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Chapter 1: Problem 9

In a famous experiment carried out in 1882 , Michelson and Newcomb obtained 66 observations on the time it took for light to travel between two locations in Washington, D.C. A few of the measurements (coded in a certain manner) were \(31,23,32,36,-2,26,27\), and 31 . a. Why are these measurements not identical? b. Is this an enumerative study? Why or why not?

Short Answer

Expert verified
a. Measurements vary due to natural variability and errors. b. No, it's not enumerative; it seeks to understand a universal property.

Step by step solution

01

Analyzing Measurement Variability

These measurements are not identical because of natural variability and measurement errors. Factors such as experimental conditions, precision of instruments, and human errors can contribute to differences in repeated measurements of the same quantity. In scientific experiments, it's common for multiple measurements to vary slightly; this helps in estimating the true value by averaging the results and considering the uncertainty in measurements.
02

Determining the Nature of the Study

To determine whether the study is enumerative, we need to understand that an enumerative study's goal is to analyze and predict characteristics of a finite, often well-defined population. However, the Michelson and Newcomb experiment aimed to measure a fundamental physical constant (the speed of light), which isn't exclusive to a finite, well-defined population and is more about understanding a universal property. Therefore, this is an analytic study rather than enumerative, as it seeks knowledge about a general law of nature rather than a specific set of objects.

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Key Concepts

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

Measurement Errors
Measurement errors are differences between observed values and true values due to various reasons. It is crucial to understand why measurements vary. Consider if you were measuring time with a stopwatch. Even tiny errors in starting or stopping the timer can lead to differences. There are several factors contributing to measurement errors:
  • Instrument Precision: Limitations of measuring devices could lead to small inaccuracies. For example, a clock not accurate to milliseconds might give slightly different results each time.
  • Environmental Conditions: Unknown factors in the environment, like temperature changes, can affect results. Experiment conditions should ideally be controlled, but variations might still occur.
  • Human Errors: Mistakes by the person conducting the experiment can contribute. For instance, errors can occur if a person doesn't repeat their actions exact each time.
Errors are naturally expected and it's part of measurement variabilities, providing a framework to consider the uncertainty and probable error scope. By averaging outcomes, scientists aim to get closer to the true value.
Enumerative Study
An enumerative study focuses on understanding features of a specific group at a specific time. Think of it like taking a snapshot of a particular population. In such a study, the aim is to count or catalog characteristics within a finite group, like assessing features of different types of apples in a basket. The essential characteristics of an enumerative study include:
  • Fixed Population: A clearly defined and finite group is assessed. It could be objects, people, places, etc.
  • Current Characteristics: It looks at things as they are now, not predicting future outcomes. This is ideal for quality control or inventory management.
In Michelson and Newcomb's experiment, they did not focus on a fixed group or count specific features but rather aimed to understand a universal constant, namely, the speed of light. Therefore, this experiment is not an enumerative study.
Analytic Study
An analytic study seeks to uncover or understand laws or truths that apply widely, not just to a single occurrence. This type of study is more like detective work, trying to find patterns or principles that explain general truths. Key aspects of an analytic study include:
  • Generalization: The aim is to understand broad principles or laws that apply universally, such as physical constants.
  • Explanatory Nature: Rather than just describing what is, it seeks to explain why or how things happen in the natural world.
Michelson and Newcomb's experiment was analytic because they were investigating the speed of light, a universal constant. They weren't concerned with the individual measurements as much as what they collectively revealed about this fundamental aspect of nature. Such studies advance our understanding of the universe.

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Most popular questions from this chapter

A study carried out to investigate the distribution of total braking time (reaction time plus accelerator-to-brake movement time, in ms) during real driving conditions at \(60 \mathrm{~km} /\) hr gave the following summary information on the distribution of times ("A Field Study on Braking Responses During Driving," Ergonomics, 1995: 1903鈥�1910): mean \(=535\) median \(=500 \quad\) mode \(=500\) \(\mathrm{sd}=96 \quad\) minimum \(=220 \quad\) maximum \(=925\) 5 th percentile \(=400 \quad 10\) th percentile \(=430\) 90 th percentile \(=640 \quad 95\) th percentile \(=720\) What can you conclude about the shape of a histogram of this data? Explain your reasoning.

The article "A Thin-Film Oxygen Uptake Test for the Evaluation of Automotive Crankease Lubricants" (Lubric. Engr., 1984: 75-83) reported the following data on oxidation-induction time (min) for various commercial oils: \(\begin{array}{lllllllllll}87 & 103 & 130 & 160 & 180 & 195 & 132 & 145 & 211 & 105 & 145\end{array}\) \(\begin{array}{llllllll}153 & 152 & 138 & 87 & 99 & 93 & 119 & 129\end{array}\) a. Calculate the sample variance and standard deviation. b. If the observations were reexpressed in hours, what would be the resulting values of the sample variance and sample standard deviation? Answer without actually performing the reexpression.

The first four deviations from the mean in a sample of \(n=5\) reaction times were \(.3, .9,1.0\), and \(1.3\). What is the fifth deviation from the mean? Give a sample for which these are the five deviations from the mean.

The three measures of center introduced in this chapter are the mean, median, and trimmed mean. Two additional measures of center that are occasionally used are the midrange, which is the average of the smallest and largest observations, and the midfourth, which is the average of the two fourths. Which of these five measures of center are resistant to the effects of outliers and which are not? Explain your reasoning.

Elevated energy consumption during exercise continues after the workout ends. Because calories burned after exercise contribute to weight loss and have other consequences, it is important to understand this process. The article "Effect of Weight Training Exercise and Treadmill Exercise on Post-Exercise Oxygen Consumption" (Medicine and Science in Sports and Exercise, 1998: 518-522) reported the accompanying data from a study in which oxygen consumption (liters) was measured continuously for 30 minutes for each of 15 subjects both after a weight training exercise and after a treadmill exercise. \(\begin{array}{lrrrrrrrr}\text { Subject } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\ \text { Weight }(x) & 14.6 & 14.4 & 19.5 & 24.3 & 16.3 & 22.1 & 23.0 \\ \text { Treadmill }(y) & 11.3 & 5.3 & 9.1 & 15.2 & 10.1 & 19.6 & 20.8 \\ \text { Subject } & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \text { Weight }(x) & 18.7 & 19.0 & 17.0 & 19.1 & 19.6 & 23.2 & 18.5 & 15.9 \\ \text { Treadmill }(y) & 10.3 & 10.3 & 2.6 & 16.6 & 22.4 & 23.6 & 12.6 & 4.4\end{array}\) a. Construct a comparative boxplot of the weight and treadmill observations, and comment on what you see. b. The data is in the form of \((x, y)\) pairs, with \(x\) and \(y\) measurements on the same variable under two different conditions, so it is natural to focus on the differences within pairs: \(d_{1}=x_{1}-y_{1}, \ldots, d_{n}=x_{n}-y_{n}\). Construct a boxplot of the sample differences. What
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