Final answer:
To find the probability of wait times at the dentist's office under a normal distribution with a mean of 30 minutes and a standard deviation of 6 minutes, the Z-scores are calculated for various time intervals. The results are approximately 0.1587 for less than 24 minutes, 0.1587 for more than 36 minutes, and 0.6826 for between 24 and 36 minutes. The probability of exactly 30 minutes is 0.
Step-by-step explanation:
Calculating Probabilities for Normal Distribution
To determine the probability of a randomly selected patient's wait time given a normal distribution with a mean of 30 minutes and a standard deviation of 6 minutes, we can use the standard normal distribution (Z-score). Below are the steps and solutions:
- A) Less than 24 minutes: First, calculate the Z-score using the formula: Z = (X - μ) / σ. Plugging in the numbers, we get Z = (24 - 30) / 6 = -1. From the standard normal distribution table, the probability (P) corresponding to Z = -1 is approximately 0.1587.
- B) More than 36 minutes: Using the same method, the Z-score for 36 minutes is Z = (36 - 30) / 6 = 1. The probability for Z > 1 is 1 - P(Z < 1), which is 1 - 0.8413 = 0.1587.
- C) Between 24 and 36 minutes: The probability of being between 24 and 36 minutes is the difference in probabilities for Z < 1 and Z = -1, which is 0.8413 - 0.1587 = 0.6826.
- D) Exactly 30 minutes: Since the normal distribution is continuous, the probability of observing a value exactly at the mean (or any specific value) is technically 0.
These calculations indicate the likelihood of different wait times at the dentist's office.