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Chapter 3: Problem 17

Use the following cell phone airport data speeds (Mbps) from Sprint. Find the percentile corresponding to the given data speed. $$\begin{array}{cccccccccc} 0.2 & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 & 0.4 & 0.4 & 0.4 \\ 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.6 & 0.6 & 0.7 & 0.8 & 1.0 \\ 1.1 & 1.1 & 1.2 & 1.2 & 1.6 & 1.6 & 2.1 & 2.1 & 2.3 & 2.4 \\ 2.5 & 2.7 & 2.7 & 2.7 & 3.2 & 3.4 & 3.6 & 3.8 & 4.0 & 4.0 \\ 5.0 & 5.6 & 8.2 & 9.6 & 10.6 & 13.0 & 14.1 & 15.1 & 15.2 & 30.4 \end{array}$$ $$2.4 \mathrm{Mbps}$$

Short Answer

Expert verified
58th Percentile

Step by step solution

01

Arrange the Data in Ascending Order

Ensure the data is listed in ascending order. The given data is already sorted from smallest to largest.
02

Count the Total Number of Data Points

Determine the number of data points. The data set contains 50 data points.
03

Identify the Position of the Given Data Speed

Locate the position of the given data speed (2.4 Mbps) in the sorted list. It appears as the 30th data point.
04

Use the Percentile Formula

Calculate the percentile using the formula: $$P = \frac{(N_p - 1)}{N} \times 100$$ where $$N_p$$ is the position in data set (30) and $$N$$ is the total number of data points (50).
05

Perform the Calculation

Substitute the values into the formula: $$P = \frac{(30 - 1)}{50} \times 100$$ $$P = \frac{29}{50} \times 100$$ $$P = 58 \text{th Percentile}$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Data Analysis
Data analysis involves inspecting, cleaning, and modeling data with the goal of discovering useful information and supporting decision-making. In our exercise, the data set provided comprises cell phone airport data speeds in Mbps.
Here's the analysis process:
  • Organizing Data: Before any meaningful analysis, data must be organized. Our data is already sorted in ascending order, simplifying analysis.
  • Counting Data Points: Knowing the total number of data points (50) helps us understand the data's extent.
  • Locating Specific Data Points: Identifying the position of a given data speed (2.4 Mbps) within the dataset allows us to proceed with calculations.
Thus, data analysis ensures that the data is ready for statistical operations and further calculations, like determining percentiles.
Statistics
Statistics is the discipline that involves collecting, analyzing, interpreting, presenting, and organizing data. It provides tools for data analysis and interpretation, which is crucial in our exercise.
For percentile calculation, the following statistical steps are important:
  • Percentile Formula: The formula for calculating a percentile position in a dataset is \( P = \frac{(N_p - 1)}{N} \times 100 \), where \( N_p \) is the position, and \( N \) is the total number of data points.
  • Substitution and Calculation: Substituting the values \( N_p = 30 \) and \( N = 50 \) into the formula, we get \( P = \frac{(30 - 1)}{50} \times 100 \), which simplifies to \( P = 58 \).
The percentile value tells us the relative standing of a particular data point within the entire dataset, which is a fundamental aspect of statistical analysis.
Percentile Rank
Percentile rank indicates the percentage of scores in its frequency distribution that are equal to or lower than it. It's a helpful way to understand the position of a particular score relative to all other scores in a dataset.
To calculate the percentile rank:
  • Identify Position: Locate the position of the given data speed in the sorted list (2.4 Mbps is the 30th data point).
  • Apply the Formula: Use the percentile formula \( P = \frac{(N_p - 1)}{N} \times 100 \), and substitute the required values.
  • Interpret Result: The result \( P = 58 \) tells us that 58% of the data speeds are less than or equal to 2.4 Mbps.
This informs us about how typical or atypical a given data speed is among the entire set of data speeds. Understanding percentile rank is crucial for interpreting data effectively.

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