Question 29·Medium·Inference from Sample Statistics and Margin of Error
A polling organization selected a simple random sample of 1,200 registered voters in a particular state. In the sample, 54% indicated that they support a proposed statewide bond measure. The poll reports a margin of error of percentage points.
Which of the following conclusions is best supported by the poll results?
For margin-of-error questions, first identify the population that was actually sampled (for example, registered voters in a specific state) and remember that any conclusion must be about that same population. Then take the reported sample percentage and add and subtract the margin of error to get a low and high endpoint for a plausible range. Finally, scan the choices for the one that uses both endpoints of this interval and refers to the correct population, avoiding options that claim an exact value, change the group (like using the whole nation instead of one state), or talk about just the people not sampled.
Hints
Focus on the group being studied
Look carefully at the description of the sample: who exactly did the polling organization survey? Any valid conclusion must be about that same type of people.
Use the margin of error correctly
You are told the sample proportion is 54% with a margin of error of percentage points. What two operations can you perform with 54 and 3 to get the smallest and largest likely percentages?
Check both the numbers and the population in each choice
After you find the low and high ends of the likely range, look for the answer that uses that whole range and applies it to the same group that was sampled, not to a different group or just a single percentage.
Desmos Guide
Compute the endpoints of the margin-of-error interval
In Desmos, type 54-3 on one line and 54+3 on another line. Use the two output values as the lower and upper bounds of the likely percentage range, then choose the answer that describes that entire interval for the same group that was sampled.
Step-by-step Explanation
Identify who the poll is about
The question says the polling organization took a simple random sample of 1,200 registered voters in a particular state.
That means any conclusion we draw from this poll should be about all registered voters in that same state, not about:
- all adults in the nation, or
- just the people not in the sample.
Understand what the margin of error means
The sample result is 54% support.
A margin of error of percentage points means the actual percentage in the whole population is likely to be no more than 3 percentage points away from 54%.
So the true value is likely somewhere between:
- 3 percentage points below 54%, and
- 3 percentage points above 54%.
Find the endpoints of the likely interval
To get the lowest likely value, subtract 3 from 54:
- Lower endpoint: compute .
To get the highest likely value, add 3 to 54:
- Upper endpoint: compute .
These two numbers form the interval that the poll suggests for the true percentage of support among all registered voters in the state.
Match the correct interval and population to an answer choice
The calculations give a likely interval from 51% to 57%.
This interval applies to all registered voters in the state, because that is the population the sample represents.
Looking at the answer choices, the only one that uses both this interval and the correct group (registered voters in the state) is:
D) Between 51% and 57% of all registered voters in the state support the proposed bond measure.