Question 3 - A-Level Maths

CAIE S2 2017 November — Question 3 4 marks

Exam Board	CAIE
Module	S2 (Statistics 2)
Year	2017
Session	November
Marks	4
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Confidence intervals
Type	Find CI width or confidence level
Difficulty	Standard +0.3 This is a straightforward reverse confidence interval problem requiring students to work backwards from the given interval to find the confidence level. It involves standard formulas for proportion confidence intervals and solving for the z-value, then looking up the corresponding confidence level. While it requires careful algebraic manipulation and understanding of the confidence interval structure, it's a routine application of A-level statistics techniques with no novel insight required.
Spec	5.05c Hypothesis test: normal distribution for population mean

3 After an election 153 adults, from a random sample of 200 adults, said that they had voted. Using this information, an \(\alpha \%\) confidence interval for the proportion of all adults who voted in the election was found to be 0.695 to 0.835 , both correct to 3 significant figures. Find the value of \(\alpha\), correct to the nearest integer.

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Question 3:

Answer	Marks	Guidance
Answer	Mark	Guidance
\(\frac{153}{200} + z \times \sqrt{\frac{\frac{153}{200} \times \frac{200-153}{200}}{200}} = 0.835\) \((Var(P_s) = 0.000898875)\) (s.d. \(0.02998\))	M1
\(z = 2.335\)	A1	allow \(2.33\) or \(2.34\)
\(2\Phi(z) - 1\)	M1	or equivalent method, indep
\(\alpha = 98\)	A1	allow \(98.0\) but not e.g. \(98.04\)
Total: 4

## Question 3:

| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{153}{200} + z \times \sqrt{\frac{\frac{153}{200} \times \frac{200-153}{200}}{200}} = 0.835$ $(Var(P_s) = 0.000898875)$ (s.d. $0.02998$) | M1 | |
| $z = 2.335$ | A1 | allow $2.33$ or $2.34$ |
| $2\Phi(z) - 1$ | M1 | or equivalent method, indep |
| $\alpha = 98$ | A1 | allow $98.0$ but not e.g. $98.04$ |
| **Total: 4** | | |

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3 After an election 153 adults, from a random sample of 200 adults, said that they had voted. Using this information, an $\alpha \%$ confidence interval for the proportion of all adults who voted in the election was found to be 0.695 to 0.835 , both correct to 3 significant figures. Find the value of $\alpha$, correct to the nearest integer.\\

\hfill \mbox{\textit{CAIE S2 2017 Q3 [4]}}

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