effect very close to zero? Or are there versions of the effect with different signs? However, if we report I c and I v we have evidence against Fisher’s sharp null, narrowing the plausible explanations for why zero is contained in I $_{v}$, which we’ve noted will tend to be similar to I $_{∗}$. The upper third of Figure 1, marked Γ = 1, shows 95% intervals for the football study, assuming that treatments are randomly assigned within matched sets. First, there are the three conventional intervals for τ , τ $^{a}$, and τ $^{b}$, corresponding to the three comparisons in Table 1. Each of these intervals is a 95% confidence interval on its own, but each one runs a 5% chance of error, so the chance that at least one interval fails to cover its corresponding parameter is greater than 5%. Obviously, we could make the three intervals longer, say using the Bonferroni inequality, so that the simultaneous coverage is 95%, but many investigators would find this unattractive because it would reduce the power of the conventional, primary analysis focused on τ that uses all of the controls; that is, it would make the first interval longer. In contrast, the intervals I c and I v in Figure 1 have simultaneous coverage of 95% in the sense of Proposition 1. Notably, I c = [ − 0 . 308 , 0 . 099] is the interval for τ from § 4, so consideration of I v has not reduced power to detect a constant effect. The versions I v = [ − 0 . 357 , 0 . 219] is slightly longer than I $_{c}$, but both intervals are compatible with no effect and both intervals are quite incompatible with an effect of half a word, ± 0 . 5. For comparison, recall from § 2 that average performance on the delayed word recall test declined by half a word from age 65 to age 72. In Figure 1, the 95% interval for “all controls” equals I $_{c}$, while I v is the union of the three intervals for “all controls”, “controls who played no sport”, and “controls who played another sport”. 5.3 Interval estimates in the football study 16