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Chapter 3: Problem 20

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

Short Answer

Expert verified
The percentile for 9.6 Mbps is the 84th percentile.

Step by step solution

01

- Sort the Data

First, ensure the data speeds are sorted in ascending order, which they already are in this case.
02

- Identify the Position of the Speed

Next, find the position of the given data speed (9.6 Mbps) in the sorted list. In this case, 9.6 is at position 43.
03

- Calculate the Percentile

Use the formula for percentile calculation: \ P = \(\frac{L}{N}\)*100 \ where L is the number of values less than the given data speed and N is the total number of values. Here, L = 42 (values below 9.6 Mbps) and N = 50 (total values). So, P = \(\frac{42}{50}\)*100 = 84.
04

- Interpret the Result

The calculated percentile indicates the percentage of data points that fall below the given data speed. Therefore, 84% of the data speeds are below 9.6 Mbps.

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Key Concepts

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

Percentile
A percentile is a measure used in statistics to express the relative position of a value within a dataset. In simpler terms, it tells you what percentage of data points are below a specific number. For example, if you scored in the 84th percentile on a test, it means you scored better than 84% of the other test-takers.

Simplified: Percentiles divide a dataset into 100 equal parts. This makes it easier to understand the distribution and compare different datasets.

In our cell phone data speed example, the 84th percentile means 84% of the data speeds are slower than 9.6 Mbps.
Data Sorting
Sorting data is the first and essential step in percentile calculation. This process involves arranging the data points in ascending order (from smallest to largest).

Why is this important?
  • It helps in quickly identifying the position of a specific data point.
  • Ensures that calculations for percentiles and other statistical metrics are accurate.

In our example, the data speeds are already sorted:
  • From 0.2 Mbps (the smallest) to 30.4 Mbps (the largest).
By sorting our data, we make the identification of the position straightforward.
Position Identification
Once the data is sorted, the next step is to identify the position of the specific data point in the sorted list. In our example, we are interested in the data speed of 9.6 Mbps, which is at position 43.

Why is Position Important? The position allows us to see how many data points lie below the given value. This number is represented as 'L' in the percentile formula. Here, L is the number of values less than the given data speed (42 values below 9.6 Mbps).

This step is crucial for applying the percentile calculation formula accurately.
Educational Statistics
Statistics play a significant role in education and data analysis. Understanding core concepts like percentiles, data sorting, and position identification helps students interpret data more meaningfully.

Key Points to Remember:
  • Percentiles: Tell you the relative standing of a value within a dataset.
  • Data Sorting: Essential first step for most statistical calculations.
  • Position Identification: Helps determine how many values lie below a specific data point.

Statistical literacy ensures that students can handle real-world data effectively.

By mastering these concepts, students can perform better analyses and make informed decisions based on statistical data.

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Most popular questions from this chapter

Use the given data to construct a boxplot and identify the 5-number summary. The following are the ratings of males by females in an experiment involving speed dating. \(\begin{array}{llllllllllllllllllll}2.0 & 3.0 & 4.0 & 5.0 & 6.0 & 6.0 & 7.0 & 7.0 & 7.0 & 7.0 & 7.0 & 7.0 & 8.0 & 8.0 & 8.0 & 8.0 & 9.0 & 9.5 & 10.0 & 10.0\end{array}\)

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The geometric mean is often used in business and economics for finding average rates of change, average rates of growth, or average ratios. To find the geometric mean of \(n\) values (all of which are positive), first multiply the values, then find the \(n\) th root of the product. For a 6-year period, money deposited in annual certificates of deposit had annual interest rates of \(5.154 \%, 2.730 \%, 0.488 \%, 0.319 \%, 0.313 \%\), and \(0.268 \%\). Identify the single percentage growth rate that is the same as the five consecutive growth rates by computing the geometric mean of \(1.05154,1.02730,1.00488,1.00319,1.00313\), and \(1.00268\).

The 20 subjects used in Data Set 8 "IQ and Brain Size" in Appendix B have weights with a standard deviation of \(20.0414 \mathrm{~kg}\). What is the variance of their weights? Be sure to include the appropriate units with the result.

Watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, \((b)\) median, (c) mode, (d) midrange, and then answer the given question. Listed below are selling prices (dollars) of TVs that are 60 inches or larger and rated as a "best buy" by Consumer Reports magazine. Are the resulting statistics representative of the population of all TVs that are 60 inches and larger? If you decide to buy one of these TVs, what statistic is most relevant, other than the measures of central tendency? \(\begin{array}{llllllllllll}1800 & 1500 & 1200 & 1500 & 1400 & 1600 & 1500 & 950 & 1600 & 1150 & 1500 & 1750\end{array}\)
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