Question 1 - A-Level Maths

CAIE Further Paper 4 2023 June — Question 1 4 marks

Exam Board	CAIE
Module	Further Paper 4 (Further Paper 4)
Year	2023
Session	June
Marks	4
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	T-tests (unknown variance)
Type	Single sample confidence interval t-distribution
Difficulty	Standard +0.3 This is a straightforward application of the t-distribution confidence interval formula with given summary statistics. Students need to calculate sample mean and standard deviation from the summaries, then apply the standard formula with t₁₁ critical value. It's slightly above average difficulty due to being Further Maths content and requiring careful calculation, but involves no conceptual challenges or novel problem-solving.
Spec	5.05d Confidence intervals: using normal distribution

1 The lengths of the leaves of a particular type of tree are normally distributed with mean $\mu \mathrm { cm }$. The lengths, $x \mathrm {~cm}$, of a random sample of 12 leaves of this type are recorded. The results are summarised as follows. $$\sum x = 91.2 \quad \sum x ^ { 2 } = 695.8$$ Find a 95\% confidence interval for $\mu$.

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Question 1:

Answer	Marks	Guidance
Answer	Marks	Guidance
$\bar{x} = \dfrac{91.2}{12} [= 7.6]$ and $s_x^2 = \dfrac{1}{11}\left(695.8 - \dfrac{91.2^2}{12}\right) \left[= 0.2436 \left(\dfrac{67}{275}\right)\right]$	M1	Both.
CI: $7.6 \pm t_{11}(0.975) \times \sqrt{\dfrac{0.2436}{12}}$	M1	With a $t$-value.
$2.201$ seen	B1
$7.6 \pm 0.3136 = [7.29, 7.91]$	A1
4

## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\bar{x} = \dfrac{91.2}{12} [= 7.6]$ and $s_x^2 = \dfrac{1}{11}\left(695.8 - \dfrac{91.2^2}{12}\right) \left[= 0.2436 \left(\dfrac{67}{275}\right)\right]$ | M1 | Both. |
| CI: $7.6 \pm t_{11}(0.975) \times \sqrt{\dfrac{0.2436}{12}}$ | M1 | With a $t$-value. |
| $2.201$ seen | B1 | |
| $7.6 \pm 0.3136 = [7.29, 7.91]$ | A1 | |
| | **4** | |

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1 The lengths of the leaves of a particular type of tree are normally distributed with mean $\mu \mathrm { cm }$. The lengths, $x \mathrm {~cm}$, of a random sample of 12 leaves of this type are recorded. The results are summarised as follows.

$$\sum x = 91.2 \quad \sum x ^ { 2 } = 695.8$$

Find a 95\% confidence interval for $\mu$.\\

\hfill \mbox{\textit{CAIE Further Paper 4 2023 Q1 [4]}}

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Q1 4 Q2 8 Q3 9 Q4 9 Q5 9 Q6 11