Question 1 - A-Level Maths

CAIE S2 2023 November — Question 1 3 marks

Exam Board	CAIE
Module	S2 (Statistics 2)
Year	2023
Session	November
Marks	3
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Linear combinations of normal random variables
Type	Single normal population sample mean
Difficulty	Moderate -0.5 This is a straightforward application of the sampling distribution of the mean with a normal population. Students need to recognize that the sample mean follows N(410, 400/36), then calculate a single z-score and look up a probability. It requires only standard recall and one calculation step, making it easier than average but not trivial since it tests understanding of the variance of sample means.
Spec	5.05a Sample mean distribution: central limit theorem

1 A random variable \(X\) has the distribution \(\mathrm { N } ( 410,400 )\).
Find the probability that the mean of a random sample of 36 values of \(X\) is less than 405 .

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Question 1:

Answer	Marks	Guidance
Answer	Marks	Guidance
\(\dfrac{405-410}{\frac{20}{\sqrt{36}}} = [-1.5]\)	M1	For standardising, must have \(\sqrt{36}\). Allow totals method \(\dfrac{14580-14760}{\sqrt{14400}}\). No mixed methods.
\(\phi(-1.5) = 1 - \phi(1.5)\)	M1	For area consistent with their working.
\(= 0.0668\)	A1
3

**Question 1:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{405-410}{\frac{20}{\sqrt{36}}} = [-1.5]$ | M1 | For standardising, must have $\sqrt{36}$. Allow totals method $\dfrac{14580-14760}{\sqrt{14400}}$. No mixed methods. |
| $\phi(-1.5) = 1 - \phi(1.5)$ | M1 | For area consistent with their working. |
| $= 0.0668$ | A1 | |
| | **3** | |

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1 A random variable $X$ has the distribution $\mathrm { N } ( 410,400 )$.\\
Find the probability that the mean of a random sample of 36 values of $X$ is less than 405 .\\

\hfill \mbox{\textit{CAIE S2 2023 Q1 [3]}}

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Q1 3 Q2 4 Q3 10 Q4 8 Q5 5 Q6 8 Q7 12