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Chapter 1: Problem 35

Consider a sample \(x_{1}, x_{2}, \ldots, x_{n}\) and suppose that the values of \(\bar{x}, s^{2}\), and \(s\) have been calculated. a. Let \(y_{i}=x_{i}-\bar{x}\) for \(i=1, \ldots, n\). How do the values of \(s^{2}\) and \(s\) for the \(y_{i}\) 's compare to the corresponding values for the \(x_{i}\) 's? Explain. b. Let \(z_{i}=\left(x_{i}-\bar{x}\right) / s\) for \(i=1, \ldots, n\). What are the values of the sample variance and sample standard deviation for the \(z_{i} s\) ?

Short Answer

Expert verified
For part (a), both sample variance \(s_y^2\) and standard deviation \(s_y\) for \(y_i\) are equal to those for \(x_i\). For part (b), the sample variance of \(z_i\) is 1, and the standard deviation is also 1.

Step by step solution

01

Define Transformation for Part (a)

For part (a), let the transformation be defined as \( y_i = x_i - \bar{x} \).This means that each element of the sample \( y_i \) is the deviation of \( x_i \) from the sample mean \( \bar{x} \).
02

Calculate Variance for Part (a)

The variance of the transformed variable \( y_i \) can be computed as: \[ s_y^2 = \frac{1}{n-1} \sum_{i=1}^{n} (y_i - \bar{y})^2 \].Since \( y_i = x_i - \bar{x} \), the mean of \( y_i \) is 0, making \( \sum (y_i - 0)^2 = \sum (x_i - \bar{x})^2 \).Thus, \( s_y^2 = s_x^2 \).
03

Calculate Standard Deviation for Part (a)

The standard deviation is the square root of the variance. So for \( y_i \), \[ s_y = \sqrt{s_y^2} = \sqrt{s_x^2} = s_x \].Therefore, the standard deviation of \( y_i \) is the same as that of \( x_i \).
04

Define Transformation for Part (b)

For part (b), let the transformation be defined as \( z_i = \frac{x_i - \bar{x}}{s} \).This scaling transforms each \( x_i \) into a standardized score (z-score).
05

Calculate Variance for Part (b)

The sample variance of the standardized variable \( z_i \) is computed as:\[ s_z^2 = \frac{1}{n-1} \sum_{i=1}^{n} (z_i - \bar{z})^2 \].Since the mean of the z-scores is 0, this simplifies to:\( s_z^2 = \frac{1}{n-1} \sum_{i=1}^{n} z_i^2 = \frac{1}{n-1} \sum_{i=1}^{n} \left(\frac{x_i - \bar{x}}{s}\right)^2 = \frac{1}{n-1} \frac{1}{s^2} \sum_{i=1}^{n} (x_i - \bar{x})^2 = 1 \).
06

Calculate Standard Deviation for Part (b)

The standard deviation for \( z_i \) is the square root of the variance:\[ s_z = \sqrt{s_z^2} = \sqrt{1} = 1 \].Thus, the standard deviation for \( z_i \) is 1.

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Key Concepts

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

Sample Variance
Sample variance is a key concept in statistics that measures the spread of a set of data points. It represents how much individual numbers in a data set differ from the mean of that set. Understanding this helps us determine the variability in data.
  • The calculation involves summing up the squared differences between each data point and the mean, and then dividing by the sample size minus one. This is expressed in the formula: \[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n}(x_i - \bar{x})^2 \]
  • When dealing with transformed data, such as applying a transformation like \( y_i = x_i - \bar{x} \), the sample variance remains unchanged, as shown in the original exercise solution: \( s_y^2 = s_x^2 \). This is because the transformation simply shifts the dataset without altering the data's spread.
  • This property of invariant variance ensures analyses remain consistent even when data is centered by subtracting the mean.
Standard Deviation
Standard deviation provides the average distance between each data point and the mean. It is the square root of variance and gives a clear understanding of data spread.
  • The formula for standard deviation is: \[ s = \sqrt{s^2} \]
  • This transformation property applies to standard deviation as well, as when data is centered by subtracting the mean, both variance and standard deviation remain the same: \( s_y = s_x \).
  • Standard deviation helps to appreciate how much dispersion there is in the original data, aiding in the assessment of the reliability of statistical conclusions.
Z-scores
Z-scores are a statistical way of standardizing data points by converting them into standard deviation units. This transformation is expressed by the formula:\[ z_i = \frac{x_i - \bar{x}}{s} \]
  • The main purpose of z-scores is to interpret data with a common scale. They tell us how many standard deviations away a data point is from the mean.
  • When values are converted into z-scores, the distribution has a sample mean of 0 and a sample standard deviation of 1. This standardization simplifies the comparison between different datasets.
  • The variance of the z-scores is 1, as the transformation of each data point accounts proportionally for the spread. This ensures data is analyzed impartially, regardless of the original scale.

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Most popular questions from this chapter

A sample of \(n=10\) automobiles was selected, and each was subjected to a 5 -mph crash test. Denoting a car with no visible damage by \(S\) (for success) and a car with such damage by \(F\), results were as follows: \(S S F\) S S S F F \(S \quad S\) a. What is the value of the sample proportion of successes \(x / n\) ? b. Replace each \(S\) with a 1 and each \(F\) with a 0 . Then calculate \(\bar{x}\) for this numerically coded sample. How does \(\bar{x}\) compare to \(x / n\) ? c. Suppose it is decided to include 15 more cars in the experiment. How many of these would have to be S's to give \(x / n=.80\) for the entire sample of 25 cars?

Poly(3-hydroxybutyrate) (PHB), a semicrystalline polymer that is fully biodegradable and biocompatible, is obtained from renewable resources. From a sustainability perspective, PHB offers many attractive properties though it is more expensive to produce than standard plastics. The accompanying data on melting point \(\left({ }^{\circ} \mathrm{C}\right)\) for each of 12 specimens of the polymer using a differential scanning calorimeter appeared in the article "The Melting Behaviour of Poly(3-Hydroxybutyrate) by DSC. Reproducibility Study" (Polymer Testing, 2013: 215-220). \(\begin{array}{llllll}180.5 & 181.7 & 180.9 & 181.6 & 182.6 & 181.6 \\ 181.3 & 182.1 & 182.1 & 180.3 & 181.7 & 180.5\end{array}\) Compute the following: a. The sample range b. The sample variance \(s^{2}\) from the definition [Hint: First subtract 180 from each observation.] c. The sample standard deviation d. \(s^{2}\) using the shortcut method

A study of the relationship between age and various visual functions (such as acuity and depth perception) reported the following observations on the area of scleral lamina \(\left(\mathrm{mm}^{2}\right)\) from human optic nerve heads ("Morphometry of Nerve Fiber Bundle Pores in the Optic Nerve Head of the Human," Experimental Eye Research, 1988: 559-568): \(\begin{array}{lllllllll}2.75 & 2.62 & 2.74 & 3.85 & 2.34 & 2.74 & 3.93 & 4.21 & 3.88 \\ 4.33 & 3.46 & 4.52 & 2.43 & 3.65 & 2.78 & 3.56 & 3.01 & \end{array}\) a. Calculate \(\Sigma x_{i}\) and \(\Sigma x_{i}^{2}\). b. Use the values calculated in part (a) to compute the sample variance \(s^{2}\) and then the sample standard deviation \(s\).

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.

Many universities and colleges have instituted supplemental instruction (SI) programs, in which a student facilitator meets regularly with a small group of students enrolled in the course to promote discussion of course material and enhance subject mastery. Suppose that students in a large statistics course (what else?) are randomly divided into a control group that will not participate in SI and a treatment group that will participate. At the end of the term, each student's total score in the course is determined. a. Are the scores from the SI group a sample from an existing population? If so, what is it? If not, what is the relevant conceptual population? b. What do you think is the advantage of randomly dividing the students into the two groups rather than letting each student choose which group to join? c. Why didn't the investigators put all students in the treatment group? [Note: The article 'Supplemental Instruction: An Effective Component of Student Affairs Programming" \(U\). of College Student Devel., 1997: 577-586) discusses the analysis of data from several SI programs.]
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