Question 6·Hard·Inference from Sample Statistics and Margin of Error
Poll Results
| Candidate | Number of poll votes |
|---|---|
| Angel Cruz | 483 |
| Terry Smith | 320 |
A total of 803 voters selected at random were asked which candidate they would vote for in the upcoming election. According to the poll, if 6,424 people vote in the election, by how many votes would Angel Cruz be expected to win?
For questions that use poll results to predict an election outcome, first find the difference between candidates in the sample, then determine how much larger the real group is compared to the sample and use that as a scale factor. Multiply each candidate’s poll totals (or the poll difference) by this factor, and finally take the difference if needed. This keeps calculations quick and avoids confusing total votes with the winning margin.
Hints
Start with the poll lead
First, compute how many more poll votes Angel Cruz has than Terry Smith by subtracting their poll counts.
Compare poll size to election size
You know the poll surveyed 803 people and the election will have 6,424 voters. How many times larger is 6,424 than 803?
Use proportional reasoning
Assume the poll proportions stay the same in the full election. Multiply both candidates’ poll votes by the same scale factor, then find the difference between their expected totals.
Desmos Guide
Compute the expected winning margin directly
In Desmos, type the expression 6424*(483-320)/803. The numerical value that Desmos outputs is the expected number of votes by which Angel Cruz would win.
Step-by-step Explanation
Find Cruz’s lead in the poll
From the table:
- Angel Cruz: 483 poll votes
- Terry Smith: 320 poll votes
Cruz’s lead in the poll is the difference:
So, in the sample of 803 voters, Cruz leads by 163 votes.
Relate the poll size to the total number of voters
The poll surveyed 803 voters, and the election is expected to have 6,424 voters.
We assume the poll is representative, so the same proportions will apply to the larger group.
Find how many times larger 6,424 is than 803:
So the election has 8 times as many voters as the poll.
Scale each candidate’s poll votes to the full election
If the election is 8 times larger and proportions stay the same, multiply each candidate’s poll votes by 8:
For Angel Cruz:
For Terry Smith:
These are the expected numbers of votes for each candidate in the full election.
Find the expected winning margin in the election
To find by how many votes Cruz is expected to win, subtract Smith’s expected votes from Cruz’s expected votes:
So, Angel Cruz would be expected to win by votes.