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Chapter 3: Problem 17

When data consist of rates of change, such as speeds, the harmonic mean is an appropriate measure of central tendency. for \(n\) data values, Harmonic mean \(=\frac{n}{\Sigma_{x}^{1}}\), assuming no data value is 0 Suppose you drive 60 miles per hour for 100 miles, then 75 miles per hour for 100 miles. Use the harmonic mean to find your average speed.

Short Answer

Expert verified
The average speed is approximately 66.67 mph.

Step by step solution

01

Identify the Speeds

The two speeds given in the problem are 60 miles per hour (mph) and 75 mph. Objective is to find the harmonic mean of these speeds.
02

Calculate the Reciprocal of Each Speed

Find the reciprocal of each speed. The reciprocal of 60 mph is \( \frac{1}{60} \) and the reciprocal of 75 mph is \( \frac{1}{75} \).
03

Sum the Reciprocals

Add the two reciprocals together: \( \frac{1}{60} + \frac{1}{75} \). Find a common denominator to perform this summation. The common denominator is 300, so \( \frac{1}{60} = \frac{5}{300} \) and \( \frac{1}{75} = \frac{4}{300} \). Thus, the sum is \( \frac{5}{300} + \frac{4}{300} = \frac{9}{300} \).
04

Calculate the Harmonic Mean

Use the formula for the harmonic mean: \( \text{Harmonic Mean} = \frac{n}{\Sigma_{x}^{-1}} \), where \(n\) is the number of speeds, which is 2. Substitute the values: \( \frac{2}{\frac{9}{300}} = \frac{2 \times 300}{9} = \frac{600}{9} \).
05

Simplify the Expression

Perform the division to simplify the expression: \( \frac{600}{9} \approx 66.67 \). So, the harmonic mean speed is approximately 66.67 mph.

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Key Concepts

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

Understanding Central Tendency
When you hear "central tendency," it refers to a statistical measure that identifies the center of a dataset. Imagine you have a collection of numbers, and you want to summarize them with one representative figure. Central tendency helps you do just that.

There are multiple measures of central tendency:
  • Mean: The arithmetic average of all numbers.
  • Median: The middle value when numbers are ordered.
  • Mode: The most frequently occurring number.
  • Harmonic Mean: Often used when dealing with rates or speeds.
For rates of change, like speed, the harmonic mean is especially suitable. It gives a more accurate average in situations where numbers relate to ratios or fractions, such as speeds. Unlike others, the harmonic mean provides a simple yet effective narrative when dealing with uneven distributions across rates.
Exploring Rates of Change
Rates of change describe how a quantity changes over time. A common real-world example is speed, which defines how quickly an object moves from one place to another.

In the context of the problem given, the speeds of 60 mph and 75 mph represent rates of change over distances. Calculating an average of these rates gives us insights into overall performance across different speeds.
  • Constant Rate: When speed or any other rate stays the same.
  • Variable Rate: When the rate changes over time, like different speeds.
For our speeds, we know the distances traveled are equal, making the use of harmonic mean for average speed an effective tool. Rates of change inform us about variations and are essential in describing real-world dynamic processes.
The Role of Reciprocals
In mathematics, a reciprocal of a number is 1 divided by that number. Reciprocals are handy especially when using the harmonic mean.

For example, if you know the speed is 60 mph, the reciprocal is \(\frac{1}{60}\), and for 75 mph, it is \(\frac{1}{75}\). In calculating the harmonic mean, reciprocals are summed up. This sum is utilized to balance out the results, especially when numbers are part of a fractional relationship.

Reciprocal operation is crucial when:
  • Working with rates or speeds and need a consistent approach.
  • The dataset emphasizes proportional relationships.
  • Translating multiplicative processes (like growth rates) into additive processes (like the sum of rates).
For our speeds, finding the reciprocals enables calculation of harmonic mean and ensures the average speed reflects true travel efficiency.
Calculating Average Speed with Harmonic Mean
When determining the average speed over different travel sections, you often use the harmonic mean. This method accurately represents average speed when distances are constant but speeds vary.

In the given exercise, here's a simplified step-by-step on how to use the harmonic mean for average speed:
  • Step 1: Identify all sections' speeds (60 mph and 75 mph here).
  • Step 2: Find the reciprocal of each speed: \(\frac{1}{60}\) and \(\frac{1}{75}\).
  • Step 3: Add the reciprocals together: \(\frac{1}{60} + \frac{1}{75}\).
  • Step 4: Compute the harmonic mean: \(\text{Harmonic Mean} = \frac{n}{\Sigma x^{-1}}\), where \(n\) is the number of speed values.
  • Step 5: Simplify to find the result, approximately 66.67 mph.
The use of harmonic mean ensures that each section's different speed is fairly weighted, giving a balanced average that more accurately reflects actual travel speed across varied conditions. It's a robust way to gauge central tendency for rates of change like speed.

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Most popular questions from this chapter

When computing the standard deviation, does it matter whether the data are sample data or data comprising the entire population? Explain.

In some reports, the mean and coefficient of variation are given. For instance, in Statistical Abstract of the United States, 116 th Edition, one report gives the average number of physician visits by males per year. The average reported is \(2.2\), and the reported coefficient of variation is \(1.5 \% .\) Use this information to determine the standard deviation of the annual number of visits to physicians made by males.

Kevlar epoxy is a material used on the NASA Space Shuttle. Strands of this epoxy were tested at the \(90 \%\) breaking strength. The following data represent time to failure (in hours) for a random sample of 50 epoxy strands (Reference: \(\mathrm{R} . \mathrm{E} .\) Barlow, University of California, Berkeley). Let \(x\) be a random variable representing time to failure (in hours) at \(90 \%\) breaking strength. Note: These data are also available for download online in HM StatSPACETM. \(\begin{array}{llllllllll}0.54 & 1.80 & 1.52 & 2.05 & 1.03 & 1.18 & 0.80 & 1.33 & 1.29 & 1.11 \\ 3.34 & 1.54 & 0.08 & 0.12 & 0.60 & 0.72 & 0.92 & 1.05 & 1.43 & 3.03 \\ 1.81 & 2.17 & 0.63 & 0.56 & 0.03 & 0.09 & 0.18 & 0.34 & 1.51 & 1.45 \\ 1.52 & 0.19 & 1.55 & 0.02 & 0.07 & 0.65 & 0.40 & 0.24 & 1.51 & 1.45 \\ 1.60 & 1.80 & 4.69 & 0.08 & 7.89 & 1.58 & 1.64 & 0.03 & 0.23 & 0.72\end{array}\) (a) Find the range. (b) Use a calculator to verify that \(\Sigma x=62.11\) and \(\Sigma x^{2}=164.23\). (c) Use the results of part (b) to compute the sample mean, variance, and standard deviation for the time to failure. (d) Use the results of part (c) to compute the coefficient of variation. What does this number say about time to failure? Why does a small CV indicate more consistent data, whereas a larger CV indicates less consistent data? Explain.

Critical Thinking: Data Transformation In this problem, we explore the effect on the standard deviation of adding the same constant to each data value in a data set. Consider the data set \(5,9,10,11,15\). (a) Use the defining formula, the computation formula, or a calculator to compute \(s\). (b) Add 5 to each data value to get the new data set \(10,14,15,16,20 .\) Compute \(s\). (c) Compare the results of parts (a) and (b). In general, how do you think the standard deviation of a data set changes if the same constant is added to each data value?

In this problem, we explore the effect on the mean, median, and mode of adding the same number to each data value. Consider the data set \(2,2,3,6,10\). (a) Compute the mode, median, and mean. (b) Add 5 to each of the data values. Compute the mode, median, and mean. (c) Compare the results of parts (a) and (b). In general, how do you think the mode, median, and mean are affected when the same constant is added to each data value in a set?
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