Question

Use the information given in Exercises \(4-7\) to calculate Spearman's rank correlation coefficient, where \(x_{i}\) and \(y_{i}\) are the ranks of the ith pair of observations and \(d_{i}=x_{i}-y_{i} .\) Assume that there are no ties in the ranks. \(\sum d_{i}^{2}=16 ; n=10\)

Short answer

Answer: The Spearman's rank correlation coefficient for the given data set is approximately 0.903.

Step-by-step solution

Recall the formula for Spearman's rank correlation coefficient

The formula for Spearman's rank correlation coefficient (denoted as \(r_{s}\)) is given by: \(r_{s} = 1 - \frac{6 \sum d_{i}^{2}}{n(n^2 - 1)}\) Where: - \(r_{s}\) is Spearman's rank correlation coefficient - \(n\) is the total number of observations - \(d_{i}\) is the difference between the ranks of the i-th pair of observations - \(\sum d_{i}^{2}\) is the sum of squared differences between the ranks

Substitute values into the formula

Now, let's substitute the given information into the formula: \(\sum d_{i}^{2} = 16\) \(n = 10\) Plugging these values into the formula, we get: \(r_{s} = 1 - \frac{6 \times 16}{10(10^2 - 1)}\)

Simplify the formula and calculate \(r_{s}\)

Simplify the formula: \(r_{s} = 1 - \frac{96}{10(100 - 1)}\) \(r_{s} = 1 - \frac{96}{10(99)}\) \(r_{s} = 1 - \frac{96}{990}\) Now, we can divide 96 by 990: \(r_{s} = 1 - \frac{32}{330}\) \(r_{s} = 1 - 0.09697\)

Final calculation of Spearman's rank correlation coefficient

Now let's calculate the final value for \(r_{s}\): \(r_{s} = 1 - 0.09697\) \(r_{s} = 0.90303\) So, the Spearman's rank correlation coefficient is approximately \(0.903\).

Key concepts

Correlation Coefficient Calculation

Understanding the correlation coefficient calculation is essential when studying relationships between different data sets. Spearman's rank correlation coefficient is a non-parametric measure of rank correlation that assesses how well the relationship between two variables can be described using a monotonic function. It's particularly useful when the requirements for Pearson's correlation coefficient are not met, like when the data does not have a normal distribution.

In the given exercise, the formula for Spearman's rank correlation coefficient, denoted as \(r_s\), is: \[r_s = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}\], where \(\sum d_i^2\) is the sum of the squared rank differences, and \(n\) is the number of data pairs. The calculation is straightforward yet requires attention to detail to avoid mistakes, especially when squaring differences and substituting into the formula. Simplifying the equation systematically, as shown in the solution, helps prevent calculation errors and leads to the correct Spearman's rank correlation coefficient value.

Statistics Exercises

Engaging with statistics exercises, such as calculating Spearman's rank correlation coefficient, equips students with problem-solving skills applicable in various fields, including business, healthcare, and social sciences. These exercises often involve interpreting data and drawing conclusions from the analyses.

To enhance comprehension and retainment of statistical methods, students are advised to approach exercises methodically. Starting with understanding the problem, gathering required formulas, carefully substituting values, and carrying out calculations with precision ensures the development of a strong foundation in statistics. Visual aids, like graphs or charts, often complement these exercises, aiding in the interpretation of results and enhancing overall statistical literacy.

Probability and Statistics

Probability and statistics are integral branches of mathematics that focus on analyzing random events and interpreting data. Understanding these concepts is critical for decision making in uncertain conditions. Probability aids in predicting the likelihood of events, while statistics involves collecting, analyzing, and representing data.

Core to these fields are correlation coefficients, like Spearman's rank correlation coefficient, which are used to measure the strength and direction of the association between two variables. Students learning probability and statistics should master these coefficients to effectively analyze and interpret real-world data. Pictorial representations, such as scatter plots, complement learning by providing visual insights into the correlation between the data sets.