A fire department in a rural county reports that it’s response time to fires is approximately normally distributed with a mean of 22 minutes and a standard deviation of 11.9 minutes. Assume that this claim is true. Ninety-nine percent of the time, the fire department responds to a fire within how many minutes? Round to the nearest whole minute. A fire department in a rural county reports that it’s response time to fires is approximately normally distributed with a mean of 22 minutes and a standard deviation of 11.9 minutes. Assume that this claim is true. Ninety-nine percent of the time, the fire department responds to a fire within how many minutes? Round to the nearest whole minute.
The correct answer and explanation is :
To determine the response time within which the fire department responds to 99% of fires, we need to find the 99th percentile of the normal distribution characterized by a mean (μ) of 22 minutes and a standard deviation (σ) of 11.9 minutes.
Understanding Percentiles in a Normal Distribution:
In a normal distribution, percentiles indicate the value below which a certain percentage of observations fall. The 99th percentile represents the value below which 99% of the data points lie, meaning only 1% of the data exceeds this value.
Calculating the 99th Percentile:
The formula to calculate a specific percentile (P) in a normal distribution is:
[ P = \mu + (Z \times \sigma) ]
Where:
- ( \mu ) is the mean.
- ( \sigma ) is the standard deviation.
- ( Z ) is the z-score corresponding to the desired percentile.
For the 99th percentile, the z-score (Z) is approximately 2.3263.
Plugging in the values:
[ P_{99} = 22 + (2.3263 \times 11.9) ]
Calculating the product:
[ 2.3263 \times 11.9 \approx 27.68297 ]
Adding the mean:
[ 22 + 27.68297 \approx 49.68297 ]
Rounding to the nearest whole minute:
[ P_{99} \approx 50 \text{ minutes} ]
Interpretation:
This calculation indicates that 99% of the time, the fire department responds to fires within approximately 50 minutes. Conversely, there’s a 1% chance that their response time will exceed 50 minutes.
Visual Representation:
To visualize this, consider a bell-shaped curve representing the normal distribution of response times. The mean (22 minutes) is at the center. The area under the curve to the left of the 99th percentile (approximately 50 minutes) encompasses 99% of all response times. This area represents the proportion of responses that occur within this time frame.
Practical Implications:
Understanding this percentile is crucial for emergency response planning. It helps in setting realistic expectations and benchmarks for performance. If the goal is to respond to 99% of fires within a certain time, resources and strategies can be aligned to achieve this target.
Conclusion:
By leveraging the properties of the normal distribution and using the mean and standard deviation of the fire department’s response times, we’ve determined that they respond to 99% of fires within approximately 50 minutes. This statistical insight aids in performance evaluation and strategic planning for emergency services.