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Chapter 11: Problem 1

To apply the sign test, do you need independent or I dependent (matched pair) data?

Short Answer

Expert verified
The sign test requires dependent (matched pair) data.

Step by step solution

01

Understand the Sign Test

The sign test is a non-parametric test used to determine if there is a difference between the medians of two related groups. It is often used when the assumptions for the paired t-test are not met, particularly when data is not normally distributed.
02

Identify the Type of Data

The sign test requires matched pairs of data. Each subject or entity in the study should have two measurements, under two different conditions, or at two different times.
03

Clarify Matched Pairs Data

Matched pairs data means that the data points are dependent. Each pair is a related observation, such as before and after measurements on the same individual, making the data dependent. This comparison is essential for the sign test.
04

Conclusion on Data Type for Sign Test

Since the sign test is used with matched pairs data, it requires dependent data. This dependency is the basis for comparing two related groups to determine if there is a significant difference.

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Key Concepts

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

Matched Pairs
Matched pairs refer to a statistical design where subjects are paired based on certain criteria. This design is crucial in ensuring that comparisons are valid. Each pair contains two related observations or measurements. For example, imagine a study measuring blood pressure before and after an intervention. Each participant provides a pair: a before measurement and an after measurement.
This pairing is key to eliminating variability that is not of interest. By focusing on differences within pairs, we reduce the influence of factors that could skew results. This method ensures the integrity of the data comparison.
Non-Parametric Test
The sign test is a non-parametric test. Non-parametric tests are statistical tests that do not assume a specific distribution for the data. This characteristic makes them especially useful when data do not meet the assumptions required for parametric tests like the t-test.
Generally, non-parametric tests are considered more flexible and less sensitive to outliers. They do not require data to follow a normal distribution, making them applicable in a wider range of scenarios, particularly with ordinal data or when sample sizes are small. This flexibility is why the sign test is a popular choice when dealing with matched pairs.
Dependent Data
Dependent data is integral to matched pairs analyses. Dependency occurs when the data points in a study are linked. For instance, measurements taken from the same individual at different times are dependent because they are inherently related.
  • Dependence ensures meaningful comparisons by linking observations within pairs.
  • It accounts for individual variability, focusing the analysis purely on treatment effects or between-condition differences.
In the context of a sign test, having dependent data means that each pair provides a basis for comparison, crucial for determining changes or effects.
Medians Comparison
The sign test is fundamentally about medians comparison. When analyzing matched pairs, the sign test determines if two related medians are different. Unlike the mean, the median is the middle value in a data set, unaffected by extremely high or low values.
The focus on medians is significant because medians provide a robust measure of central tendency. This is especially important when data are skewed or when outliers are present.
By using the sign test, researchers can make informed decisions based on median comparisons, ensuring that conclusions drawn are both valid and reliable, even in the absence of normally distributed data.

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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 $$\begin{array}{|l|rrrrrrrrrrrr|} \hline \text { Reading } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\\ \hline \text { Inlet }(\mathrm{mg} / \mathrm{L}) & 8.0 & 7.1 & 24.2 & 47.7 & 50.1 & 63.9 & 66.0 & 15.1 & 37.2 & 93.1 & 53.7 & 73.3 \\ \hline \text { Outlet }(\mathrm{mg} / \mathrm{L}) & 2.4 & 3.6 & 4.5 & 14.9 & 7.4 & 7.4 & 6.7 & 3.6 & 5.9 & 8.2 & 6.2 & 18.1 \\ \hline \end{array}$$

How do the average weekly incomes of lawyers and architects compare? A random sample of 18 regions in the United States gave the following information about average weekly incomes (in dollars) (Reference: U.S. Department of Labor, Bureau of Labor Statistics). $$\begin{array}{|l|ccccccccc} \hline \text { Region } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \text { Lawyers } & 709 & 898 & 848 & 1041 & 1326 & 1165 & 1127 & 866 & 1033 \\\ \hline \text { Architects } & 859 & 936 & 887 & 1100 & 1378 & 1295 & 1039 & 888 & 1012 \\ \hline & & & & & & & & & \\ \hline \text { Region } & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 \\ \hline \text { Lawyers } & 718 & 835 & 1192 & 992 & 1138 & 920 & 1397 & 872 & 1142 \\ \hline \text { Architects } & 794 & 900 & 1150 & 1038 & 1197 & 939 & 1124 & 911 & 1171 \\ \hline \end{array}$$ Does this information indicate that architects tend to have a larger average weekly income? Use \(\alpha=0.05\)

When applying the rank-sum test, do you need independent or dependent samples?

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. Demographics: Police and Fire Protection Is there a relation between police protection and fire protection? A random sample of large population areas gave the following information about the number of local police and the number of local firefighters (units in thousands) (Reference: Statistical Abstract of the United States). $$\begin{array}{|l|ccccccccccccc} \hline \text { Area } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\\ \hline \text { Police } & 11.1 & 6.6 & 8.5 & 4.2 & 3.5 & 2.8 & 5.9 & 7.9 & 2.9 & 18.0 & 9.7 & 7.4 & 1.8 \\ \hline \text { Firefighters } & 5.5 & 2.4 & 4.5 & 1.6 & 1.7 & 1.0 & 1.7 & 5.1 & 1.3 & 12.6 & 2.1 & 3.1 & 0.6 \\ \hline \end{array}$$ (i) Rank-order police using 1 as the largest data value. Also rank-order firefighters 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 \(5 \%\) level of significance to test the claim that there is a monotone relationship (either way) between the ranks of number of police and number of firefighters.
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