For $n=59$ and $X=27$, find the exact proportions $\widehat{p}$ and $\hat{q}$.
\[
\widehat{p}=\square \quad \hat{q}=\square
\]

Step 1 ：Given that $n=59$ and $X=27$, we can assume that $X$ is the number of successes.

Step 2 ：Therefore, the proportion of successes, denoted as $\widehat{p}$, would be calculated as $\frac{X}{n}$.

Step 3 ：Substituting the given values, we get $\widehat{p} = \frac{27}{59} = 0.4576271186440678$.

Step 4 ：The proportion of failures, denoted as $\hat{q}$, would be calculated as $1 - \widehat{p}$.

Step 5 ：Substituting the value of $\widehat{p}$, we get $\hat{q} = 1 - 0.4576271186440678 = 0.5423728813559322$.

Step 6 ：Final Answer: The exact proportions are $\widehat{p}=\boxed{0.4576271186440678}$ and $\hat{q}=\boxed{0.5423728813559322}$.