Problem 32 - Algebra 2

Question

For a daily airline flight between two cities, the number of pieces of checked luggage has a mean of 380 and a standard deviation of \(20 .\) Use this information for Exercises \(32-33 .\) On what percent of the flights would you expect from 340 to 420 pieces of checked luggage? \(\begin{array}{llll}{\text { A. } 34 \%} & {\text { B. } 47.5 \%} & {\text { C. } 68 \%} & {\text { D. } 95 \%}\end{array}\)

Step-by-Step Solution

Answer

D. 95%

Step 1: Net transformations to Z-Score

The problem has given means (μ) as 380 and standard deviation (σ) as 20. The Z-Score for an individual datum can be determined by the formula \(Z = \frac {(X-μ)} {σ}\). Here, ‘X’ is the individual datum. Compute the Z-Scores for individual data points: \nFor 340, \n\( Z = \frac {(340-380)} {20} = -2 \) \nFor 420, \n\( Z = \frac {(420-380)} {20} = 2 \)

Step 2: Get the areas under curve from Z-table

Turn now to the Z-table to find the areas. For Z = -2, the area to the left of it is 0.0228. For Z = 2, the area to the left is 0.9772. To get the area between these two, subtract the area corresponding to Z = -2 from the area corresponding to Z = 2, which is 0.9772 - 0.0228 = 0.9544 or 95.44%.

Key Concepts

Z-Score

The Z-Score is a statistical measurement that describes a value's relationship to the mean of a group of values. Essentially, it represents how many standard deviations a certain point is from the mean. You calculate the Z-Score using the formula:

• \(Z = \frac{X - \mu}{\sigma}\)
• \(X\) is the value or score you're examining.
• \(\mu\) is the mean of the dataset.
• \(\sigma\) is the standard deviation of the dataset.

The Z-Score is important in understanding data distribution, especially when determining how extreme or normal a datapoint is within a dataset. For example, a Z-Score of 0 indicates that the data point is exactly at the mean, while a Z-Score of 2 means it's two standard deviations away from the mean. Calculating Z-Scores helps in comparing different scores across different datasets or in evaluating probabilities using the standard normal distribution.

Standard Deviation

Standard Deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, whereas a high standard deviation indicates that the values are spread out over a wider range. The formula for calculating standard deviation is:

• \(\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (X_i - \mu)^2}\)
• \(X_i\) is each individual value in the dataset.
• \(\mu\) is the mean of the dataset.
• \(N\) is the number of values in the dataset.

Understanding standard deviation is crucial for grasping the spread of the dataset. For example, in the context of the original exercise, knowing that the standard deviation is 20 enables you to determine how much variation exists around the mean of 380 pieces of luggage checked on the flight.

Mean

The mean, often called the average, is the central value of a dataset. It is calculated by adding up all the numbers in a dataset and then dividing by the count of those numbers. The formula is:

• \(\mu = \frac{1}{N} \sum_{i=1}^{N} X_i\)
• \(X_i\) represents each individual value.
• \(N\) is the total number of values.

The mean provides a simple yet informative snapshot of the data's overall trend. In the airline baggage example, the mean of 380 pieces tells you that, on average, this is the number expected per flight. This allows you to compare this typical value against specific instances or variations, using tools like the standard deviation and Z-Score.

Probability

Probability is a way to quantify the likelihood of a specific event occurring. In the context of a normal distribution, which is a continuous probability distribution, probabilities are often associated with areas under the curve. For example, when you want to find the probability of a value falling within a certain range, you use the Z-Score to look up corresponding probabilities on the Z-table. Key points to understand:

• Probabilities range from 0 to 1, where 0 means an event will not happen, and 1 means it will happen for sure.
• The total area under a normal distribution curve equals 1 (or 100%).
• In the exercise, the probability of pieces of luggage being between 340 and 420 can be found by calculating the area under the normal distribution curve between those Z-Scores.

Knowing how to calculate and interpret probabilities helps in making informed predictions and decisions based on statistical data. For example, the calculated probability of 95.44% indicates that it is very likely, based on the normal distribution, for the number of checked bags to fall between 340 and 420 pieces in this scenario.