Problem 3 - Basic Statistics for Business and Economics

Question

America West Airlines reports the flight time from Los Angeles International Airport to Las Vegas is 1 hour and 5 minutes, or 65 minutes. Suppose the actual flying time is uniformly distributed between 60 and 70 minutes. a. Show a graph of the continuous probability distribution. b. What is the mean flight time? What is the variance of the flight times? c. What is the probability the flight time is less than 68 minutes? d. What is the probability the flight takes more than 64 minutes?

Step-by-Step Solution

Answer

Mean is 65 minutes, variance is 8.33, P(X<68) is 0.8, P(X>64) is 0.6.

Step 1: Understand the Uniform Distribution

The problem states that the actual flying time is uniformly distributed between 60 and 70 minutes. In a uniform distribution, every interval of the same length within the range is equally likely. This means that any amount of flight time between 60 minutes and 70 minutes has an equal chance of occurring.

Step 2: Graph the Probability Distribution

Uniform distributions are represented as a rectangle over the range of possible values. In this case, the rectangle starts at 60 minutes and ends at 70 minutes, with a constant height, since probability density is constant across the interval.

Step 3: Calculate the Mean of the Uniform Distribution

The mean (average) of a uniform distribution between two values, \( a \) and \( b \), is calculated using the formula:\[\mu = \frac{a + b}{2}\]Substitute \( a = 60 \) and \( b = 70 \):\[\mu = \frac{60 + 70}{2} = 65\]Thus, the mean flight time is 65 minutes.

Step 4: Calculate the Variance of the Uniform Distribution

The formula for the variance of a uniform distribution is:\[\sigma^2 = \frac{(b - a)^2}{12}\]Substitute \( a = 60 \) and \( b = 70 \):\[\sigma^2 = \frac{(70 - 60)^2}{12} = \frac{100}{12} = \frac{25}{3} \approx 8.33\]The variance of the flight times is approximately 8.33.

Step 5: Calculate the Probability for Less than 68 Minutes

The probability of the flight time being less than 68 minutes in a uniform distribution is the area under the distribution curve from 60 to 68 minutes:\[P(X < 68) = \frac{68 - 60}{70 - 60} = \frac{8}{10} = 0.8\]Thus, the probability is 0.8.

Step 6: Calculate the Probability for More than 64 Minutes

The probability of the flight taking more than 64 minutes is the area from 64 to 70 minutes:\[P(X > 64) = \frac{70 - 64}{70 - 60} = \frac{6}{10} = 0.6\]Thus, the probability is 0.6.

Key Concepts

Continuous Probability Distribution

In mathematics, a continuous probability distribution is a type of probability distribution where the set of possible outcomes can take any value within a given range. This is different from discrete probability distributions, where the possible outcomes are distinct and separate values.
For a continuous uniform distribution, like the one described in the flight times from Los Angeles to Las Vegas, any value between the minimum and maximum boundary is equally likely to occur. This means that this type of distribution is represented graphically by a flat, rectangular shape.
Here, every minute between 60 and 70 has an equal chance, creating a rectangle that starts at 60 and finishes at 70 on the number line, showcasing that the probability density across this interval is constant.

Mean Calculation

In statistics, the mean provides a measure of the central tendency of a set of numbers. For a uniform distribution, this will be the midpoint of the defined interval.
The formula for calculating the mean \( \, \mu \, \) in a uniform distribution where the range is from \( a \) to \( b \) is:
\[\mu = \frac{a + b}{2}\]
Substituting the flight time values \( a = 60 \) and \( b = 70 \):
\[\mu = \frac{60 + 70}{2} = 65\]
This provides us with a mean flight time of 65 minutes, indicating that, on average, flights take this time to travel between Los Angeles and Las Vegas.

Variance Calculation

Variance is a statistical measurement that helps us understand the spread or spread of a set of numbers. In a uniform distribution, variance demonstrates how much the numbers in the interval deviate from the mean.
The formula for variance \(\sigma^2 \) in a uniform distribution over an interval \([a, b]\) is:
\[\sigma^2 = \frac{(b - a)^2}{12}\]
So, for this exercise, the variance is calculated as:
\[\sigma^2 = \frac{(70 - 60)^2}{12} = \frac{100}{12} = \frac{25}{3} \approx 8.33\]
This means the flight times deviate from the mean of 65 minutes by about 8.33 minutes squared, indicating a moderate dispersion in flight times.

Probability Calculation

Probability helps us determine the likelihood of a specific event occurring. In the context of continuous uniform distributions, probabilities relate to the length of the interval within the possible range.
The probability calculation involves finding the area of the desired interval divided by the total length of the distribution interval.
For the flight times:

• For a flight time of less than 68 minutes: \[P(X < 68) = \frac{68 - 60}{70 - 60} = \frac{8}{10} = 0.8\]
• For flight times exceeding 64 minutes: \[P(X > 64) = \frac{70 - 64}{70 - 60} = \frac{6}{10} = 0.6\]

These calculations show the percentage of flights likely to take less than 68 minutes (80%) and more likely to be longer than 64 minutes (60%). Understanding probabilities in this way helps to see the likelihood of different flight times and what to expect when planning travel.