Math Problem Statement
The scheduled arrival time for a daily flight from Boston to New York is 9:35 am. Historical data show that the arrival time follows the continuous uniform distribution with an early arrival time of 9:13 am and a late arrival time of 9:47 am.
After converting the time data to a minute scale, calculate the mean and the standard deviation for the distribution.
Note: Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.
Solution
To solve this problem, we will first convert the arrival times to minutes, then calculate the mean and standard deviation for a uniform distribution.
Step 1: Convert the Time Data to Minutes
-
Convert 9:13 am to minutes:
-
Convert 9:47 am to minutes:
So, the arrival times in minutes are from 553 minutes to 587 minutes.
Step 2: Mean of the Uniform Distribution
For a continuous uniform distribution, the mean is calculated as follows:
Where:
- (the earliest time),
- (the latest time).
Substitute the values:
Step 3: Standard Deviation of the Uniform Distribution
The standard deviation for a uniform distribution is calculated with the formula:
Substitute the values:
Now, calculate and then divide:
Rounded to two decimal places, the standard deviation is:
Final Answer
- Mean: minutes
- Standard Deviation: minutes
Would you like further explanation on any of these steps?
Here are some additional questions that build on this concept:
- How would you calculate the probability of an arrival between 9:20 am and 9:40 am?
- If the earliest arrival time changed to 9:00 am, how would that affect the mean and standard deviation?
- How does a uniform distribution differ from a normal distribution in terms of shape and spread?
- Could this uniform distribution model be used for predicting arrival times of flights on other routes?
- How would you set up a hypothesis test to determine if future arrival times follow a uniform distribution?
Tip: Remember that for any continuous uniform distribution, probabilities are found by calculating the area under the distribution curve, which is constant for any given interval.
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Math Problem Analysis
Mathematical Concepts
Continuous Uniform Distribution
Descriptive Statistics
Formulas
Mean of Uniform Distribution: μ = (a + b) / 2
Standard Deviation of Uniform Distribution: σ = (b - a) / sqrt(12)
Theorems
Uniform Distribution Theorem
Suitable Grade Level
Grades 10-12
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