Question

In Exercise 10 (Section 15.2), you used the sign test to determine whether the data provided sufficient evidence to indicate that assessor A tends to give higher assessments than assessor \(\mathrm{B}\), using the data shown in the table. $$ \begin{array}{ccc} \hline \text { Property } & \text { Assessor A } & \text { Assessor B } \\ \hline 1 & 276.3 & 275.1 \\ 2 & 288.4 & 286.8 \\ 3 & 280.2 & 277.3 \\ 4 & 294.7 & 290.6 \\ 5 & 268.7 & 269.1 \\ 6 & 282.8 & 281.0 \\ 7 & 276.1 & 275.3 \\ 8 & 279.0 & 279.1 \\ \hline \end{array} $$ a. Use the Wilcoxon signed-rank test for a paired experiment to test the null hypothesis that there is no difference in the distributions of property assessments between assessors \(A\) and \(B\) against the onetailed alternative, using a value of \(\alpha\) near. \(05 .\) b. Compare the conclusion of the test in part a with the conclusions derived from the results of the \(t\) -test and the sign test in Exercises \(10(\mathrm{a})\) and \(10(\mathrm{~b})\) (Section 15.2). Explain why these test conclusions are (or are not) consistent.

Short answer

Question: Perform the Wilcoxon signed-rank test for the given data of property assessments by two assessors, A and B. Compare the conclusions with the t-test and sign test. Answer: After performing the Wilcoxon signed-rank test, we found sufficient evidence to suggest that Assessor A tends to give higher assessments than Assessor B. This conclusion is consistent with the findings from the t-test and sign test performed in previous exercises, as all of these tests are non-parametric tests with similar purposes, focusing on differences between paired observations.

Step-by-step solution

Calculate the differences

First, calculate the differences between the assessments of each property by Assessor A and Assessor B: $$ \begin{array}{cccc} \hline \textbf{Property} & \textbf{Assessor A} & \textbf{Assessor B} & \textbf{Difference}\\ \hline 1 & 276.3 & 275.1 & 1.2 \\ 2 & 288.4 & 286.8 & 1.6 \\ 3 & 280.2 & 277.3 & 2.9 \\ 4 & 294.7 & 290.6 & 4.1 \\ 5 & 268.7 & 269.1 & -0.4 \\ 6 & 282.8 & 281.0 & 1.8 \\ 7 & 276.1 & 275.3 & 0.8 \\ 8 & 279.0 & 279.1 & -0.1 \\ \hline \end{array} $$

Calculate the ranks of the absolute differences

Next, we need to rank the absolute differences. Ignore any zero differences and assign the smallest non-zero difference the rank 1. $$ \begin{array}{cc} \hline \textbf{Ranked Differences} & \textbf{Rank}\\ \hline 0.1 & 1 \\ 0.4 & 2 \\ 0.8 & 3 \\ 1.2 & 4 \\ 1.6 & 5 \\ 1.8 & 6 \\ 2.9 & 7 \\ 4.1 & 8 \\ \hline \end{array} $$

Calculate the sum of positive and negative ranks

Add up the ranks of the positive and negative differences separately: - Positive ranks sum: \(4 + 5 + 7 + 8 + 3 + 6 = 33\) - Negative ranks sum: \(1 + 2 = 3\) Since the alternative hypothesis is that Assessor A tends to give higher assessments than Assessor B, we're interested in the sum of the positive ranks: \(W = 33\).

Determine the critical value

For a one-tailed test at a significance level of \(\alpha \approx 0.05\), with \(n = 8\) pairs, the critical value of W using a Wilcoxon signed-rank table is \(W_\text{critical} = 7\).

Compare the test statistic and critical value

Since \(W = 33 > W_\text{critical} = 7\), we reject the null hypothesis. This means that there is sufficient evidence to suggest that Assessor A tends to give higher assessments than Assessor B. b. Compare the conclusions The conclusion from the Wilcoxon signed-rank test is consistent with the conclusions derived from both the \(t\)-test and the sign test in Exercises 10(a) and 10(b) (Section 15.2) as all of them suggest that Assessor A tends to give higher assessments than Assessor B. This consistency is expected as all these tests are non-parametric tests with similar purposes, focusing on differences between paired observations.

Key concepts

Non-Parametric Statistics

When it comes to analyzing data that doesn't necessarily conform to the common assumptions of standard parametric tests, non-parametric statistics come in handy. One of the primary features of non-parametric methods is that they don't assume a particular distribution for the data, such as normality. This makes them particularly useful when dealing with small sample sizes, ordinal data, or data that is not evenly distributed.

Non-parametric tests can be applied to various types of data and research questions. The Wilcoxon signed-rank test, an example of such a test, is used when comparing two related samples or repeated measurements on a single sample to assess whether their population mean ranks differ. It's an alternative to the paired sample t-test which requires data to follow a normal distribution. In our exercise, it allows us to examine the assessments made by two appraisers without making assumptions about the normality of the differences in their assessments.

Advantages of Non-Parametric Tests

• No need for data to fit a normal distribution.
• Useful for small sample sizes.
• Can handle ordinal data and non-numeric data well.

By utilizing the ranks of data rather than the data itself, non-parametric methods manage to provide strong statistical tools when the assumptions for parametric tests are not met.

Hypothesis Testing

Hypothesis testing is a fundamental concept in statistics, providing a way to make inferences about populations based on sample data. There are two hypotheses in this framework: the null hypothesis (\( H_0 \)) and the alternative hypothesis (\( H_a \) or \( H_1 \))). The null hypothesis usually posits no effect or no difference, while the alternative hypothesis represents what the researcher is aiming to support.

The procedure involves using a test statistic to decide whether to reject the null hypothesis. This decision is made with reference to a pre-chosen level of significance \( \alpha \), which is the probability of incorrectly rejecting the null hypothesis, known as a Type I error. Common significance levels are 0.05 or 0.01.

In the case of the Wilcoxon signed-rank test, the null hypothesis might state that there's no difference in the median of the differences between assessments from Assessor A and B. Based on the computed test statistic, and comparing this with a critical value or p-value, we can reject or fail to reject the null hypothesis.

Steps in Hypothesis Testing

• State the null and alternative hypotheses.
• Determine the appropriate test statistic and its distribution.
• Choose a significance level \( \alpha \).
• Calculate the test statistic based on the sample data.
• Make a decision to reject or not reject the null hypothesis based on the comparison of the test statistic to a critical value.

Paired Experimental Design

Paired experimental design, also known as matched-pairs or repeated-measures design, is a robust approach in experiments where the participants are inherently linked or matched in some way. This can be the same individual measured over time or pairs of individuals who are closely matched on key characteristics.

This design reduces the variability due to differences between subjects, as each pair serves as its own control. By comparing measurements within pairs, researchers are able to more accurately attribute any differences in the outcomes to the experimental manipulation rather than to individual differences.

In our textbook exercise, each property is evaluated by both Assessor A and Assessor B. The paired design allows us to use each property as its own control to compare the two assessors’ assessments.

Key Elements of Paired Experimental Design

• Reduces variability and improves accuracy.
• Each subject or unit is matched, minimizing the differences within each pair.
• Especially useful in before-and-after studies, or when two treatments must be compared within the same subject.

The Wilcoxon signed-rank test is particularly fitting for analyzing data from a paired experimental design as it specifically tests differences between pairs, in line with the design's intent to control for individual variabilities.