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Chapter 11: Problem 2

Suppose your data consist of a sequence of numbers. To apply a runs test for randomness about the median, what process do you use to convert the numbers into two distinct symbols?

Short Answer

Expert verified
Convert numbers greater than the median to '+' and less than the median to '-'.

Step by step solution

01

Identify the Median

First, calculate the median of the given sequence of numbers. The median is the middle value when the numbers are arranged in ascending order. If there is an even number of observations, the median is the average of the two middle numbers.
02

Compare Each Number to the Median

For each number in the sequence, compare it to the median. Determine if it's greater than, less than, or equal to the median.
03

Convert to Symbols

Assign a '+' symbol to numbers that are greater than the median and a '-' symbol to numbers that are less than the median. If any number equals the median, you may exclude it from the run test conversion, or choose either symbol consistently.

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Key Concepts

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

Understanding the Median in Statistical Tests
In statistics, the median is a measure of central tendency, which is often used to determine the middle value in a data set. Unlike the mean, the median isn't distorted by outliers or skewed data, making it a reliable tool for understanding data distribution. To find the median:
  • Arrange your data in ascending order.
  • If the number of data points is odd, the median is the middle number.
  • If the number of data points is even, the median is the average of the two middle numbers.
For example, in the data set [3, 1, 5, 2], first arrange it as [1, 2, 3, 5]. Since there are four numbers, the median is (2+3)/2 = 2.5. Finding the median is crucial for various statistical tests, including the runs test for randomness, because it acts as a benchmark to compare other numbers in the set.
The Runs Test for Randomness Explained
The runs test for randomness is a non-parametric statistical test used to evaluate whether a sequence of numbers is randomly ordered. It helps determine if there are any patterns within the data. This test is often employed in quality control and other applications where randomness determines consistency or fairness. To implement the runs test:
  • Identify the median of your data set.
  • Convert all values to symbols based on their relation to the median. Assign '+' to numbers greater than the median and '-' to numbers less than the median.
  • Count the number of runs in your sequence. A 'run' is a consecutive sequence of identical symbols (e.g., '+++' or '---').
The number and length of runs are analyzed to infer randomness. A surprising number of runs could suggest a non-random distribution, prompting further investigation.
Symbol Conversion in the Context of Statistical Tests
Symbol conversion is a crucial step in several statistical methods, including the runs test for randomness. It involves transforming quantitative data into qualitative symbols to streamline analysis. Here's how symbol conversion works in practice:
  • After identifying the median of your data set, evaluate each number relative to the median.
  • Assign a '+' symbol if a number is greater than the median and a '-' if it is less.
  • Numbers equal to the median can be treated consistently by assigning the same symbol or excluding them from analysis.
Why use symbol conversion? It simplifies complex numerical data into a form that鈥檚 easier to analyze for patterns. Once data is converted, you can apply further statistical analyses or visual inspections to understand distribution, trends, or anomalies.

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Most popular questions from this chapter

Consider the Spearman rank correlation coefficient \(r_{s}\) for data pairs \((x, y) .\) What is the monotone relationship, if any, between \(x\) and \(y\) implied by a value of (a) \(r_{s}=0 ?\) (b) \(r\) close to \(1 ?\) (c) \(r_{s}\) close to \(-1 ?\)

Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Compute the sample test statistic. (c) Find or estimate the \(P\) -value of the sample test statistic. (d) Conclude the test. (e) Interpret the conclusion in the context of the application. Ecology: Wetlands Turbid water is muddy or cloudy water. Sunlight is necessary for most life forms; thus turbid water is considered a threat to wetland ecosystems. Passive filtration systems are commonly used to reduce turbidity in wetlands. Suspended solids are measured in \(\mathrm{mg} / \mathrm{L} .\) Isthere a relation between input and output turbidity for a passive filtration system and, if so, is it statistically significant? At a wetlands environment in Illinois, the inlet and outlet turbidity of a passive filtration system have been measured. A random sample of measurements follow $$\begin{array}{|l|rrrrrrrrrrrr|} \hline \text { Reading } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\\ \hline \text { Inlet }(\mathrm{mg} / \mathrm{L}) & 8.0 & 7.1 & 24.2 & 47.7 & 50.1 & 63.9 & 66.0 & 15.1 & 37.2 & 93.1 & 53.7 & 73.3 \\ \hline \text { Outlet }(\mathrm{mg} / \mathrm{L}) & 2.4 & 3.6 & 4.5 & 14.9 & 7.4 & 7.4 & 6.7 & 3.6 & 5.9 & 8.2 & 6.2 & 18.1 \\ \hline \end{array}$$

Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Compute the sample test statistic. (c) Find or estimate the \(P\) -value of the sample test statistic. (d) Conclude the test. (e) Interpret the conclusion in the context of the application. FBI Report: Child Abuse and Runaway Children Is there a relation between incidents of child abuse and number of runaway children? A random sample of 15 cities (over 10,000 population) gave the following information about the number of reported incidents of child abuse and the number of runaway children (Reference: Federal Bureau of Investigation, U.S. Department of Justice). $$\begin{array}{|l|ccccccccccccccc} \hline \text { City } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline \text { Abuse cases } & 49 & 74 & 87 & 10 & 26 & 119 & 35 & 13 & 89 & 45 & 53 & 22 & 65 & 38 & 29 \\ \hline \text { Runaways } & 382 & 510 & 581 & 163 & 210 & 791 & 275 & 153 & 491 & 351 & 402 & 209 & 410 & 312 & 210 \\ \hline \end{array}$$ (i) Rank-order abuse using 1 as the largest data value. Also rank-order runaways using 1 as the largest data value. Then construct a table of ranks to be used for a Spearman rank correlation test. (ii) Use a \(1 \%\) level of significance to test the claim that there is a monotoneincreasing relationship between the ranks of incidents of abuse and number of runaway children.

Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Compute the sample test statistic. (c) Find or estimate the \(P\) -value of the sample test statistic. (d) Conclude the test. (e) Interpret the conclusion in the context of the application. Economics: Stocks As an economics class project, Debbie studied a random sample of 14 stocks. For each of these stocks, she found the cost per share (in dollars) and ranked each of the stocks according to cost. After 3 months, she found the earnings per share for each stock (in dollars). Again, Debbie ranked each of the stocks according to earnings. The way Debbie ranked, higher ranks mean higher cost and higher earnings. The results follow, where \(x\) is the rank in cost and \(y\) is the rank in earnings. $$\begin{array}{|l|cccccccccccccc} \hline \text { Stock } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\ \hline x \text { rank } & 5 & 2 & 4 & 7 & 11 & 8 & 12 & 3 & 13 & 14 & 10 & 1 & 9 & 6 \\ \hline y \text { rank } & 5 & 13 & 1 & 10 & 7 & 3 & 14 & 6 & 4 & 12 & 8 & 2 & 11 & 9 \\ \hline \end{array}$$ Using a 0.01 level of significance, test the claim that there is a monotone relation, cither way, between the ranks of cost and earnings.

To apply a runs test for randomness as described in this section to a sequence of symbols, how many different symbols are required?
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