CAIE S2 2023 November — Question 2 4 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2023
SessionNovember
Marks4
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Mark schemeDownload PDF ↗
TopicConfidence intervals
TypeFind alpha from CI width
DifficultyStandard +0.3 This is a straightforward confidence interval question requiring students to work backwards from the upper bound to find the confidence level. It involves standard formulas for proportion confidence intervals and solving for the z-value, then converting to a percentage. The calculation is routine with no conceptual challenges beyond knowing the confidence interval formula for proportions.
Spec5.05d Confidence intervals: using normal distribution
2 In a survey of 300 randomly chosen adults in Rickton, 134 said that they exercised regularly. This information was used to calculate an \(\alpha \%\) confidence interval for the proportion of adults in Rickton who exercise regularly. The upper bound of the confidence interval was found to be 0.487 , correct to 3 significant figures. Find the value of \(\alpha\) correct to the nearest integer.
Question 2:
AnswerMarks Guidance
AnswerMarks Guidance
\(\dfrac{134}{300} + z\sqrt{\dfrac{\frac{134}{300} \times \frac{166}{300}}{300}} = 0.487\)M1 For expression of the correct form.
\(z = 1.405\)A1 Accept 1.404, or anything that rounds to 1.39 to 1.41.
\(\phi^{-1}(1.405) = 0.9199\) or \(0.92\); \(1 - 2(1-0.92)\)M1 Attempt area above or below their 1.405 and convert to a confidence level.
\(\alpha = 84\)A1 Allow \(\alpha = 84\%\). CWO. Note: final answer 0.84 scores A0.
4 
**Question 2:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{134}{300} + z\sqrt{\dfrac{\frac{134}{300} \times \frac{166}{300}}{300}} = 0.487$ | M1 | For expression of the correct form. |
| $z = 1.405$ | A1 | Accept 1.404, or anything that rounds to 1.39 to 1.41. |
| $\phi^{-1}(1.405) = 0.9199$ or $0.92$; $1 - 2(1-0.92)$ | M1 | Attempt area above or below their 1.405 and convert to a confidence level. |
| $\alpha = 84$ | A1 | Allow $\alpha = 84\%$. CWO. Note: final answer 0.84 scores A0. |
| | **4** | |
2 In a survey of 300 randomly chosen adults in Rickton, 134 said that they exercised regularly. This information was used to calculate an $\alpha \%$ confidence interval for the proportion of adults in Rickton who exercise regularly. The upper bound of the confidence interval was found to be 0.487 , correct to 3 significant figures.

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

\hfill \mbox{\textit{CAIE S2 2023 Q2 [4]}}
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