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Chapter 2: Problem 27

a. For runners in a race, a higher speed means a faster run. Is it more desirable to have a speed with a high or a low percentile when running a race? b. The 40th percentile of speeds in a particular race is 7.5 miles per hour. Write a sentence interpreting the 40th percentile in the context of the situation.

Short Answer

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a. A higher percentile speed is more desirable. b. 40% of runners have speeds less than or equal to 7.5 mph.

Step by step solution

01

Understanding Percentiles and Speed

In any context, including running, a higher speed indicates a faster run. Thus, when we consider percentiles, a higher percentile indicates that the speed is greater than that achieved by most of the participants. Therefore, having a speed with a high percentile is more desirable as it indicates superior performance relative to others.
02

Deciphering the Meaning of the 40th Percentile in Speed

When we say that a speed of 7.5 miles per hour is the 40th percentile, it means that 40% of the runners have speeds less than or equal to 7.5 miles per hour, while 60% of runners have speeds greater than 7.5 miles per hour. This situates a runner with this speed below average in this particular race.

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Key Concepts

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

Interpreting Percentiles
Percentiles are a way to understand how a particular value in a dataset compares to the rest. When you hear about the 40th percentile, it means that the value in question is greater than 40% of all other values. In practical terms, if we are talking about speeds in a race, being in the 40th percentile for speed indicates that a runner's speed is faster than 40% of the other participants. However, it also means that 60% of the participants ran faster.
This can be very useful in different contexts, such as grades, test scores, and other performance measures.

\( P(x) \), the percentile of a value \( x \), is calculated as follows:

\[ P(x) = \left( \frac{\text{Number of values less than } x}{\text{Total number of values}} \right) \times 100 \]
This formula gives a clear picture of where a specific value stands in relation to others, which helps evaluate performance effectively.
Speed in Running Context
Speed plays a crucial role in determining the performance of a runner during a race. The higher the speed, the faster the runner completes the race, which is usually the desired outcome.

In running, speed can be measured in different units such as miles per hour or kilometers per hour. Let's consider miles per hour for simplicity.
A speed of 7.5 miles per hour being in the 40th percentile means:
  • 40% of runners have speeds of 7.5 mph or less, implying a significant number of runners were slower or as fast.
  • 60% of the runners were faster, indicating that 7.5 mph is below the average speed.
This interpretation is crucial for runners to understand their standing in relation to their peers and to devise training strategies to improve their performance.
Percentiles and Performance Comparison
When talking about percentiles in the context of performance, it's all about seeing where you stand compared to the competition. A high percentile is favorable because it means you surpass a large percentage of participants.

In the race example, achieving a high percentile for speed indicates excellent performance. If a runner's speed is in the 80th percentile, it means the runner outran 80% of their peers. On the flip side, being in a lower percentile, such as the 40th percentile, shows there is room for improvement since most participants performed better.

Understanding how to compare performance using percentiles helps individuals set realistic goals and track their progress over time. It serves as a benchmarking tool that can be motivational, as you can actively see your status among peers and work towards improving your percentile.

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Use the two frequency tables to compare the life expectancy of men and women from 20 randomly selected countries. Include an overlayed frequency polygon and discuss the shapes of the distributions, the center, the spread, and any outliers. What can we conclude about the life expectancy of women compared to men? $$\begin{array}{|c|c|}\hline \text { Life Expectancy at Birth - women } & {\text { Frequency }} \\ \hline 49-55 & {3} \\ \hline 56-62 & {3} \\ \hline 63-62 & {1} \\ \hline 70-76 & {3} \\ \hline 77-83 & {8} \\ \hline 84-90 & {2} \\ \hline\end{array}$$ Table 2.47 $$\begin{array}{|c|c|}\hline \text { Life Expectancy at Birth - Men } & {\text { Frequency }} \\ \hline 49-55 & {3} \\ \hline 56-62 & {3} \\ \hline 63-62 & {1} \\ \hline 70-6 & {1} \\ \hline 77-83 & {7} \\ \hline 84-90 & {5} \\\ \hline\end{array}$$ Table 2.48

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Construct a times series graph for (a) the number of male births, (b) the number of female births, and (c) the total number of births. $$\begin{array}{|l|l|l|l|l|l|}\hline \text { Sex/Year } & {1855} & {1856} & {1857} & {1858}& {1859} & {1860} & {1861} \\ \hline \text { Female } & {45,545} & {49,582} & {50,257} & {50,324}& {51,915} & {51,220} & {52,403} \\\ \hline \text { Male } & {47,804} & {52,239} & {53,158} & {53,694}& {54,628} & {54,409} & {54,606} \\ \hline \text { Total } & {93,349} & {101,821} & {103,415} & {104,018}& {106,543} & {105,629} & {107,009}\\\ \hline\end{array}$$ Table 2.49 $$\begin{array}{|l|l|l|l|l|l|}\hline \text { Sex/Year } & {1862} & {1863} & {1864} & {1865}& {1866} & {1867} & {1868}& {1869} \\ \hline \text { Female } & {51,812} & {53,115} & {54,959} & {54,850}& {55,307} & {55,527} & {56,292}& {55,033} \\ \hline \text { Male } & {55,257} & {56,226} & {57,374} & {58,220}& {58,360} & {58,517} & {59,222}& {58,321} \\ \hline \text { Total } & {107,069} & {109,341} & {112,333} & {113,070}& {113,667} & {114,044} & {115,514} & {113,354}\\\ \hline\end{array}$$ Table 2.50 $$\begin{array}{|l|l|l|l|l|l|}\hline \text { Sex/Year } & {1871} & {1870} & {1872} & {1871}& {1872} & {1827} & {1874}& {1875} \\ \hline \text { Female } & {56,099} & {56,431} & {57,472} & {56,099}& {57,472} & {58,233} & {60,109}& {60,146} \\ \hline \text { Male } & {60,029} & {58,959} & {61,293} & {60,029}& {61,293} & {61,467} & {63,602}& {63,432} \\ \hline \text { Total } & {116,128} & {115,390} & {118,765} & {116,128}& {118,765} & {119,700} & {123,711} & {123,578}\\\ \hline\end{array}$$ Table 2.51
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