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. Subway and Commuter Rail Passengers Six cities are randomly selected, and the number of daily passenger trips (in thousands) for subways and commuter rail service is obtained. At \(\alpha=0.05,\) is there a relationship between the variables? Suggest one reason why the transportation authority might use the results of this study. $$ \begin{array}{l|rrrrrr} \text { City } & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Subway } & 845 & 494 & 425 & 313 & 108 & 41 \\ \hline \text { Rail } & 39 & 291 & 142 & 103 & 33 & 38 \end{array} $$

Short answer

The Spearman rank correlation coefficient indicates if there's a significant correlation between subway and rail use. If \( |r_s| > 0.886 \), there's a significant correlation, implying the transportation authority might adjust services accordingly.

Step-by-step solution

Rank the Data

First, we assign ranks to the subway and rail data for each city. For subway data, rank the values from largest to smallest: 1 for 845, 2 for 494, 3 for 425, 4 for 313, 5 for 108, and 6 for 41. For the rail data, 1 for 291, 2 for 142, 3 for 103, 4 for 39, 5 for 38, and 6 for 33.

Calculate Differences in Ranks

Next, calculate the difference between the ranks of subway and rail for each city, denoted by \(d\). Then, compute \(d^2\) for each. For example, if Subway Rank = 1 and Rail Rank = 4, then \(d = 1-4 = -3\) and \(d^2 = 9\).

Compute Sum of Squared Differences

Sum all the \(d^2\) values to get \(\sum d^2\). This is an essential step for later calculations of the Spearman rank correlation coefficient.

Calculate Spearman Rank Correlation Coefficient

Use the formula: \( r_s = 1 - \frac{6\sum d^2}{n(n^2-1)} \), where \(n\) is the number of pairs (cities). Substitute \(\sum d^2\) and \(n=6\) to find \(r_s\).

State the Hypotheses

Formulate the null hypothesis \(H_0\): There is no correlation between subway and rail passenger numbers. The alternative hypothesis \(H_a\): There is a correlation between subway and rail passenger numbers.

Find the Critical Value

Find the critical value for Spearman's rank correlation using a significance level \(\alpha=0.05\) and \(n=6\). The critical value from statistical tables for a two-tailed test at this significance level is approximately \(0.886\).

Make the Decision

Compare the calculated \(r_s\) with the critical value. If \(|r_s| > 0.886\), reject \(H_0\); otherwise, do not reject \(H_0\).

Summarize the Results

If \(H_0\) is rejected, we conclude there is a significant correlation between subway and rail passenger trips. The transportation authority could use these results to improve coordination between subway and rail services based on passenger trends.

Key concepts

Hypothesis Testing

Hypothesis testing is a statistical method that allows us to make inferences or draw conclusions about a population based on sample data. The core idea is to test a claim or hypothesis statistically using collected data.
The process begins with stating two hypotheses: the null hypothesis (\( H_0 \)), often a statement of no effect or no difference, and the alternative hypothesis (\( H_a \)), which is what you aim to show. For example, in a transportation study comparing subway and rail data, \( H_0 \) might state there is no correlation between subway and rail passenger trips, while \( H_a \) would suggest there is a correlation.
The significant level (\( \alpha \)), such as 0.05, denotes the probability of committing a Type I error, which is incorrectly rejecting the null hypothesis.
Hypothesis testing helps to ensure that conclusions made from data are reliable and not due to random chance.

Critical Value

The critical value is a key component of hypothesis testing that determines the threshold at which we decide whether to reject the null hypothesis.
For Spearman's rank correlation, the critical value is derived from statistical tables based on the chosen significance level (\( \alpha \)) and the number of data pairs (\( n \)). In our example with six cities, and using \( \alpha = 0.05 \), the critical value for a two-tailed test is approximately 0.886.
This value acts as a boundary:

• If the absolute value of the calculated Spearman rank correlation coefficient (\(|r_s|\)) is greater than the critical value, we reject the null hypothesis, suggesting a statistically significant correlation exists between variables.
• If \(|r_s|\) is less than or equal to the critical value, we do not reject the null hypothesis, indicating no significant correlation.

Understanding critical values is essential in determining the statistical significance of your findings.

Statistical Analysis

Statistical analysis involves the series of techniques we apply to understand, interpret, and draw conclusions from data. This process is vital for revealing patterns and relationships which are not immediately obvious.
In a transportation study, statistical analysis helps determine if there's a correlation between subway and rail passenger numbers. Here, the Spearman rank correlation is a non-parametric test used to identify the strength and direction of association between ranked variables.
The process involves several steps:

• First, rank the data points for each variable independently.
• Then, calculate the differences in ranks for each pair of data points.
• Next, compute the square of these rank differences and sum them up.
• Finally, use these values in the Spearman rank correlation formula to find \( r_s \), with larger values indicating stronger relationships.

This analysis aids in decision-making, essential for effective planning and management.

Transportation Study

Transportation studies analyze and inform strategies to improve transit systems in cities. They often explore the patterns of passenger usage to optimize services and resources.
For instance, examining the correlation between subway and rail passenger numbers can provide insights into usage patterns. These insights help in improving coordination between services, leading to better scheduling and more efficient use of capacity.
A transportation authority might use a study like this one to:

• Identify potential areas for synchronized scheduling between subways and rail services.
• Determine peak usage times and adjust capacity accordingly.
• Explore opportunities for improving the passenger experience by reducing wait times and enhancing connectivity.

By using data-driven insights from studies, authorities can implement targeted strategies for a more efficient public transportation network. This ensures both resource optimization and improved passenger service, benefiting the urban commuting experience overall.