Question

The data in the table below are the changes in the amount of space available to standing passengers at the 19 stops between 1987 and 1988.

Changes AM	Changes PM
-0.4	-5.1
-1.1	-1.5
0	0.3
4.6	8.1
-0.7	3.3
0	0.5
3	-1.2
-1.8	-1.2
1	-3.2
0.8	-0.4
-3	5.3
-3	16.9
-0.9	-0.1
-0.3	-0.5
-0.5	0.6
0.2	-0.2
-0.3	-0.4
-0.3	-1.1
0.4	0.6

In the table below, summary information is presented for these data.

time	lower quartile	median	upper quartile
AM	-0.9	-0.3	0.4
PM	-1.2	-0.2	0.6

time	mean	standard deviation
AM	-0.121	1.766
PM	1.089	4.784

1. (a) Using the raw data and summary information presented in the tables above, construct modified box plots to compare the changes in available space the morning and afternoon. (Reminder: Don't forget to check for outliers!)

2. (b) The Transit System wishes to know if their efforts to improve the standing space were successful. (Remember, more space is better!) Their engineers had suggested that the changes in the system would, on average, be more successful at increasing the available space in the morning than in the afternoon. Does the data support this initial belief? What specific aspects of the plot in part (a) support your answer?

3. (c) Using your box plots in part (a), write a short paragraph for the New York Times describing the success the Transit System had in increasing the available space per passenger. Note any differences in success between the morning rush and the afternoon rush.

Answer 1

(a)

From the box plots, there are 4 outliers in Changes AM and they are: -3,-3, 3, 4.6 and 4 outliers in Changes PM and they are:

-5.1, 5.3, 8.1, 16.9.

(b)

From the above probability plot it is seem that normality assumption does not hold for these data sets. So for testing the claim that the changes in the system would, on average, be more successful at increasing the available space in the morning than in the afternoon, we use Wilcoxon's sign rank test (since the data is bivariate in nature).

Wilcoxon Signed Rank Test: Difference (AM-PM)

Test of median = 0.000000 versus median > 0.000000

N for Wilcoxon Estimated
N Test Statistic P Median
Difference (AM-PM) 19 19 84.0 0.678 -0.2000

Since p-value=0.678>0.05 so the data do not support that the changes in the system would, on average, be more successful at increasing the available space in the morning than in the afternoon.

From box plots it is observed that Median of AM change is greater than the median of PM change hence these plots suggest that the changes in the system would, on average, be more successful at increasing the available space in the morning than in the afternoon.

(c)

From box plots, first quartile of the data of morning rush is greater than third quartile of the data of afternoon i.e. 75% observations of Afteroon rush lies below the 25% observations of morning rush hence the changes in the system would be more successful at increasing the available space in the morning than in the afternoon significantly.