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Chapter 1: Problem 52

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.

Short Answer

Expert verified
Fifth deviation is -3.5; sample possible as [1.3, 1.9, 2.0, 2.3, -2.5].

Step by step solution

01

Understanding Deviations

The deviations from the mean of a sample should sum to zero. For the deviations given (.3, .9, 1.0, and 1.3), we need to determine the fifth deviation that balances these to zero.
02

Calculating Sum of Given Deviations

Add the first four deviations: \[ 0.3 + 0.9 + 1.0 + 1.3 = 3.5 \]
03

Setting Up the Equation for Equality

To ensure the total deviations sum to zero, set up the equation: \[ 3.5 + x = 0 \] where \( x \) represents the fifth deviation.
04

Solving for the Fifth Deviation

Subtract 3.5 from both sides of the equation to solve for \( x \): \[ x = -3.5 \] Thus, the fifth deviation is \(-3.5\).
05

Constructing the Sample

To find a sample where these are the deviations, start with any mean, say 1. Add each deviation to this mean:- For 0.3: \(1 + 0.3 = 1.3\)- For 0.9: \(1 + 0.9 = 1.9\)- For 1.0: \(1 + 1.0 = 2.0\)- For 1.3: \(1 + 1.3 = 2.3\)- For -3.5: \(1 - 3.5 = -2.5\)Thus, one possible sample is \([1.3, 1.9, 2.0, 2.3, -2.5]\).

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Key Concepts

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

Sample Mean
In statistics, the sample mean is a measure of central tendency. It's the average value of a set of numbers from a sample. To find the sample mean, you add up all the individual values and then divide by the number of values.
It's important because it provides a snapshot of the data's overall trend.
In this context, using a sample of reaction times, we calculate deviations to understand how far each observation is from the average.
Sum of Deviations
When dealing with deviations from the mean, the sum of all deviations in a sample equals zero. This property is crucial in most statistical calculations.
If you add up the deviations from the mean, positive and negative, they balance each other out to total zero.
For example, given the deviations of 0.3, 0.9, 1.0, and 1.3 from a sample mean, the sum is 3.5.
To balance this to zero, the fifth deviation must be -3.5.
  • Positive deviations indicate values above the mean.
  • Negative deviations indicate values below the mean.
Reaction Times
Reaction times, in this case, are considered as the data points or observations from which deviations are calculated.
By looking at the deviations, you can see how each reaction time compares to the sample mean.
This comparison helps in understanding the speed of reaction in relation to the average for that group.
Statistics Calculation
Statistical calculations often involve various steps to ensure data is properly analyzed.
In this problem, after knowing deviations sum to zero, calculations like adding deviations and solving for unknowns bring clarity.
Let's consider the process:
  • Sum known deviations, calculate the missing one (ensuring total is zero).
  • Apply each deviation to a chosen mean to reconstruct the sample.
These steps break down complex data analysis into manageable tasks, essential for achieving accurate results in statistics.

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Most popular questions from this chapter

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. Is this an enumerative or analytic study? Explain your reasoning.

The California State University (CSU) system consists of 23 campuses, from San Diego State in the south to Humboldt State near the Oregon border. A CSU administrator wishes to make an inference about the average distance between the hometowns of students and their campuses. Describe and discuss several different sampling methods that might be employed. Would this be an enumerative or an analytic study? Explain your reasoning.

In 1997 a woman sued a computer keyboard manufacturer, charging that her repetitive stress injuries were caused by the keyboard (Genessy v. Digital Equipment Corp.). The injury awarded about \(\$ 3.5\) million for pain and suffering, but the court then set aside that award as being unreasonable compensation. In making this determination, the court identified a "normative" group of 27 similar cases and specified a reasonable award as one within two standard deviations of the mean of the awards in the 27 cases. The 27 awards were (in \(\$ 1000\) s) \(37,60,75,115,135,140,149,150,238,290\), \(340,410,600,750,750,750,1050,1100,1139,1150,1200\), \(1200,1250,1576,1700,1825\), and 2000 , from which \(\sum x_{i}=\) \(20,179, \sum x_{i}^{2}=24,657,511\). What is the maximum possible amount that could be awarded under the two-standarddeviation rule?

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.

Consider numerical observations \(x_{1}, \ldots, x_{n^{*}}\) It is frequently of interest to know whether the \(x_{i} \mathrm{~s}\) are (at least approximately) symmetrically distributed about some value. If \(n\) is at least moderately large, the extent of symmetry can be assessed from a stem-and-leaf display or histogram. However, if \(n\) is not very large, such pictures are not particularly informative. Consider the following alternative. Let \(y_{1}\) denote the smallest \(x_{i}, y_{2}\) the second smallest \(x_{i}\), and so on. Then plot the following pairs as points on a two-dimensional coordinate system: \(\left(y_{n}-\tilde{x}, \tilde{x}-y_{1}\right),\left(y_{n-1}-\tilde{x}, \tilde{x}-y_{2}\right),\left(y_{n-2}-\tilde{x}\right.\), \(\left.\tilde{x}-y_{3}\right), \ldots\) There are \(n / 2\) points when \(n\) is even and \((n-1) / 2\) when \(n\) is odd. a. What does this plot look like when there is perfect symmetry in the data? What does it look like when observations stretch out more above the median than below it (a long upper tail)? b. The accompanying data on rainfall (acre-feet) from 26 seeded clouds is taken from the article "A Bayesian Analysis of a Multiplicative Treatment Effect in Weather Modification" (Technometrics, 1975: 161-166). Construct the plot and comment on the extent of symmetry or nature of departure from symmetry. \(\begin{array}{rrrrrrr}4.1 & 7.7 & 17.5 & 31.4 & 32.7 & 40.6 & 92.4 \\ 115.3 & 118.3 & 119.0 & 129.6 & 198.6 & 200.7 & 242.5 \\ 255.0 & 274.7 & 274.7 & 302.8 & 334.1 & 430.0 & 489.1 \\ 703.4 & 978.0 & 1656.0 & 1697.8 & 2745.6 & & \end{array}\)
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