CAIE S2 (Statistics 2) 2020 Specimen

3 Th m brg calls receid d at a small call cen re \(\mathbf { h }\) s a Pósso d strib in with mean

4 Tb lifetimes, in b s, 6 L ie lig \(\mathbf { b }\) b ad Ee rlw lig \(\mathbf { b }\) b \(\mathbf { b }\) \& tb id \(\mathbf { P } \mathbf { d }\) n id strib in \(\mathrm { N } \left( \mathrm { LS } ^ { 2 } \right)\) adN ( \(\mathrm { L } ^ { 2 }\) ) resp ctie ly.

1. Fid th pb b lity th t to to al 6 th lifetimes 6 fie rach ly cb en \(L \mathbf { b }\) ie \(\mathbf { b }\) b is less th \(\mathrm { HB } \quad \mathrm { Ch } \quad \mathrm { S }\).
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2. Fid th pb b lity th tth lifetime 6 a rach lycb en En rlw b b is at least th ee times th t 6 a rach lyc b erL b ir b b

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\includegraphics[max width=\textwidth, alt={}, center]{ffc7febd-0df7-4cb6-ac6c-63779e032617-08_270_648_251_712} Th diag am sh s th g a to th pb ab lityd nsityf n tiff, to a rach \& riab e \(X\),w b re $$f ( x ) = \begin{cases} \frac { 2 } { 9 } \left( 3 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$

1. State th \& le \(6 \mathrm { E } ( X )\) aff id \(\operatorname { Var } ( X )\).
2. State th le \(6 \mathrm { P } (
   ) \leqslant X \leqslant 4\(.
3. Giv it h \)\mathrm { P } \left( 1 \leqslant X \leqslant \mathcal { P } = \frac { 13 } { 27 } \right.\(, f idP \)( X > \mathcal { P }$.

7 Th mean weit 6 bg 6 carrb s is \(\mu \mathrm { k }\) lg ams. An in p cto wish s to test wh th \(\mathrm { r } \mu = 20\) He weits a rand sampe 6 tb g ach s resh ts are sm marised s fb low s. $$\Sigma x = 430 \quad \Sigma x ^ { 2 } = \theta$$ Carryd th test at the \%o sig fican e lee 1. If B e th follw ig lin dpg to cm p ete th an wer(s) to ay q stin (s), th q stin \(\mathrm { m } \quad \mathbf { b } \quad \mathrm { r } ( \mathrm { s } )\) ms tb clearlys n n