Once you have both ranges, imagine them on a number line. Do the ranges partly cover the same section of the line, or are they completely separate?

Ask yourself: Does the sample information guarantee a conclusion, or does it only suggest what is plausible? Be careful with choices that sound 100% certain.

In Desmos, plot the endpoints of each interval as points: for survey X, enter (16.1, 0) and (18.5, 0); for survey Y, enter (16.2, 1) and (21.2, 1). This lets you see where each survey’s plausible range begins and ends along the x-axis.

You can type segment((16.1,0),(18.5,0)) for survey X and segment((16.2,1),(21.2,1)) for survey Y. Then look at the x-axis to see whether the two horizontal segments cover any of the same x-values, and use that visual overlap to reason about the answer choices.

The mean and margin of error give a range (confidence interval) of plausible values for the true mean.

• For survey X: mean $17.3$, margin of error $1.2$.

  • Lower bound: $17.3 - 1.2 = 16.1$.
  • Upper bound: $17.3 + 1.2 = 18.5$.
  • So X suggests the true mean is between $16.1$ and $18.5$ hours.

• For survey Y: mean $18.7$, margin of error $2.5$.

  • Lower bound: $18.7 - 2.5 = 16.2$.
  • Upper bound: $18.7 + 2.5 = 21.2$.
  • So Y suggests the true mean is between $16.2$ and $21.2$ hours.

Each interval is a likely range for the true mean at the $95\%$ confidence level, not a guaranteed range.

Write the intervals clearly:

• Survey X: $[16.1, 18.5]$
• Survey Y: $[16.2, 21.2]$

Look for overlap:

• The overlap is from $16.2$ to $18.5$ (numbers that are in both intervals).

This means there are values that are consistent with both surveys’ data at the $95\%$ confidence level.

Use what we know:

• Each survey gives a likely range for the same district’s true mean screen time.
• The ranges overlap.
• A $95\%$ confidence level does not mean the true mean is definitely in the interval; it just means the method usually captures the true value.

Now check each statement:

• A) “The true mean weekly screen time for the district is between 16.1 and 21.2 hours.”
  This claims certainty (“is between”) based only on sample data. We cannot be sure; the true mean could lie outside these combined bounds, even though it is unlikely.

• B) “The true mean weekly screen time for the population represented by survey Y is greater than that for survey X.”
  The intervals overlap, so there are plausible values where the true means are equal or even where X’s is bigger. The data do not prove that Y’s true mean is greater.

• D) “Survey Y must have been based on a larger sample size than survey X.”
  A larger margin of error (2.5 vs 1.2) usually suggests more uncertainty, which tends to come from a smaller sample, not a larger one. Also, we don’t have enough information to say anything “must” be true about the sample sizes.

• C) “It is possible that the two surveys estimate the same true mean weekly screen time.”
  Because the confidence intervals overlap, there are values that work for both surveys. This makes it completely reasonable that both surveys are estimating the same true mean.

Therefore, the best-supported statement is C) It is possible that the two surveys estimate the same true mean weekly screen time.