in new york city the average response time for calls to the first depatrment is a bout 5 minutes with a standarrd deviation of 1.5 minuties suppose response follow a normal distribuation.
1-what is the probability that a response time exceeds 9 minutes/
2-compute the top 10 th percentile for response times ?
http://davidmlane.com/hyperstat/z_table.html
To answer both questions, we need to use the concept of Z-scores and the standard normal distribution. Here's how you can calculate the probability and the top 10th percentile for response times in New York City:
1. Probability that a response time exceeds 9 minutes:
First, we need to calculate the Z-score for 9 minutes using the formula: Z = (x - μ) / σ
where x is the value we want to calculate the probability for, μ is the mean (average response time), and σ is the standard deviation.
Z = (9 - 5) / 1.5 = 4 / 1.5 = 2.67
Next, we need to find the probability associated with the Z-score. We can use a standard normal distribution table or a calculator to find this probability. The probability of a Z-score being greater than 2.67 is approximately 0.0038 or 0.38%.
Therefore, the probability that a response time exceeds 9 minutes is approximately 0.0038 or 0.38%.
2. Compute the top 10th percentile for response times:
The top 10th percentile represents the value below which 10% of the response times fall. In other words, we need to find the Z-score at the 10th percentile.
Using the standard normal distribution table, we can find the Z-score associated with the 10th percentile, which is approximately -1.28.
Now, we can use the Z-score formula to find the response time corresponding to this Z-score:
Z = (x - μ) / σ
-1.28 = (x - 5) / 1.5
Solving for x:
-1.92 = x - 5
x = 5 - 1.92
x ≈ 3.08
Therefore, the top 10th percentile for response times is approximately 3.08 minutes.