Problems
Drill
1. Five within-session readings of an untouched memory: 54, 61, 58, 63, 59. Compute the mean, the sample standard deviation, the standard error of a single within-session Δ, and the 95% minimum detectable effect. State in one sentence what a Δ of 7 means.
2. A practitioner runs eight OAT probes; within-session noise is ε = 4.0. The reductions are: brightness 4, size 8, colour 12, distance 18, location 7, motion 16, volume 3, tempo 10. (a) Classify at the uncorrected 95% threshold. (b) Re-classify with a Bonferroni correction across the family of eight. (c) How many false positives should you expect at the uncorrected threshold if none of the eight submodalities does anything?
3. A client's log gives μ = 55, σ = 14, and a fortnight-apart r = 0.55. She arrives reading X₁ = 82; a fortnight after the session she reads 64. (a) What reading was expected with no intervention? (b) How many of the eighteen points are attributable to the work? (c) Compute the conditional SD and the corresponding z. (d) What may be claimed?
4. Δa = 16, Δb = 14, Δ(a,b) = 18, baseline S₀ = 70, and the client reads a neutral memory of the same age at 6. Compute I and R, perform the floor check, and state the verdict and the next probe.
Applied
5. For a given client the occasion-plus-noise variance is ω² + ε² = 100. (a) With equal numbers of baseline and follow-up occasions, how many occasions per side are needed to certify a 15-point effect at 80% power, two-tailed α = 0.05? (b) The Ch. 6 composite averages two indicators correlating r = 0.55. Give the composite's reliability by the Spearman–Brown formula, and the reliability if a third comparable indicator is added. (c) Which of (a) or (b) buys you more, and why?
6. A contrast yields k = 7 differing submodalities. Compute: the size of the complete intervention space; the number of OAT probes; the number of probes in a screen-then-top-3-pairs design; the fraction of pairs that design tests; the fraction of triples it tests. Then write the one sentence you owe the client about what your design cannot see.
7. You suspect that size and distance are confounded in a client's imagery. Design the pair of isolation probes that separates them, and write out the readings each of two hypotheses predicts: (H1) the driver is apparent visual angle; (H2) the driver is felt egocentric proximity independent of angle. Baseline S₀ = 64, size-alone Δ = 20, distance-alone Δ = 22, ε = 4.5. Give the predicted Δ under each hypothesis for the probe "recede the image while scaling it up so its apparent size is unchanged," and state which observed values would leave you unable to decide.
8. A client reports no visual imagery at all — he knows where the memory's events stand in relation to him, and can say which is nearer, but sees nothing. (a) Explain why the standard visual sweep is not merely difficult but invalid here, and what it would produce if you ran it anyway. (b) Redesign the driver hunt. (c) Name the one thing about his report that the redesign must take seriously rather than treat as a workaround.
9. A firefighter's most charged memory returns a driver: distance, Δ = 27, D = 4.9, clean isolation probe. Under the stop rule he answers: (1) "It makes me walk the building before I commit anyone." (2) "I'd commit faster." (3) "...I don't know." State what you do, in order, and what you would need before the change becomes admissible. Then state the symmetrical error — the failure mode of applying the stop rule too readily.
Extension
10. Derive E[X₂ | X₁ = x] = μ + r(x − μ) from the bivariate normal density. Carry every step: write the joint density, form the conditional as joint over marginal, complete the square in the exponent, and read off the conditional mean and variance. Then state, in one sentence each, (a) what r = 0 implies, (b) what r = 1 implies, and (c) why a practitioner who selects clients on an extreme baseline reading will observe apparent improvement at a rate that has nothing to do with technique.
11. Three drivers survive a sweep: Δa = 22, Δb = 20, Δc = 15, ε = 3.9. The pairwise redundancy coefficients are R(a,b) = 0.91, R(a,c) = 0.12, R(b,c) = 0.08. (a) How many independent dimensions does the data support, and which submodalities load on which? (b) Under the model "fully redundant within a cluster, additive across clusters," predict Δ(a,b,c). (c) The observed Δ(a,b,c) is 34. Compute the residual, express it in standard errors, and state whether the model survives. (d) Propose a name for the latent dimension that {a,b} share, and the single probe that would test the name rather than the maths.
12. Construct a falsification design for the central claim of this chapter — that submodality parameters, and not merely attention, expectancy, and regression, drive state change. Specify: the sham condition, what can and cannot be blinded and why, the pre-registered predictions, the sample and the analysis, and — the part most designs omit — the result that would oblige you to abandon the model. Then state the strongest available objection to submodality work at full strength, and answer it or concede it.