CAIE Further Paper 4 (Further Paper 4) 2020 Specimen

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1. State b iefly tb circm stan es d r wh cha a \(\quad\) p rametric test \(\mathbf { 6 }\) sig fican e sh lid be \(\mathbf { b }\) ed rath r thara \(\mathbf { P }\) rametric test. Th le l6 p ltu in ira rie r was measu ed at \(\mathbb { r }\) ach lyc \(\mathbf { b }\) serl o atin . Tb resli ts, iı ù tab e i ts,a re sh wib lo , w b reh g r le s rep esen g eater p ltu in B 56 B 158 58 日 9
2. Use a Wilc sig d rah test to test wh th r th a rag p ltu in le l in th rie r is mo e th iU se a \(\%\) sig fican e le 1 .
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2 Each \(\mathbf { 6 }\) Q id \(n\) ically \(\mathbf { b }\) ased d ce is th \(n\) rep ated \(\mathrm { y } n\) il an eq \(n\) b \(\mathbf { r }\) is \(\mathbf { b }\) ain d Th \(\mathbf { m }\) b \(\mathbf { r }\) 6 th se ed d s reco d d d b resu ts are sm marised it b fb low ig ab e. Carry \(\mathbf { a }\) a ss \(\mathbf { 6 }\) fit test, at th \(\% _ { 0 }\) sig fican e lev l, to test wh th r Ge( Đ is a satisfacto y md lfo th d ta.
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3 Empø es at ap rtich ar comp y \(\mathbf { h }\) \textbackslash ＆ b en wo kg seqnb s each \(\mathbf { d y }\) ，frm 9 am to 4 p ．To try to red e ab en e，th cm \(\mathbf { p } \mathrm { y }\) d cid s to in rd e ＇flex－time＇ad all emp 甲 es to wo k th ir see nb s each d y at ay time b tween 7 am ad 9 p ．Fo a rach sampe \(\mathbf { 6 } \mathbf { 0 }\) emp \(\boldsymbol { \varphi }\) es，th m b rs 6 b s of ab en e in th \(y\) arb fo e ad th \(y\) ar after th in rd tin 6 flex－time are g t rit t fb low ig ab e． Test，at th \textbackslash ％sig ficance le l，wh th r th \(\mathbf { p }\) atin mean m br \(\mathbf { 6 }\) b s \(\mathbf { 6 }\) ab en e \(\mathbf { s }\) d creased b lw ig b in rd tim flex－time，s tatig pr ssm p it b tm ak．［ア

4 Th m b r, \(x , 6\) a certain ty sea sh ll was co ed at \(\theta\) rach ly cb en sites, each \(\mathbf { a }\) metre seq re, alg th co stlie in co ry \(A\). Th \(m \quad \mathbf { b } , y , \boldsymbol { 6 }\) th same \(\quad \mathbf { 6 }\) sea sh ll was co ed at \(\theta\) rach ly cb en sites, each \(\mathbf {} { } _ { \text {t } }\) metre sq re, alog th co stlie in co ry \(B\). Tb results are sm marised as fb lw s,w b re \(\bar { x }\) ad \(\bar { y } \mathbf { d } \mathbf { h }\) e th samp e meas \(\mathbf { 6 } x\) ad \(y\) resp ctiv ly. $$\bar { x } = 9 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = \mathbf { 3 } \quad \bar { y } = \mathbf { 4 } \quad \Sigma ( y - \bar { y } ) ^ { 2 } = \text { 日 }$$ \includegraphics[max width=\textwidth, alt={}]{0df58f9d-6700-46cc-bcf0-903e94cccc02-06_58_1667_539_239} metre,it b co stlin sirc \(\mathbf { b }\) ry \(A\) ad inc \(\mathbf { b }\) ry \(B\).

5 Th co in rach \& riab e \(X \mathbf { h }\) s prb b lity e \(\mathbf { s }\) ityf \(\mathbf { n }\) tiff \(\dot { \mathrm { g } }\) \& $$f ( x ) = \begin{cases} 0 & x < 0
\frac { 6 } { 5 } x & 0 \leqslant x \leqslant 1
\frac { 6 } { 5 } x ^ { - 4 } & x > 1 \end{cases}$$

1. FidP \(( X > 1\).
2. Fid b med arm le \(6 X\).
3. Gie it \(\mathbf { h } \mathrm { t } \mathrm { E } ( X ) =\), fif id \(\mathbf { b }\) riance \(6 X\).
4. Fide \(( \sqrt { X } )\).

6 Aish h sab g co ainig 3 red b lls ad 3 wh te b lls. Sb selects a b ll at rach , b es its cb o ad return it to th b g th same p o ess is rep ated twice mo e. Tb m brd red b lls selected b Aish is d no edy \(X\).

1. Fid \(\mathbf { b } \mathbf { p b }\) b lityg \(\mathbf { a }\) ratig \(\mathbf { a }\) ting \({ } _ { X } ( t ) \boldsymbol { 6 } X\). Basan also s sab g co ain g 3 red balls ad 3 wh te b lls. He selects th ee b lls at rach , with rep acemen, frm hsbg Th m brg red lls selectedB asan is d n edy \(Y\).
2. Fid \(\mathbf { b } \mathbf { p } \mathbf { b }\) b litys \(\mathbf { a }\) ratif \(\mathbf { a }\) ting \({ } _ { Y } ( t ) \underset { \text { b } } { } Y\). Th rad \(m\) riab e \(Z\) is to to alm brø reb lls selected y Aish adB asan.
3. Fid \(\mathbf { b } \mathbf { p } \mathbf { b }\) b lityg \(\mathbf { e }\) ratig \(\mathbf { u }\) tim \(Z\), essig as wer as \(\mathrm { p } \mathbf { p }\) ial. [β
4. Use th p b b lityg \(\mathbf { B }\) ratig u tim \(Z\) tof idE ( \(Z\) ) ad \(\operatorname { Var } ( Z )\). [\$ 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