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 \(\boxed{0.0874}\).