a We know that the Congress can override veto by a two-thirds majority vote in each house. Five new organizations conduct individual random surveys of U.S. Senators to find whether they will vote to override the veto. The results are given in the table.
Sample Size
Number of Votes to Override Veto
Percent of Votes to Override Veto
7
6
85.7 %
22
16
72.7 %
28
21
75 %
31
17
54.8 %
49
27
55.1 %
We will consider whether the Senate will vote to override the veto by looking at the results of the first two surveys.
Sample Size
Number of Votes to Override Veto
Percent of Votes to Override Veto
7
6
85.7 %
22
16
72.7 %
28
21
75 %
31
17
54.8 %
49
27
55.1 %
We will use the information that a two-thirds majority vote is necessary in each house to override a veto. Let's write the required vote ratio as a percent.
Two-Thirds of the Votes&=2/3of the Votes
&≈ 66.7 %
Therefore, if the percent of the votes is greater that 66.7 %, the Senate will vote to override the veto.
Sample Size
Number of Votes to Override Veto
Percent of Votes to Override Veto
7
6
85.7 %> 66.7 %
22
16
72.7 %> 66.7 %
28
21
75 %
31
17
54.8 %
49
27
55.1 %
Consequently, yes! The Senate will vote to override the veto.
b In this exercise we will take account of the whole table, and we will think about whether the Senate will vote to override the veto. From the table we can see that when the sample size increases, the percent of votes to override veto decreases. Let's consider the survey with the larger sample space.
Sample Size
Number of Votes to Override Veto
Percent of Votes to Override Veto
7
6
85.7 %
22
16
72.7 %
28
21
75 %
31
17
54.8 %
49
27
55.1 %
We found in Part A that the percent of votes needs to be more than 66.7 % to override veto. In the last survey this percent is 55.1 %, which is less than 66.7 %. Therefore, the percent is insufficient to override veto. The answer is no.
