Question

Perform these steps. a. Find the Spearman rank correlation coefficient. b. State the hypotheses. c. Find the critical value. Use \(\alpha=0.05\). d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Drug Prices Shown are the price for a human dose of several randomly selected prescription drugs and the price for an equivalent dose for animals. At \(\alpha=0.10\), is there a relationship between the variables? $$ \begin{array}{l|llllllll} \text { Humans } & 0.67 & 0.64 & 1.20 & 0.51 & 0.87 & 0.74 & 0.50 & 1.22 \\ \hline \text { Animals } & 0.13 & 0.18 & 0.42 & 0.25 & 0.57 & 0.58 & 0.49 & 1.28 \end{array} $$

Short answer

There is no significant correlation between human and animal drug prices at \(\alpha = 0.10\).

Step-by-step solution

Rank the Data

First, we need to rank the data for humans and animals separately. Assign a rank of 1 to the smallest value, 2 to the next smallest, and so on. If there are ties, assign the average rank to each tied value.

Calculate Differences in Ranks

Now, calculate the difference (\(d\)) between the ranks of each paired human and animal value. Also, compute the squared differences \(d^2\).

Find the Spearman Rank Correlation Coefficient

Using the formula \(r_s = 1 - \frac{6 \sum d^2}{n(n^2-1)}\), where \(n\)is the number of pairs, calculate the Spearman rank correlation coefficient. Substitute the values obtained from Step 2.

State the Hypotheses

The null hypothesis \(H_0\) is that there is no correlation between drug prices for humans and animals (\(r_s = 0\)). The alternative hypothesis \(H_a\) is that there is a correlation (\(r_s eq 0\)).

Find the Critical Value

Refer to a Spearman rank correlation critical value table at \(\alpha = 0.10\) for \(n = 8\) to find the critical values. A two-tailed test is required, so consider the upper and lower bounds.

Make the Decision

Compare the calculated Spearman rank correlation coefficient to the critical values. If the coefficient is outside the critical value ranges, reject the null hypothesis; otherwise, do not reject it.

Summarize the Results

Interpret the decision in the context of the problem. If you rejected the null hypothesis, state that there is a statistically significant relationship between human and animal drug prices at \(\alpha = 0.10\). Otherwise, state that there is not a statistically significant relationship.

Key concepts

hypothesis testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It involves defining two competing hypotheses:

• The null hypothesis (\(H_0\)) which suggests that there is no effect or relationship, often represented as "no correlation." For our exercise, \(H_0: r_s = 0\) implies no relationship between human and animal drug prices.
• The alternative hypothesis (\(H_a\)), indicating that there is an effect or relationship. Here, it is \(H_a: r_s eq 0\), meaning there is a correlation between the drug prices for humans and animals.

The choice between the null and alternative hypothesis is made based on statistical evidence provided by the analysis of your data. We set a level of significance, \(\alpha\), usually \(0.05\) or \(0.10\), which represents the probability of rejecting the null hypothesis if it is true.
If the evidence against \(H_0\) is strong enough, we reject \(H_0\) in favor of \(H_a\). Otherwise, we do not reject \(H_0\). This methodical approach provides a structured framework to make inferential decisions.

critical value

The critical value is a threshold that helps to decide whether to reject the null hypothesis in a hypothesis test. It depends on the significance level (\(\alpha\)) and the sample size. For a Spearman rank correlation test, the critical values can be found from statistical tables specific to rank correlation.
In this exercise, our sample size (\(n\)) is 8, and \(\alpha\) is 0.10. This calls for a two-tailed test, meaning that both extreme ends of the probability distribution are considered. Check the critical values for \(n = 8\) and \(\alpha = 0.10\) in a Spearman rank correlation critical value table.
If the calculated correlation coefficient exceeds these critical value limits, it indicates that the correlation observed is strong enough to suggest a real relationship at the chosen significance level, leading to the rejection of the null hypothesis. Otherwise, the null hypothesis cannot be rejected.

rank data

Ranking data is a vital step in calculating the Spearman Rank Correlation. To start, each piece of data is ordered from the smallest to the largest, and a rank is assigned to each value.
In cases where two or more values are identical, an average rank is assigned to these values. This ranking simplifies data, converting raw scores into rank orders to facilitate non-parametric correlation estimation.

• For example, in this exercise, we examine human and animal drug prices. Each set of data points for humans and animals must be ranked separately.
• After ranking, you compare these ranks between the two data sets to find patterns in order.

Ranking handles data that doesn't adhere to normal distribution assumptions, making it flexible for many practical scenarios.
By converting the original observations into ranks, you can effectively measure the degree of association between two variables, which is essential for the subsequent calculation of the Spearman rank correlation coefficient.

correlation coefficient

The correlation coefficient, specifically the Spearman rank correlation coefficient (\(r_s\)), provides a measure of the strength and direction of association between two ranked variables. Spearman's \(r_s\) ranges between -1 and 1.
An \(r_s\) of 1 suggests a perfect positive correlation, meaning that as one variable increases, the other does as well. Conversely, an \(r_s\) of -1 indicates a perfect negative correlation where one variable increases and the other decreases.
The calculation involves the differences in the ranks of each pair of observations. The formula used is:\[r_s = 1 - \frac{6 \sum d^2}{n(n^2-1)}\]where \(d\) is the difference between the ranks of each pair and \(n\) is the number of pairs.

• For the given data in the exercise, the ranks for both human and animal drug prices are calculated.
• Then, the differences in those ranks are squared and summed up.
• These calculations plug into the formula to yield \(r_s\).

Understanding the correlation coefficient's value tells us how strongly the variables are related, guiding the decision on the null hypothesis.