Question

An experiment was conducted to study the relationship between the ratings of a tobacco leaf grader and the moisture content of the tobacco leaves. Twelve leaves were rated by the grader on a scale of \(1-10\), and corresponding readings of moisture content were made. $$\begin{array}{ccc}\hline \text { Leaf } & \text { Grader's Rating } & \text { Moisture Content } \\\\\hline 1 & 9 & .22 \\\2 & 6 & .16 \\\3 &7 & .17 \\\4 & 7 & .14 \\\5 & 5 & .12 \\\6 & 8 & .19 \\\7 & 2 & .10 \\\8 & 6 & .12 \\\9 & 1 & .05 \\\10 & 10 & .20 \\\11 & 9 & .16 \\\12 & 3 & .09 \\\\\hline\end{array}$$ a. Calculate \(r_{s}\) b. Do the data provide sufficient evidence to indicate an association between the grader's ratings and the moisture contents of the leaves?

Short answer

Answer: __________

Step-by-step solution

Ranking the Data

First, we need to rank the grader's ratings and the moisture content of the leaves. Assign a rank based on the value of each observed variable, starting with the smallest. For example, Leaf 1 has a 9 rating and Leaf 9 has a 1 rating; hence, Leaf 9 will be ranked 1 and Leaf 1 will be ranked 11. Similarly, assign ranks for the moisture content. After ranking the data, create a table with columns for the leaf number, the grader's rating rank, and the moisture content rank.

Calculate \(d_i\) and \(d_i^2\)

Next, calculate the differences between the ranks for each leaf. This difference is represented as \(d_i\). Then, compute the squared difference, represented as \(d_i^2\). Add these two columns to the table.

Calculate \(r_s\) using the formula

After finding the differences and their squares, calculate the Spearman's rank correlation coefficient (\(r_s\)) using the following formula: \(r_s = 1-\frac{6\sum d_i^2}{n(n^2-1)}\) Where \(n\) is the number of data points (in this case, 12).

Test the Hypothesis

To determine if the data provide sufficient evidence to indicate an association between the grader's ratings and the moisture content, we need to test the null hypothesis (\(H_0\)) that there is no association (or \(r_s = 0\)) against the alternative hypothesis (\(H_1\)) that there is an association (or \(r_s ≠ 0\)). If the absolute value of the calculated \(r_s\) is greater than its critical value corresponding to the given significance level and degrees of freedom (which can be found in a critical value table for the Spearman rank correlation), we reject the null hypothesis, indicating a significant association between the grader's ratings and the moisture content. Now, after following these steps, determine whether the data show a significant association between the grader's ratings and the moisture content of the leaves.

Key concepts

Ranked Data Analysis

Ranked data analysis is an essential tool in non-parametric statistics, where numerical data are replaced by their rank orders. By converting raw scores to ranks, we can investigate relationships in ordinal data or continuous data that does not meet the assumptions of parametric tests, such as normal distribution or homoscedasticity.

Consider the case of the tobacco leaf grader's ratings and the moisture contents: the rankings transform the original numerical values into a sequence that reflects their relative standing. This conversion simplifies the complexity of the data and neutralizes the impact of outliers. It also allows us to use Spearman's rank correlation coefficient, which is robust against non-normal data distributions and provides insights into the monotonic relationship between two variables.

Steps in Ranked Data Analysis

• Assign ranks to the data based on their value.
• Calculate differences between ranks of paired observations if necessary.
• Use ranks in statistical formulas, such as the Spearman rank correlation coefficient, to assess relationships.

By meticulously following these steps, the statistical analysis maintains the order of the values while discarding less essential information, like the exact difference between values, which is often not required for non-parametric tests.

Hypothesis Testing in Statistics

Hypothesis testing is a fundamental aspect of inferential statistics used to make decisions about the properties of a population based on sample data. The test begins with two opposing hypotheses: the null hypothesis ((H_0)) and the alternative hypothesis ((H_1)).

The null hypothesis typically asserts that no effect, difference, or association exists, while the alternative hypothesis suggests the opposite. Using a test statistic, calculated from the data, one evaluates the likelihood of observing the sample data if the null hypothesis were true.

Process of Hypothesis Testing

• Formulate the null and alternative hypotheses.
• Calculate a test statistic based on the sample data.
• Determine the p-value or compare the test statistic to the critical value.
• Decide whether to reject or fail to reject the null hypothesis.

In our tobacco leaf example, hypothesis testing is used to decide whether the observed Spearman's rank correlation coefficient suggests a true association between the grader's ratings and moisture content of the leaves or if the observed correlation could be due to random chance.

Association Between Variables

Determining the association between variables is pivotal in many fields as it helps in understanding how one variable may predict or affect another. An association can take various forms, such as linear, monotonic, or nonlinear relationships.

In the context of the tobacco leaves, we are concerned with a monotonic association, which means that as one variable increases or decreases, the other variable tends to move in the same direction, but not necessarily at a constant rate.

Measuring Association

• Spearman's rank correlation coefficient for monotonic relationships.
• Pearson's correlation coefficient for linear relationships.
• Chi-square tests for categorical variables.

Spearman’s rank coefficient ((r_s)) specifically measures the strength and direction of association between two ranked variables. High magnitude of ((r_s)) indicates a strong association, while a value near zero suggests little to no association. By testing this statistic against a critical value, we infer whether the observed association is statistically significant, that is, unlikely to have arisen by chance alone.