Problem 44 - Basic Statistics for Business and Economics

Question

The current model Boeing 737 has a capacity of 189 passengers. Suppose Delta Airlines uses this equipment for its Atlanta to Houston flights. The distribution of the number of seats sold for the Atlanta to Houston flights follows the normal distribution with a mean of 155 seats and a standard deviation of 15 seats. a. What is the likelihood of selling more than 134 seats? b. What is the likelihood of selling less than 173 seats? c. What is the likelihood of selling more than 134 seats but less than 173 seats? d. What percent of the time would Delta be able to sell more seats than there are seats actually available?

Step-by-Step Solution

Answer

a. 0.9192 b. 0.8849 c. 0.8041 d. 0.012

Step 1: Understanding the Problem

We need to calculate probabilities associated with a normal distribution. The random variable, the number of seats sold, is normally distributed with a mean (\( \mu \)) of 155 and a standard deviation (\( \sigma \)) of 15.

Step 2: Finding Probability of Selling More Than 134 Seats

To calculate the probability of selling more than 134 seats, we first standardize 134 to a Z-score using the formula: \[ Z = \frac{X - \mu}{\sigma} \]where \( X = 134 \), \( \mu = 155 \), \( \sigma = 15 \). Calculate: \[ Z = \frac{134 - 155}{15} = \frac{-21}{15} \approx -1.4 \]Using the Z-table, find the probability corresponding to a Z-score of -1.4, which is approximately 0.0808. Therefore, the probability of selling more than 134 seats is \( 1 - 0.0808 = 0.9192 \).

Step 3: Finding Probability of Selling Less Than 173 Seats

Standardize 173 to a Z-score: \[ Z = \frac{173 - 155}{15} = \frac{18}{15} = 1.2 \]Using the Z-table, find the probability for a Z-score of 1.2, which is approximately 0.8849. Therefore, the probability of selling less than 173 seats is 0.8849.

Step 4: Finding Probability Between 134 and 173 Seats

We have found probabilities for more than 134 seats and less than 173 seats. To find the probability between these values, subtract the probability of selling more than 173 seats from the probability of selling more than 134 seats: \[ P(134 < X < 173) = P(X < 173) - P(X < 134) = 0.8849 - 0.0808 = 0.8041 \]

Step 5: Probability of Selling More Than 189 Seats

Standardize 189 to find the likelihood of selling more seats than available: \[ Z = \frac{189 - 155}{15} = \frac{34}{15} \approx 2.27 \]Using the Z-table, the probability for a Z-score of 2.27 is approximately 0.988. Thus, the probability of selling more than 189 seats is \( 1 - 0.988 = 0.012 \).

Step 6: Concluding Remarks

The answers to the questions are as follows: a. Probability of selling more than 134 seats is approximately 0.9192. b. Probability of selling less than 173 seats is approximately 0.8849. c. Probability of selling between 134 and 173 seats is approximately 0.8041. d. Probability of selling more seats than the 189 available is approximately 0.012.

Key Concepts

Probability Calculation

Probability calculation is a fundamental concept in statistics that helps determine the likelihood of various outcomes. In the context of a normal distribution, calculating probabilities often involves transforming data points (like the number of seats sold) into a standard form to make them easy to compare.
This standard form is typically referred to as the Z-score, which helps measure how far a particular data point is from the mean in terms of standard deviations.To calculate probability, follow these basic steps:

• Identify the mean (\( \mu \)) and standard deviation (\( \sigma \)) of your data set.
• Convert your specific point of interest (such as 134 or 173 seats) into a Z-score.
• Use the Z-table to find the probability that corresponds to your Z-score.
• If you want the probability of a range, you might need to subtract two probabilities.

Understanding these steps will make it easier for you to tackle a variety of probability-related questions involving normally distributed data.

Z-score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. Expressed in terms of standard deviations, a Z-score indicates how many standard deviations away a given value is from the mean.
For example, a Z-score of 1 indicates that the value is one standard deviation away from the mean, while a positive or negative sign tells you whether the value is above or below the mean.Here's the formula to calculate a Z-score:\[ Z = \frac{X - \mu}{\sigma} \]where:

• \( X \) is your data point of interest.
• \( \mu \) is the mean of the data.
• \( \sigma \) is the standard deviation.

Using Z-scores allows you to convert different sets of data into a standardized format, enabling an easy comparison of different data points across various datasets.
This becomes particularly valuable when you are handling normally distributed data, making such calculations not just common but essential.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.
In normal distributions, standard deviation is a key component for calculating probabilities.In mathematical terms, the standard deviation (\( \sigma \)) is calculated using the equation:\[ \sigma = \sqrt{\frac{\sum (X_i - \mu)^2}{N}} \]where:

• \( X_i \) are individual data points.
• \( \mu \) is the mean of the data points.
• \( N \) is the number of data points.

Standard deviation is crucial for understanding how your data is distributed. It helps in interpreting whether a certain value is typical or unusual within the context of the data set.
In the scenario of Delta Airlines' flights, knowing the standard deviation allows us to understand how variable the number of seats sold is, assisting in predicting potential outcomes.

Mean in Statistics

In statistics, the mean is a measure of central tendency that provides an average value from a set of numbers. It represents a central or typical value for a probability distribution.
To find the mean, you simply sum all the data points and then divide by the number of points.Here's the formula for calculating the mean (\( \mu \)):\[ \mu = \frac{\sum X}{N} \]where:

• \( \sum X \) is the sum of all data points.
• \( N \) is the number of data points.

The mean gives you a quick snapshot of where the data centers, and it's vital for understanding broader statistical concepts like normal distribution, standard deviation, and Z-scores.
In our example, the mean number of seats sold is 155, which places it as the value around which we gauge variability, calculate probabilities, and predict sales outcomes for the flights.