AP Statistics 4.7 Constructing a Confidence Interval for the Difference Between Two Population Means- Exam Style Questions - FRQs - New Syllabus

(a)
First check conditions for a two-sample \(t\)-interval: the two groups of urban children were assigned at random and independently to play inside or outside, and dotplots of each group’s data show no strong skew or outliers, so it’s reasonable to treat the underlying populations as approximately normal.

Summary statistics for the urban sample:
\( n_{\text{in}}=9,\quad \bar{x}_{\text{in}}=4.56,\quad s_{\text{in}}=0.846 \)
\( n_{\text{out}}=9,\quad \bar{x}_{\text{out}}=17.56,\quad s_{\text{out}}=4.61 \)
The two-sample \(t\)-confidence interval formula is
\( (\bar{x}_{\text{in}}-\bar{x}_{\text{out}})\pm t^*\sqrt{\dfrac{s_{\text{in}}^2}{n_{\text{in}}}+\dfrac{s_{\text{out}}^2}{n_{\text{out}}}} \)
Using the conservative degrees of freedom \(df=\min(n_{\text{in}}-1,\,n_{\text{out}}-1)=8\), so \(t^*=2.306\):
\( (4.56-17.56)\pm 2.306\sqrt{\dfrac{(0.846)^2}{9}+\dfrac{(4.61)^2}{9}} \)
\( -13.00\pm 2.306(1.564) \)
\( -13.00\pm 3.61 \)
\( \boxed{(-16.60,\ -9.40)\text{ mcg}} \)
Interpretation: we are 95% confident that, for the population of urban day-care children, the mean amount of lead on the dominant hand after an hour of play inside is between 9.40 and 16.60 mcg lower than after an hour of play outside. Since this interval doesn’t contain zero, the difference is meaningful — urban children who play outside pick up noticeably more lead on their hands.

(b)

Plot the points \((\text{Suburban},3.75)\) and \((\text{Urban},4.56)\), connect them with a line labeled “inside.” Then plot \((\text{Suburban},5.65)\) and \((\text{Urban},17.56)\), connect them with a line labeled “outside.” The “inside” line should be nearly flat and low on the graph, while the “outside” line should rise sharply from suburban to urban.
\( \boxed{\text{Inside line: nearly flat, low values; Outside line: steep increase from suburban to urban}} \)

(c)
Setting (inside vs. outside): In both suburban and urban environments, children who played outside ended up with more lead on their hands than children who played inside. This is supported by the fact that all four endpoints of the two confidence intervals (inside minus outside) are negative, and the graph shows the “outside” line sitting above the “inside” line everywhere.
Environment (suburban vs. urban): For both inside and outside play, urban children had more lead on their hands on average than suburban children. The graph shows both lines sloping upward from suburban to urban.
Relationship between the two: The effect of going inside versus outside depends heavily on the environment. In the suburban setting, the inside and outside means are fairly close together (3.75 vs. 5.65), but in the urban setting the gap is much larger (4.56 vs. 17.56). In other words, playing outside makes a much bigger difference in lead exposure in the urban environment than in the suburban environment.
\( \boxed{\text{Outside > Inside in both settings; Urban > Suburban in both settings; the inside/outside gap is much larger for urban than suburban}} \)