CAIE S2 2022 June — Question 1 3 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2022
SessionJune
Marks3
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TopicLinear combinations of normal random variables
TypeNon-normal population sample mean (CLT)
DifficultyModerate -0.5 This is a straightforward application of the Central Limit Theorem with a normal population. Students need only recognize that the sample mean has distribution N(1250, 480²/100), then calculate a single probability using standardization. The population is already normal (making CLT unnecessary), and it's a direct one-step calculation with no conceptual subtlety.
Spec5.05a Sample mean distribution: central limit theorem
1 The number of characters in emails sent by a particular company is modelled by the distribution \(\mathrm { N } \left( 1250,480 ^ { 2 } \right)\). Find the probability that the mean number of characters in a random sample of 100 emails sent by the company is more than 1300 .
Question 1:
AnswerMarks Guidance
\[\frac{1300 + \frac{1}{200} - 1250}{\frac{480}{10}}\] or \[\frac{1300 - 1250}{\frac{480}{10}}\] \([= 1.042]\)M1 Allow with incorrect or omitted continuity correction. Must have 10. Accept totals method.
\(1 - \Phi(1.042)\)M1 For area consistent with *their* values
\(0.149\) (3 s.f.)A1 
**Question 1:**

$$\frac{1300 + \frac{1}{200} - 1250}{\frac{480}{10}}$$ or $$\frac{1300 - 1250}{\frac{480}{10}}$$ $[= 1.042]$ | **M1** | Allow with incorrect or omitted continuity correction. Must have 10. Accept totals method.

$1 - \Phi(1.042)$ | **M1** | For area consistent with *their* values

$0.149$ (3 s.f.) | **A1** |

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1 The number of characters in emails sent by a particular company is modelled by the distribution $\mathrm { N } \left( 1250,480 ^ { 2 } \right)$.

Find the probability that the mean number of characters in a random sample of 100 emails sent by the company is more than 1300 .\\

\hfill \mbox{\textit{CAIE S2 2022 Q1 [3]}}
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