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Chapter 1: Problem 39

a. If a constant \(c\) is added to each \(x_{i}\) in a sample, yielding \(y_{i}=x_{i}+c\), how do the sample mean and median of the \(y_{i}\) 's relate to the mean and median of the \(x_{i}\) 's? Verify your conjectures. b. If each \(x_{i}\) is multiplied by a constant \(c\), yielding \(y_{i}=c x_{i}\), answer the question of part (a). Again, verify your conjectures.

Short Answer

Expert verified
Adding a constant to data increases both mean and median by the constant. Multiplying data by a constant scales both mean and median by the constant.

Step by step solution

01

Understanding the Effect of Adding a Constant

The problem asks what happens when you add a constant, \(c\), to each data point in a sample. If \(x_i\) represents each data point, then \(y_i = x_i + c\). Our goal is to find the relationship between the mean and median of \(y_i\) and those of \(x_i\).
02

Calculating New Mean after Adding a Constant

The mean of the original data set is given by \(\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i\). After adding \(c\) to each element, the new mean \(\bar{y}\) becomes: \[ \bar{y} = \frac{1}{n} \sum_{i=1}^n y_i = \frac{1}{n} \sum_{i=1}^n (x_i + c) = \frac{1}{n} \sum_{i=1}^n x_i + \frac{1}{n} \sum_{i=1}^n c \] Simplifying this gives \(\bar{y} = \bar{x} + c\).
03

Calculating New Median after Adding a Constant

The median is the middle value when the data set is ordered. Adding a constant \(c\) to each data element arranges the set in the same order, but each value is \(c\) units higher. Thus, the median of \(y_i\)'s becomes the median of \(x_i\)'s plus \(c\).
04

Understanding the Effect of Multiplying by a Constant

Now consider multiplying each \(x_i\) by \(c\), yielding \(y_i = c x_i\). We need to explore how the mean and median are affected by this operation.
05

Calculating New Mean after Multiplying by a Constant

The mean of the original data set is given by \(\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i\). For the new data, \(y_i = c x_i\), hence: \[ \bar{y} = \frac{1}{n} \sum_{i=1}^n (c x_i) = c \times \frac{1}{n} \sum_{i=1}^n x_i = c \times \bar{x} \] This shows that the new mean is the original mean multiplied by \(c\).
06

Calculating New Median after Multiplying by a Constant

When multiplying each data point by \(c\), the relative order of values isn't changed, but each value is scaled by \(c\). Therefore, the median of \(y_i\)'s becomes the median of \(x_i\)'s multiplied by \(c\).

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Key Concepts

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

Sample Mean
The sample mean is a statistical measure that provides the average value of a dataset. To calculate the sample mean, you sum all the data points and divide by the number of data points. This calculation can be expressed as \( \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \), where \( n \) is the sample size, and \( x_i \) represents each data point.
  • Adding a Constant: If a constant \( c \) is added to each data point in the dataset, the entire dataset shifts by that constant. The new mean \( \bar{y} \) will be the original mean \( \bar{x} \) plus the constant \( c \). This is because the increase by a uniform amount keeps the distribution balanced but shifted.
  • Multiplying by a Constant: When each data point is multiplied by a constant \( c \), the sample mean is scaled. The new mean \( \bar{y} \) becomes the original mean multiplied by the constant \( c \), or \( c \bar{x} \). Thus, both the mean and the spread of the dataset increase proportionally.
Sample Median
The sample median is another measure of central tendency, representing the middle value of an ordered dataset. It best describes the center of a dataset in a way that is resistant to extreme values. To find the median:
  • Sort the dataset in ascending order.
  • If the number of data points is odd, the median is the middle point.
  • If even, it is the average of the two middle values.
Adding or multiplying each data point affects the median as follows:
  • Adding a Constant: The median will increase by the constant \( c \) added to each data point. This happens because the order of the dataset does not change.
  • Multiplying by a Constant: The median will be the original median scaled by the constant, \( c \), since multiplication does not alter the sorted order of the dataset.
Statistics
Statistics is the branch of mathematics dealing with data analysis, interpretation, and presentation. It uses various measures of central tendency, like the mean and the median, to summarize datasets. Here's what to keep in mind:
  • Understanding Data: Statistical analysis helps make sense of complex data by providing simplified numerical summaries.
  • Application: Decisions in real-world scenarios are often supported by statistical insights, which are derived from measures like the sample mean and median.
  • Data Characteristics: When data is modified by adding or multiplying by constants, statistics help us predict how these operations impact mean and median, aiding in effective data transformation.
Data Transformation
Data transformation involves changing the form of data through operations such as adding or multiplying by a constant. Transforming data can make datasets more suitable for analysis or improve interpretability.
  • Purpose: Helps in normalizing data, handling missing values, or preparing data for certain statistical operations.
  • Effect on Mean and Median: Simple transformations like addition and multiplication apply consistently across datasets and directly affect mean and median.
  • Use Cases:
    • Adding a constant can help shift distributions for cleaning data and avoiding negative values.
    • Multiplying by a constant may standardize data, ensuring consistency across different units of measure.
Understanding transformations allows statisticians and data analysts to effectively prep data for further analysis.

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Most popular questions from this chapter

The minimum injection pressure (psi) for injection molding specimens of high amylose corn was determined for eight different specimens (higher pressure corresponds to greater processing difficulty), resulting in the following observations (from "Thermoplastic Starch Blends with a Polyethylene-Co-Vinyl Alcohol: Processability and Physical Properties,"Polymer Engrg. \& Sci., 1994: 17-23): \(\begin{array}{llllllll}15.0 & 13.0 & 18.0 & 14.5 & 12.0 & 11.0 & 8.9 & 8.0\end{array}\) a. Determine the values of the sample mean, sample median, and \(12.5 \%\) trimmed mean, and compare these values. b. By how much could the smallest sample observation, currently \(8.0\), be increased without affecting the value of the sample median? c. Suppose we want the values of the sample mean and median when the observations are expressed in kilograms per square inch (ksi) rather than psi. Is it necessary to reexpress each observation in ksi, or can the values calculated in part (a) be used directly? [Hint: \(\mathrm{l} \mathrm{kg}=2.2 \mathrm{lb} .]\)

A sample of 77 individuals working at a particular office was selected and the noise level (dBA) experienced by each one was determined, yielding the following data ("Acceptable Noise Levels for Construction Site Offices, Build. Serv. Engr. Res. Technol., 2009: 87-94). \(\begin{array}{lllllllll}55.3 & 55.3 & 55.3 & 55.9 & 55.9 & 55.9 & 55.9 & 56.1 & 56.1 \\ 56.1 & 56.1 & 56.1 & 56.1 & 56.8 & 56.8 & 57.0 & 57.0 & 57.0 \\\ 57.8 & 57.8 & 57.8 & 57.9 & 57.9 & 57.9 & 58.8 & 58.8 & 58.8 \\ 59.8 & 59.8 & 59.8 & 62.2 & 62.2 & 63.8 & 63.8 & 63.8 & 63.9 \\ 53.9 & 63.9 & 64.7 & 64.7 & 64.7 & 65.1 & 65.1 & 65.1 & 65.3 \\ 65.3 & 65.3 & 65.3 & 67.4 & 67.4 & 67.4 & 67.4 & 68.7 & 68.7 \\ 58.7 & 68.7 & 69.0 & 70.4 & 70.4 & 71.2 & 71.2 & 71.2 & 73.0 \\ 73.0 & 73.1 & 73.1 & 74.6 & 74.6 & 74.6 & 74.6 & 79.3 & 79.3 \\ 79.3 & 79.3 & 83.0 & 83.0 & 83.0 & & & & \end{array}\) Use various techniques discussed in this chapter to organize, summarize, and describe the data.

a. Give three different examples of concrete populations and three different examples of hypothetical populations. b. For one each of your concrete and your hypothetical populations, give an example of a probability question and an example of an inferential statistics question.

The article "Oxygen Consumption During Fire Suppression: Error of Heart Rate Estimation" (Ergonomics, 1991: 1469-1474) reported the following data on oxygen consumption \((\mathrm{mL} / \mathrm{kg} /\) min) for a sample of ten firefighters performing a fire-suppression simulation: \(\begin{array}{lllllllllll}29.5 & 49.3 & 30.6 & 28.2 & 28.0 & 26.3 & 33.9 & 29.4 & 23.5 & 31.6\end{array}\) Compute the following: a. The sample range b. The sample variance \(s^{2}\) from the definition (by first computing deviations, then squaring them, etc.) c. The sample standard deviation d. \(s^{2}\) using the shortcut method

In a study of warp breakage during the weaving of fabric (Technometrics, 1982: 63), 100 specimens of yarn were tested. The number of cycles of strain to breakage was determined for each yarn specimen, resulting in the following data: \(\begin{array}{rrrrrrrrrr}86 & 146 & 251 & 653 & 98 & 249 & 400 & 292 & 131 & 169 \\ 175 & 176 & 76 & 264 & 15 & 364 & 195 & 262 & 88 & 264 \\ 157 & 220 & 42 & 321 & 180 & 198 & 38 & 20 & 61 & 121 \\ 282 & 224 & 149 & 180 & 325 & 250 & 196 & 90 & 229 & 166 \\ 38 & 337 & 65 & 151 & 341 & 40 & 40 & 135 & 597 & 246 \\ 211 & 180 & 93 & 315 & 353 & 571 & 124 & 279 & 81 & 186 \\ 497 & 182 & 423 & 185 & 229 & 400 & 338 & 290 & 398 & 71 \\ 246 & 185 & 188 & 568 & 55 & 55 & 61 & 244 & 20 & 284 \\ 393 & 396 & 203 & 829 & 239 & 236 & 286 & 194 & 277 & 143 \\ 198 & 264 & 105 & 203 & 124 & 137 & 135 & 350 & 193 & 188\end{array}\) a. Construct a relative frequency histogram based on the class intervals \(0-100,100-200, \ldots\), and comment on features of the distribution. b. Construct a histogram based on the following class intervals: \(0-50,50-100,100-150\), \(150-200, \quad 200-300, \quad 300-400, \quad 400-500\), \(500-600,600-900 .\) c. If weaving specifications require a breaking strength of at least 100 cycles, what proportion of the yam specimens in this sample would be considered satisfactory?
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