Final answer:
The probability that a randomly selected customer will have to wait less than 41 minutes at a restaurant, where wait times are normally distributed with a mean of 46 minutes and a standard deviation of 4 minutes, is approximately 0.106, to the nearest thousandth.
Step-by-step explanation:
To calculate the probability that a randomly selected customer will have to wait less than 41 minutes for their food at a local restaurant, where the wait times are normally distributed with a mean of 46 minutes and a standard deviation of 4 minutes, we need to find the z-score and then use the standard normal distribution to find the probability.
First, we calculate the z-score using the following formula:
Z = (X - μ) / σ
Where:
- X is the value we are comparing to the mean (41 minutes),
- μ is the mean (46 minutes),
- σ is the standard deviation (4 minutes).
Plugging in the values we get:
Z = (41 - 46) / 4 = -5 / 4 = -1.25
To find the probability of a z-score being less than -1.25, we refer to the standard normal distribution table (z-table), which gives us the probability that a standard normally distributed variable is less than -1.25. This is equivalent to the cumulative probability up to that point.
Looking up the z-score of -1.25 on the z-table gives us a probability of approximately 0.1056. Therefore, the probability that a randomly selected customer will have to wait less than 41 minutes is 0.106, to the nearest thousandth.