Extra Credit: Suppose that the amount of revenue that a food truck makes per day is normally distributed with a mean of $\$ 800$ and a standard deviation of $\$ 175$. Calculate the following probabilities:
(a) The probability that on a single day the food truck makes more than $\$ 900$ of revenue.
(b) Assuming each day is independent, the probability that the food truck makes more than $\$ 900$ of revenue for 5 days in a row.
(c) The probability that the food truck averages more than $\$ 900$ of revenue over the span of 5 days.
Answer
Final Answer: The probability that the food truck makes more than \$900 in a single day is approximately \(\boxed{0.284}\) or \(\boxed{28.4\%}\).
Step 1 ：The problem is asking for probabilities related to a normal distribution. The normal distribution is defined by two parameters: the mean and the standard deviation. In this case, the mean is \(800\) and the standard deviation is \(175\).
Step 2 ：To find the probability that the food truck makes more than \$900 in a single day, we need to find the z-score for \$900 and then find the area to the right of that z-score on the standard normal distribution. The z-score is calculated as \((X - \text{mean}) / \text{standard deviation}\).
Step 3 ：Substitute the given values into the z-score formula: \(z = (900 - 800) / 175 = 0.5714285714285714\).
Step 4 ：The probability corresponding to this z-score is approximately 0.2838545830986763.
Step 5 ：Final Answer: The probability that the food truck makes more than \$900 in a single day is approximately \(\boxed{0.284}\) or \(\boxed{28.4\%}\).
