Standard +0.3 This is a straightforward application of the confidence interval formula for proportions. Students must recall the formula for width (2z*√(p(1-p)/n)), substitute known values (p=0.35, z=2.326 for 98%), and solve algebraically for n. It's slightly above average difficulty due to the algebraic manipulation required and the need to recall the correct z-value, but remains a standard textbook exercise with no novel insight needed.
35% of a random sample of \(n\) students walk to college. This result is used to construct an approximate 98% confidence interval for the population proportion of students who walk to college. Given that the width of this confidence interval is 0.157, correct to 3 significant figures, find \(n\). [5]
Rearrange to form \(n = \ldots\) from a correct equation in \(n\), but allow any \(z\) and/or factor of "2" errors
\(n = 200\)
A1
[5] cao
$2 \times z \times \sqrt{\frac{0.35 \times 0.65}{n}} = 0.157$ | M1, M1 | For $\sqrt{p(q/n)}$ in equation; For equation of the form $2 \times z \times f(n) = 0.157$
$z = 2.326$ | B1 |
$n = 4 \times 2.326^2 \times 0.35 \times 0.65 + 0.157^2 \quad (=199.738\ldots)$ | M1 | Rearrange to form $n = \ldots$ from a correct equation in $n$, but allow any $z$ and/or factor of "2" errors
$n = 200$ | A1 | [5] cao
35% of a random sample of $n$ students walk to college. This result is used to construct an approximate 98% confidence interval for the population proportion of students who walk to college. Given that the width of this confidence interval is 0.157, correct to 3 significant figures, find $n$. [5]
\hfill \mbox{\textit{CAIE S2 2011 Q2 [5]}}