Final answer:
The probability that the waitress will earn at least $400 over a weekend is 0.50, and for the best 1% of weekends, she earns approximately $388.59.
Step-by-step explanation:
To answer the question of estimating the probability that the waitress will earn at least $400 over a weekend and how much she earns on the best 1% of such weekends, we need to understand that tips have a model that is slightly skewed to the right, with a mean of $10.50 per party and a standard deviation of $5.70. Since she waits on about 30 parties, we can calculate the mean and standard deviation for the total tips over the weekend.
First, let's calculate the mean total tips for the weekend:
Mean total tips = Mean per party × Number of parties = $10.50 × 30 = $315.
Next, the standard deviation for the weekend:
Standard deviation of total tips = Standard deviation per party × √(Number of parties) = $5.70 × √30 ≈ $31.41.
a) Since the sample size is 30, by the Central Limit Theorem, we can assume the distribution of the total tips is approximately normally distributed even though the initial distribution is slightly skewed. To find the probability of earning at least $400:
- First, find the z-score:
Z = (X - Mean total tips) / (Standard deviation of total tips) = ($400 - $315) / $31.41 ≈ 2.70. - Next, look up the z-score in a z-table or use technology to find the probability that Z is greater than 2.70, which corresponds to the probability of earning less than $400. Subtract this value from 1 to find the probability of earning at least $400. (1-0.496533 = 0.50)
b) To find how much she earns on the best 1% of weekends, we need to find the z-score that corresponds to the top 1% of a normal distribution. This z-score is approximately 2.33. Using this z-score:
- Calculate the tip amount using the formula:
Tip amount = Mean total tips + (z-score × Standard deviation of total tips) = $315 + (2.33 × $31.41) ≈ $388.59.
Therefore, on the best 1% of weekends, she earns approximately $388.59.