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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 , \(32,36,22,26,27\), and 31 . a. Why are these measurements not identical? b. Does this study involve sampling an existing population or a conceptual population?

Short Answer

Expert verified
Measurements vary due to experimental errors. The study involves a conceptual population.

Step by step solution

01

Understanding the Experiment

The measurements represent experimental data obtained from measuring the time it took for light to travel between two locations. Variability in these measurements can arise due to factors such as experimental errors, equipment limitations, or environmental conditions.
02

Identifying Variability

The measurements (31, 23, 32, 36, 22, 26, 27, and 31) are not identical because every measurement experiment is subject to some form of error and variability. This may include random errors in measurement, fluctuations in experimental conditions, or limitations in precision of the measuring instruments.
03

Sampling Consideration

In analyzing this study, consider whether the data comes from a finite set of pre-existing observations or represents a theoretical framework of all possible observations. Given the setup of this experiment, it most likely samples from a conceptual population.

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Key Concepts

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

Variability in Measurements
Variability in measurements occurs due to differences in repeated observations of the same phenomenon. It's natural to expect variations when conducting experiments involving measurements. Some common sources of variability include:
  • Random Errors: These are unpredictable fluctuations that can occur during measurement. They could arise from changes in environmental conditions or minor fluctuations in instrument readings.
  • Instrument Precision: No measuring device is perfectly precise. Limitations in precision can cause slight differences in measurements.
  • Human Factors: Even when automated systems are in place, human involvement or error can contribute to variability.

Variability isn't necessarily a flaw but an inherent aspect of experimental work. It highlights the importance of taking multiple measurements to achieve more reliable and accurate results. By analyzing a range of data points, researchers can better understand the nature of the phenomena they are studying and draw more precise conclusions.
Experimental Error
Experimental error refers to the deviations that occur when an experiment is repeated under the same conditions but yields different outcomes. This is a normal part of scientific investigations.

There are two primary types of experimental errors:
  • Systematic Errors: These errors occur consistently, following a predictable pattern. They could be due to faulty equipment calibration or measurement methods. Systematic errors can be identified and often corrected with careful calibration and methodology review.
  • Random Errors: Unlike systematic errors, random errors don't have a consistent effect but cause variations in measurements unpredictably. They result from unknown variables affecting the conditions and can be minimized by increasing the sample size.

Understanding the sources and types of errors in experiments can help in designing better studies, improving measurement techniques, and ensuring more reliable data analysis.
Conceptual Population
A conceptual population is an abstract idea where researchers consider all possible outcomes or measurements, rather than a finite set of collected data. This concept is especially valuable in experiments where all potential data cannot be gathered.

Here's how it applies to certain types of studies:
  • In the Michelson and Newcomb experiment, instead of relying on data from a known, finite group, they treated each measurement as a sample from a theoretical distribution that represents all possible measurements of light travel times.
  • Such a population helps researchers generalize findings by considering the variability and potential outcomes not directly measurable or accessible.

Utilizing the idea of a conceptual population assists in tackling theoretical investigations and large-scale hypothesis testing, where observing every possible outcome isn't feasible.

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Most popular questions from this chapter

Fire load \(\left(\mathrm{MJ} / \mathrm{m}^{2}\right)\) is the heat energy that could be released per square meter of floor area by combustion of contents and the structure itself. The article "Fire Loads in Office Buildings" \((J .\) Struct. Engrg., 1997: \(365-368\) ) gave the following cumulative percentages (read from a graph) for fire loads in a sample of 388 rooms: \(\begin{array}{lccccc}\text { Value } & 0 & 150 & 300 & 450 & 600 \\ \text { Cumulative \% } & 0 & 19.3 & 37.6 & 62.7 & 77.5 \\ \text { Value } & 750 & 900 & 1050 & 1200 & 1350 \\ \text { Cumulative \% } & 87.2 & 93.8 & 95.7 & 98.6 & 99.1 \\ \text { Value } & 1500 & 1650 & 1800 & 1950 & \\ \text { Cumulative \% } & 99.5 & 99.6 & 99.8 & 100.0\end{array}\) a. Construct a relative frequency histogram and comment on interesting features. b. What proportion of fire loads are less than 600 ? At least 1200 ? c. What proportion of the loads are between 600 and 1200 ?

a. Let \(a\) and \(b\) be constants and let \(y_{i}=a x_{i}+b\) for \(i=1,2, \ldots, n\). What are the relationships between \(\bar{x}\) and \(\bar{y}\) and between \(s_{x}^{2}\) and \(s_{y}^{2}\) ? b. The Australian army studied the effect of high temperatures and humidity on human body temperature (Neural Network Training on Human Body Core Temperature Data, Technical Report DSTO TN-0241, Combatant Protection Nutrition Branch, Aeronautical and Maritime Research Laboratory). They found that, at \(30^{\circ} \mathrm{C}\) and \(60 \%\) relative humidity, the sample average body temperature for nine soldiers was \(38.21^{\circ} \mathrm{C}\), with standard deviation \(.318^{\circ} \mathrm{C}\). What are the sample average and the standard deviation in \({ }^{\circ} \mathrm{F}\) ?

Let \(\bar{x}_{n}\) and \(s_{n}^{2}\) denote the sample mean and variance for the sample \(x_{1}, \ldots, x_{n}\) and let \(\bar{x}_{n+1}\) and \(s_{n+1}^{2}\) denote these quantities when an additional observation \(x_{n+1}\) is added to the sample. a. Show how \(\bar{x}_{n+1}\) can be computed from \(\bar{x}_{n}\) and \(x_{n+1}\). b. Show that $$ n s_{n+1}^{2}=(n-1) s_{n}^{2}+\frac{n}{n+1}\left(x_{n+1}-\bar{x}_{n}\right)^{2} $$ so that \(s_{n+1}^{2}\) can be computed from \(x_{n+1}, \bar{x}_{n}\), and \(s_{n^{-}}^{2}\). c. Suppose that a sample of 15 strands of drapery yarn has resulted in a sample mean thread elongation of \(12.58 \mathrm{~mm}\) and a sample standard deviation of \(.512 \mathrm{~mm}\). A 16 th strand results in an elongation value of \(11.8\). What are the values of the sample mean and sample standard deviation for all 16 elongation observations?

Temperature transducers of a certain type are shipped in batches of 50 . A sample of 60 batches was selected, and the number of transducers in each batch not conforming to design specifications was determined, resulting in the following data: \(\begin{array}{llllllllllllllllllll}2 & 1 & 2 & 4 & 0 & 1 & 3 & 2 & 0 & 5 & 3 & 3 & 1 & 3 & 2 & 4 & 7 & 0 & 2 & 3 \\ 0 & 4 & 2 & 1 & 3 & 1 & 1 & 3 & 4 & 1 & 2 & 3 & 2 & 2 & 8 & 4 & 5 & 1 & 3 & 1 \\ 5 & 0 & 2 & 3 & 2 & 1 & 0 & 6 & 4 & 2 & 1 & 6 & 0 & 3 & 3 & 3 & 6 & 1 & 2 & 3\end{array}\) a. Determine frequencies and relative frequencies for the observed values of \(x=\) number of nonconforming transducers in a batch. b. What proportion of batches in the sample have at most five nonconforming transducers? What proportion have fewer than five? What proportion have at least five nonconforming units? c. Draw a histogram of the data using relative frequency on the vertical scale, and comment on its features.

The amount of flow through a solenoid valve in an automobile's pollution- control system is an important characteristic. An experiment was carried out to study how flow rate depended on three factors: armature length, spring load, and bobbin depth. Two different levels (low and high) of each factor were chosen, and a single observation on flow was made for each combination of levels. a. The resulting data set consisted of how many observations? b. Does this study involve sampling an existing population or a conceptual population?
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