The amount of syrup that people put on their pancakes is normally distributed with mean $65 \mathrm{~mL}$ and standard deviation $7 \mathrm{~mL}$. Suppose that 40 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 $X ? X-N(65$
b. What is the distribution of $\bar{x} ? \bar{x}-\mathrm{N}$
65 $1.625 \times 0$ \% $o^{8} 1.1068$
C. If a single randomly selected individual is observed, find the probability that this person consumes is between $66.2 \mathrm{~mL}$ and $67.8 \mathrm{~mL}$. 14.2857
d. For the group of 40 pancake eaters, find the probability that the average amount of syrup is between $66.2 \mathrm{~mL}$ and $67.8 \mathrm{~mL}$.
e. For part d), is the assumiptiontinat of decimal number, accurate to at least 4 decimal places [more.. fo
Hint:

Step 1 ：Given that the amount of syrup that people put on their pancakes is normally distributed with mean \(65 \mathrm{~mL}\) and standard deviation \(7 \mathrm{~mL}\).

Step 2 ：We are asked to find the probability that a single individual consumes between \(66.2 \mathrm{~mL}\) and \(67.8 \mathrm{~mL}\) of syrup.

Step 3 ：We can use the formula for the z-score to find the z-scores for \(66.2 \mathrm{~mL}\) and \(67.8 \mathrm{~mL}\). The z-score is given by the formula: \[z = \frac{x - \mu}{\sigma}\] where x is the value we are interested in, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Step 4 ：Substituting the given values into the formula, we get \(z1 = 0.1714\) and \(z2 = 0.4\).

Step 5 ：We can use a z-table to find the probabilities associated with these z-scores. The probability that a value is between two z-scores is given by the difference of the probabilities associated with these z-scores.

Step 6 ：From the z-table, we find that the probabilities associated with \(z1\) and \(z2\) are approximately \(0.5681\) and \(0.6554\) respectively.

Step 7 ：The probability that a single individual consumes between \(66.2 \mathrm{~mL}\) and \(67.8 \mathrm{~mL}\) of syrup is therefore \(0.6554 - 0.5681 = 0.0873\).

Step 8 ：Final Answer: The probability that a single individual consumes between \(66.2 \mathrm{~mL}\) and \(67.8 \mathrm{~mL}\) of syrup is approximately \(\boxed{0.0874}\).