CAIE S2 2020 March — Question 2 5 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2020
SessionMarch
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCentral limit theorem
TypeKnown variance confidence intervals
DifficultyStandard +0.3 This is a straightforward confidence interval question requiring knowledge of the formula for interval width (2z × σ/√n) and reverse lookup of z-values. The calculation is direct with no conceptual complications, making it slightly easier than average for S2 level.
Spec5.05d Confidence intervals: using normal distribution
2 Lengths of a certain species of lizard are known to be normally distributed with standard deviation 3.2 cm . A naturalist measures the lengths of a random sample of 100 lizards of this species and obtains an \(\alpha \%\) confidence interval for the population mean. He finds that the total width of this interval is 1.25 cm . Find \(\alpha\).
Question 2:
AnswerMarks Guidance
AnswerMark Guidance
\(2 \times z \times \frac{3.2}{10} = 1.25\)M1 OE Allow without '\(2\times\)'
\(z = 1.953\)A1 SOI
\(\phi(\text{their } 1.953) (= 0.9746)\)M1
\(= 1 - 2(1 - 0.9746)\) \(= 0.9492\)M1 OE
\(\alpha = 94.9\) or \(95\)A1 CWO
5 
## Question 2:

| Answer | Mark | Guidance |
|--------|------|----------|
| $2 \times z \times \frac{3.2}{10} = 1.25$ | **M1** | OE Allow without '$2\times$' |
| $z = 1.953$ | **A1** | SOI |
| $\phi(\text{their } 1.953) (= 0.9746)$ | **M1** | |
| $= 1 - 2(1 - 0.9746)$ $= 0.9492$ | **M1** | OE |
| $\alpha = 94.9$ or $95$ | **A1** | CWO |
| | **5** | |
2 Lengths of a certain species of lizard are known to be normally distributed with standard deviation 3.2 cm . A naturalist measures the lengths of a random sample of 100 lizards of this species and obtains an $\alpha \%$ confidence interval for the population mean. He finds that the total width of this interval is 1.25 cm .

Find $\alpha$.\\

\hfill \mbox{\textit{CAIE S2 2020 Q2 [5]}}
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