Once you have both ranges, look carefully to see whether they overlap or are completely separate. Think about what overlap means for the possible true values.

For each answer choice, ask yourself: "Does this statement have to be true for all possible values inside those intervals, or is it going beyond what the intervals guarantee?" Be especially careful with words like "at least" and specific difference ranges like "between 1% and 10%."

A larger margin of error makes an estimate less precise, but does that automatically mean the poll is unreliable? Consider what information you do and do not have about how the polls were conducted.

In Desmos, type 45-3 on one line and 45+3 on another. Note the two outputs; they are the lower and upper bounds of City X’s likely range of support (in percent).

Type 41-4 on a new line and 41+4 on the next line. These two outputs are the lower and upper bounds of City Y’s likely range of support (in percent).

Look at the two pairs of numbers from Desmos and think: Do the intervals for City X and City Y overlap, or are they completely separate? Use that comparison to decide which statement about the possible true percents in the two cities is safest and fully supported by the data.

The problem says the table gives sample percents and margins of error for 95% confidence intervals.

That means for each city, the true population percent of supporters is likely to be in the interval:

• (sample percent) ± (margin of error).

For City X:

• Sample percent: $45\%$
• Margin of error: $3\%$

So the 95% confidence interval for City X is

The true percent of supporters in City X is likely between $42\%$ and $48\%$.

For City Y:

• Sample percent: $41\%$
• Margin of error: $4\%$

So the 95% confidence interval for City Y is

The true percent of supporters in City Y is likely between $37\%$ and $45\%$.

Now compare the intervals:

• City X: $42\%$ to $48\%$
• City Y: $37\%$ to $45\%$

These intervals overlap between $42\%$ and $45\%$. That means:

• City X might be higher than City Y,
• City Y might be higher than City X, or
• They might be the same (for example, both could be $45\%$).

Because all of these possibilities are consistent with the data and the margins of error, we cannot conclude from the table that support in City X is definitely higher or that the difference is at least a certain amount. The only justified conclusion is that it is possible the true percents of supporters in the two cities are the same in the populations.