CAIE S2 2021 November — Question 3 5 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2021
SessionNovember
Marks5
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TopicConfidence intervals
TypeFind alpha from CI width
DifficultyStandard +0.3 This is a straightforward inverse confidence interval problem requiring students to work backwards from interval width to find the confidence level. It involves standard formulas for proportion confidence intervals and solving for the z-value, but the steps are routine and mechanical with no conceptual challenges beyond recalling the formula.
Spec5.05d Confidence intervals: using normal distribution
3 A random sample of 75 students at a large college was selected for a survey. 15 of these students said that they owned a car. From this result an approximate \(\alpha \%\) confidence interval for the proportion of all students at the college who own a car was calculated. The width of this interval was found to be 0.162 . Calculate the value of \(\alpha\) correct to 2 significant figures.
Question 3:
AnswerMarks Guidance
AnswerMarks Guidance
\(\text{est}(p) = 0.2\), accept \(\frac{15}{75}\)B1 SOI
\(2 \times z \times \sqrt{\frac{0.2 \times 0.8}{75}} = 0.162\)M1 Expression of the correct form. Condone missing \(2x\).
\(z \left[= 0.081 \times \sqrt{\frac{75}{0.2 \times 0.8}}\right] = 1.754\)A1 Correct \(z\). Condone 3sf accuracy.
\(\Phi(\text{'1.754'}) = 0.96[03]\); \(\text{'0.96'} - (1 - \text{'0.96'})\)M1 OE. Using *their* \(z\) to find alpha.
\(\alpha = 92\)A1 Following correct working.
5 
## Question 3:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{est}(p) = 0.2$, accept $\frac{15}{75}$ | B1 | SOI |
| $2 \times z \times \sqrt{\frac{0.2 \times 0.8}{75}} = 0.162$ | M1 | Expression of the correct form. Condone missing $2x$. |
| $z \left[= 0.081 \times \sqrt{\frac{75}{0.2 \times 0.8}}\right] = 1.754$ | A1 | Correct $z$. Condone 3sf accuracy. |
| $\Phi(\text{'1.754'}) = 0.96[03]$; $\text{'0.96'} - (1 - \text{'0.96'})$ | M1 | OE. Using *their* $z$ to find alpha. |
| $\alpha = 92$ | A1 | Following correct working. |
| | **5** | |

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3 A random sample of 75 students at a large college was selected for a survey. 15 of these students said that they owned a car. From this result an approximate $\alpha \%$ confidence interval for the proportion of all students at the college who own a car was calculated. The width of this interval was found to be 0.162 .

Calculate the value of $\alpha$ correct to 2 significant figures.\\

\hfill \mbox{\textit{CAIE S2 2021 Q3 [5]}}
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