CAIE S2 2024 June — Question 3 4 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2024
SessionJune
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicConfidence intervals
TypeFind alpha from CI width
DifficultyChallenging +1.2 This question requires working backwards from a confidence interval to find the confidence level, involving manipulation of the normal approximation formula for a proportion. While it tests understanding of confidence intervals beyond routine calculation, the algebraic steps are straightforward once the formula is set up correctly. It's moderately harder than average due to the reverse-engineering aspect, but remains a standard S2 technique.
Spec5.05d Confidence intervals: using normal distribution
3 A student wishes to estimate the proportion, \(p\), of students at her college who have exactly one brother. She surveys a random sample of 50 students at her college and finds that 18 of them have exactly one brother. She calculates an approximate \(\alpha \%\) confidence interval for \(p\) and finds that the lower limit of the confidence interval is 0.244 correct to 3 significant figures. Find \(\alpha\) correct to the nearest integer.
Question 3:
AnswerMarks Guidance
AnswerMark Guidance
\(\frac{18}{50} - z \times \sqrt{\frac{\frac{18}{50}\times(1-\frac{18}{50})}{50}} = 0.244\)M1 Use of correct equation
\(z = 1.709\) or \(1.708\)A1 Accept 1.71 if nothing better seen
\(\phi^{-1}(1.709) = 0.956\); \(1 - 2(1 - 0.956)\) \([= 0.912]\)M1 Attempt area above or below their 1.709 and use correct method to find \(\alpha\)
\(\alpha = 91\)A1 Allow \(\alpha = 91\%\). 0.91 or 91.2 score A0
Total: 4 
## Question 3:

| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{18}{50} - z \times \sqrt{\frac{\frac{18}{50}\times(1-\frac{18}{50})}{50}} = 0.244$ | M1 | Use of correct equation |
| $z = 1.709$ or $1.708$ | A1 | Accept 1.71 if nothing better seen |
| $\phi^{-1}(1.709) = 0.956$; $1 - 2(1 - 0.956)$ $[= 0.912]$ | M1 | Attempt area above or below their 1.709 and use correct method to find $\alpha$ |
| $\alpha = 91$ | A1 | Allow $\alpha = 91\%$. 0.91 or 91.2 score A0 |
| **Total: 4** | | |

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3 A student wishes to estimate the proportion, $p$, of students at her college who have exactly one brother. She surveys a random sample of 50 students at her college and finds that 18 of them have exactly one brother. She calculates an approximate $\alpha \%$ confidence interval for $p$ and finds that the lower limit of the confidence interval is 0.244 correct to 3 significant figures.

Find $\alpha$ correct to the nearest integer.\\

\hfill \mbox{\textit{CAIE S2 2024 Q3 [4]}}
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