The amount of syrup that people put on their pancakes is normally distributed with mean $60 \mathrm{~mL}$ and standard deviation $8 \mathrm{~mL}$. Suppose that 6 randomly selected people are observed pouring syrup on their pancakes. Round all answers to 4 decimal places where possible.
a. What is the distribution of
b. What is the distribution of $\bar{x} ? \bar{x} \sim N($
c. If a single randomly selected individual is observed, find the probability that this person consumes is between $56.6 \mathrm{~mL}$ and $61.4 \mathrm{~mL}$.
d. For the group of 6 pancake eaters, find the probability that the average amount of syrup is between $56.6 \mathrm{~mL}$ and $61.4 \mathrm{~mL}$.
e. For part d), is the assumption that the distribution is normal necessary? No Yes

Step 1 ：Given that the amount of syrup that people put on their pancakes is normally distributed with a mean of \(60 \mathrm{~mL}\) and a standard deviation of \(8 \mathrm{~mL}\).

Step 2 ：We are asked to find the probability that a single randomly selected individual consumes between \(56.6 \mathrm{~mL}\) and \(61.4 \mathrm{~mL}\) of syrup.

Step 3 ：To solve this, we can use the Z-score formula to standardize the values and find the corresponding probabilities from the standard normal distribution.

Step 4 ：The Z-score for the lower bound \(56.6 \mathrm{~mL}\) is calculated as \(-0.425\).

Step 5 ：The Z-score for the upper bound \(61.4 \mathrm{~mL}\) is calculated as \(0.175\).

Step 6 ：Using the standard normal distribution, we find the probability that a single randomly selected individual consumes between \(56.6 \mathrm{~mL}\) and \(61.4 \mathrm{~mL}\) of syrup is approximately \(0.2340\).

Step 7 ：Thus, the final answer is \(\boxed{0.2340}\).