Question

The following table shows the means of transportation to work for the U.S. in 1990 and 2000. The data values in the columns of “Percentage in 1990” and “Percentage in 2000” are the percentages of each travel mode in 1990 and in 2000 respectively using two independent and random samples. We want to do a hypothesis test and see whether the travel modes have been significantly changed from 1990 to 2000.

Table: The means of transportation to work for the US: 1990 and 2000. (Data source: US Census)

Travel Mode	Percentage in 1990	Percentage in 2000
Private vehicle	86.5	87.5
Public transportation	5.3	5.2
Motorcycle	0.2	0.1
Bicycle	0.4	0.4
Walk only	3.9	2.7
Other means	0.7	0.9
Work at home	3.0	3.2

Use the most appropriate method to evaluate the hypothesis by hand. Include the step of hypothesis testing.

Answer 1

Since we are doing repeated measure on the same subject, the paired sample t-test is used to test the following hypothesis

Hypothesis

The null hypothesis and the alternative hypothesis are defined as,

This is a two-tailed test. The significance level of the test is 0.05.

Test statistic

The t-statistic is obtained using the formula,

From the data values,

Percentage in 1990	Percentage in 2000	Difference
86.5	87.5	1
5.3	5.2	-0.1
0.2	0.1	-0.1
0.4	0.4	0
3.9	2.7	-1.2
0.7	0.9	0.2
3	3.2	0.2
.	Sum	0
.	Average	0
.	Std Dev	0.6506
.	n	7

P-value

The P-value for t = 0 is obtained from t-distribution table for degree of freedom = n - 1 = 6 and two-sided alternative hypothesis,

Conclusion:

Since the P-value for the two-tailed alternative hypotheses is greater than 0.05 at a 5% significance level, the null hypothesis is not rejected. Hence there is not sufficient evidence to conclude that the percentage of transportation mode has changed between 1990 and 2000.