Let's analyze the table and the details it provides, particularly focusing on the totals in the last row:
[tex]\[
\begin{tabular}{llll}
Weather condition & On-time & Delayed & Total \\
\hline
Sunny & 167 & 3 & 170 \\
Cloudy & 115 & 5 & 120 \\
Rainy & 40 & 15 & 55 \\
Snowy & 8 & 12 & 20 \\
\hline
Total & 330 & 35 & 365 \\
\end{tabular}
\][/tex]
To answer the question, we need to understand what the last row represents:
1. Total On-time Trains: This is the sum of all on-time trains across all weather conditions (Sunny, Cloudy, Rainy, Snowy).
[tex]\[
167 + 115 + 40 + 8 = 330
\][/tex]
2. Total Delayed Trains: This is the sum of all delayed trains across all weather conditions.
[tex]\[
3 + 5 + 15 + 12 = 35
\][/tex]
3. Overall Total: This is the sum of total trains across all weather conditions.
[tex]\[
170 + 120 + 55 + 20 = 365
\][/tex]
Based on the definition of marginal distribution, it is the distribution of either variable by itself, ignoring other variables. In this context:
- [tex]\(\textbf{Marginal distribution of arrival status}\)[/tex] refers to the distribution of on-time versus delayed status across all weather conditions.
- [tex]\(\textbf{Marginal distribution of weather type}\)[/tex] would involve the total counts for each weather type alone, which is noted in the individual sums of each weather condition (170 for Sunny, 120 for Cloudy, etc.).
- [tex]\(\textbf{Conditional distribution}\)[/tex] refers to the distribution of one variable under the condition that the other variable has a specific value, which would require comparisons within subsets rather than total sums.
Here, the last row provides the total number of on-time trains, the total number of delayed trains, and the overall total number of trains. Therefore, it encapsulates the on-time and delayed train statuses over all conditions, making this a description of the marginal distribution of the arrival status.
Thus, the best description of the distribution boxed in the last row of the table is:
A. This is the marginal distribution of arrival status.
