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

  1. Convert 9:13 am to minutes: 9:13 am=9×60+13=540+13=553 minutes9:13 \text{ am} = 9 \times 60 + 13 = 540 + 13 = 553 \text{ minutes}

  2. Convert 9:47 am to minutes: 9:47 am=9×60+47=540+47=587 minutes9:47 \text{ am} = 9 \times 60 + 47 = 540 + 47 = 587 \text{ 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:

μ=a+b2\mu = \frac{a + b}{2}

Where:

  • a=553a = 553 (the earliest time),
  • b=587b = 587 (the latest time).

Substitute the values:

μ=553+5872=11402=570 minutes\mu = \frac{553 + 587}{2} = \frac{1140}{2} = 570 \text{ minutes}

Step 3: Standard Deviation of the Uniform Distribution

The standard deviation for a uniform distribution is calculated with the formula:

σ=ba12\sigma = \frac{b - a}{\sqrt{12}}

Substitute the values:

σ=58755312=3412\sigma = \frac{587 - 553}{\sqrt{12}} = \frac{34}{\sqrt{12}}

Now, calculate 12\sqrt{12} and then divide:

  1. 123.4641\sqrt{12} \approx 3.4641

  2. σ=343.46419.8138\sigma = \frac{34}{3.4641} \approx 9.8138

Rounded to two decimal places, the standard deviation is:

σ9.81 minutes\sigma \approx 9.81 \text{ minutes}

Final Answer

  • Mean: 570570 minutes
  • Standard Deviation: 9.819.81 minutes

Would you like further explanation on any of these steps?


Here are some additional questions that build on this concept:

  1. How would you calculate the probability of an arrival between 9:20 am and 9:40 am?
  2. If the earliest arrival time changed to 9:00 am, how would that affect the mean and standard deviation?
  3. How does a uniform distribution differ from a normal distribution in terms of shape and spread?
  4. Could this uniform distribution model be used for predicting arrival times of flights on other routes?
  5. 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