit about which term in the ANOVA each hypothesis loads onto.
(b) A random-effects decomposition of the resulting scores yields σ²_person = 0.4, σ²_task = 8.6, σ²_error = 3.0. Compute the proportion of total variance attributable to stable person differences.
(c) A person-level "preferred system" score is formed by averaging k independent task observations per person. Its reliability is given by the Spearman–Brown form
R(k) = k σ²_person / (k σ²_person + σ²_error)
Compute R(1), R(10), and R(40). How many observations are needed for R ≥ 0.80?
(d) Using (b) and (c), write the three-sentence verdict on the preferred-representational-system claim that is more accurate than "it was disproved."