Functionally important traits in animals tend to vary little from one individual to the next within populations, possible because individuals that deviate too much from the mean die sooner or leave fewer offspring in the long run. If so, does variance in a trait rise after it becomes less functionally important? Billet et al. (2012) investigated this question with the semicircular canals (SC) of the inner ear of the three-toed sloth (Bradypus variegatus). Sloths move very slowly and infrequently, and the authors suggested that this behavior reduces the functional demands on the SC, which usually provide information on angular head movement to the brain. Indeed, the motion signal from the SC to the brain may be very weak in sloths as compared to faster-moving animals. The following numbers are measurements of the length to the width of the anterior semicircular canals in seven sloths. Assume that this represents a random sample. 1.52, 1.06, 0.93, 1.38, 1.47, 1.20, 1.16 a. In related, faster-moving animals, the standard deviation of the ratio of the length to the width of the anterior semicircular canals is known to be 0.09. What is the estimate of the standard deviation of this measurement in three toed sloths? b. Based on these data, what is the most plausible range of values for the population standard deviation in the three-toed sloth? Does this range include the known value of the standard deviation in related, faster-moving species? c. What additional assumption is required for your answer in (b)? What do you know about how sensitive the confidence interval calculation is when the assumption is not met? Solution.

1. Functionally important traits in animals tend to vary little from one individual to the next within
populations, possible because individuals that deviate too much from the mean die sooner or
leave fewer offspring in the long run. If so, does variance in a trait rise after it becomes less
functionally important? Billet et al. (2012) investigated this question with the semicircular canals
(SC) of the inner ear of the three-toed sloth (Bradypus variegatus). Sloths move very slowly and
infrequently, and the authors suggested that this behavior reduces the functional demands on the
SC, which usually provide information on angular head movement to the brain. Indeed, the
motion signal from the SC to the brain may be very weak in sloths as compared to faster-moving
animals. The following numbers are measurements of the length to the width of the anterior
semicircular canals in seven sloths. Assume that this represents a random sample.
1.52, 1.06, 0.93, 1.38, 1.47, 1.20, 1.16
a. In related, faster-moving animals, the standard deviation of the ratio of the length to the width
of the anterior semicircular canals is known to be 0.09. What is the estimate of the standard
deviation of this measurement in three toed sloths?
b. Based on these data, what is the most plausible range of values for the population standard
deviation in the three-toed sloth? Does this range include the known value of the standard
deviation in related, faster-moving species?
c. What additional assumption is required for your answer in (b)? What do you know about how
sensitive the confidence interval calculation is when the assumption is not met?
Solution