A margin of error of 3 percentage points around 54% creates a range of likely values. What is that range?

Does the margin of error let you say the true proportion is exactly a certain value, or that some values are impossible? Or does it tell you which values would be more or less expected based on the sample?

In Desmos, type 680/1250 on a line. Note the decimal result (about $0.544$), and recognize that this corresponds to about 54.4%, which matches the article’s "about 54%" estimate.

On separate lines in Desmos, type 0.54 - 0.03 and 0.54 + 0.03. The outputs give the lower and upper bounds (in decimal form) of the likely interval for the true proportion; convert them to percentages to see the corresponding percent range.

Type 0.59 on another line to represent 59%. Compare this value to the two bounds you found in the previous step. Decide whether 59% falls inside or outside that range, and then look back at the answer choices for the one that correctly describes how a value like this relates to the margin-of-error interval.

The survey found that 680 out of 1,250 registered voters in the sample support the policy.

Compute the sample proportion:

This matches the article’s statement that about 54% of registered voters support the policy.

The article gives a margin of error of 3 percentage points around the reported 54%.

That means the likely range for the true proportion is:

• Lower end: $54\% - 3\% = 51\%$
• Upper end: $54\% + 3\% = 57\%$

So the survey suggests the actual proportion of all registered voters who support the policy is likely between 51% and 57%.

A margin of error does not tell us the exact true percentage. Instead, it tells us a range of values that are consistent with the sample results.

Key ideas:

• We cannot say the true proportion is exactly $54\%$.
• We cannot say values below 51% or above 57% are impossible, only that they would be unlikely or unexpected based on this survey.
• The margin of error says nothing about people lying or changing their answers; it only describes sampling variability.

From the margin-of-error range (51% to 57%) and its meaning:

• Any statement that claims an exact percentage for all voters is too strong.
• Any statement that says some values are impossible (like “cannot be less than 51%”) also goes beyond what the margin of error allows.
• A statement that talks about some values being unexpected or unlikely when they are outside this 51%–57% range is the kind of claim that is supported by the idea of margin of error.

Now compare the choices:

• One choice claims an exact 54%—too strong.
• Another says the proportion cannot be less than 51%—also too strong.
• Another misinterprets the 3% as about people answering incorrectly.
• The remaining choice says it would be unexpected if the actual proportion were 59%, which is 5 percentage points above 54% and outside the 51%–57% margin-of-error range.

Therefore, the best-supported statement is: It would be unexpected if the actual proportion of all registered voters who support the policy were 59%.