CAIE A-Level Mathematics 6.4.8 Proportion confidence intervals

Practise forming approximate confidence intervals for population proportions and using interval limits to interpret claims.

Exam points

calculate a proportion interval using p̂ ± z√(p̂(1−p̂)/n) from a large sample
set up equations from interval width or a limit to find z, p̂, or the confidence level
use whether a claimed proportion lies in the interval to comment on the claim

[Maximum number: 3]

In a survey, a random sample of 250 adults in Fromleigh were asked to fill in a questionnaire about their travel.

It was found that 102 adults in the sample travel by bus. Find an approximate 90% confidence interval for the proportion of all the adults in Fromleigh who travel by bus.

$\hat p=\frac{102}{250}=0.408, \qquad \frac{\hat p(1-\hat p)}{250}=\frac{102}{250}\frac{148}{250}\frac{1}{250}=0.000966144.$ $\hat p\pm z\sqrt{0.000966144},\qquad z=1.645.$
Confidence interval: 0.357 to 0.459 (3 s.f.).
M1 for correct form using a z-value; B1 for z=1.645; A1 for the interval.

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6.4.8—Proportion confidence intervals question 1