Answer: To find the exact proportions ^p and ^q, we need to use the formulas for sample proportions:
^p = X / n
^q = (n - X) / n
where:
^p = proportion of successes (in this case, the proportion of successes is represented by X)
^q = proportion of failures (1 - ^p)
X = number of successes (in this case, X = 31)
n = total number of trials (in this case, n = 58)
Let's calculate ^p and ^q:
^p = X / n = 31 / 58 ≈ 0.5345 (rounded to four decimal places)
^q = (n - X) / n = (58 - 31) / 58 = 27 / 58 ≈ 0.4655 (rounded to four decimal places)
So, the exact proportions are approximately ^p = 0.5345 and ^q = 0.4655.