<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/b2963c8c-cf34-4aa4-947d-879773bcb628/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls14 wsc"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls14 wsc">1 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls14 wsc"> </div></div><div class="t m0 x1 h4 y5 ff2 fs1 fc1 sc0 ls14 wsc">Estatística Aplicada - Resumo </div><div class="t m0 x1 h4 y6 ff2 fs1 fc1 sc0 ls14 wsc"> </div><div class="t m0 x1 h5 y7 ff3 fs2 fc0 sc0 ls14 wsc">DISTRIBUIÇÃO DA VARIÂNCIA </div><div class="t m0 x1 h5 y8 ff3 fs2 fc0 sc0 ls14 wsc">AMOSTRAL E DISTRIBUIÇÃO </div><div class="t m0 x1 h5 y9 ff3 fs2 fc0 sc0 ls14 wsc">AMOSTRAL DA PROPORÇÃO </div><div class="t m0 x1 h5 ya ff3 fs2 fc0 sc0 ls14 wsc">AMOSTRAL </div><div class="t m0 x1 h6 yb ff2 fs3 fc1 sc0 ls14 wsc"> </div><div class="t m0 x1 h7 yc ff3 fs4 fc1 sc0 ls14 wsc">Distribuição da variância amostral </div><div class="t m0 x1 h8 yd ff2 fs3 fc0 sc0 ls14 wsc">Seja <span class="_0 blank"> </span><span class="ff4">\ue73a</span></div><div class="t m0 x3 h9 ye ff4 fs5 fc0 sc0 ls0">\ueb35<span class="fs3 ls14 ws0 v1">, \u2026 \ue73a</span></div><div class="t m0 x4 ha ye ff4 fs5 fc0 sc0 ls1">\uebe1<span class="ff2 fs3 ls14 wsc v1"> <span class="_0 blank"> </span>a.a <span class="_0 blank"> </span>de tamanho <span class="_0 blank"> </span><span class="ff4 ls2">\ue74a</span> <span class="_0 blank"> </span>de <span class="_0 blank"> </span><span class="ff4 ws1">\ue73a<span class="_1 blank"> </span>\u223c<span class="_2 blank"> </span>\ue730 (\ue7e4 ,<span class="_3 blank"> </span>\ue7ea<span class="_4 blank"> </span></span></span><span class="ls3 v2">\ueb36</span><span class="fs3 ls14 ws2 v1">)<span class="ff2 wsc">. <span class="_0 blank"> </span>Considere <span class="_0 blank"> </span><span class="ff4 ls4">\ue735</span></span></span><span class="ls5 v2">\ueb36</span><span class="fs3 ls6 v1">=</span><span class="ls14 v3">\ueb35</span></div><div class="t m0 x5 hb yf ff4 fs5 fc0 sc0 ls14 ws3">\uebe1\ueb3f\ueb35 <span class="fs3 ws4 v4">\u2211 (<span class="v5">\ue73a</span></span></div><div class="t m0 x6 h9 ye ff4 fs5 fc0 sc0 ls7">\uebdc<span class="fs3 ls14 ws5 v1">\u2212 \ue73a</span></div><div class="t m0 x7 h8 y10 ff4 fs3 fc0 sc0 ls8">\ued24<span class="ls14 ws2 v6">)<span class="fs5 v7">\ueb36</span></span></div><div class="t m0 x8 hc y11 ff4 fs5 fc0 sc0 ls14">\uebe1</div><div class="t m0 x8 hc y12 ff4 fs5 fc0 sc0 ls14 ws6">\uebdc \ueb40\ueb35<span class="_5 blank"> </span><span class="ff2 fs3 wsc v8"> <span class="_0 blank"> </span>a </span></div><div class="t m0 x1 h6 y13 ff2 fs3 fc0 sc0 ls14 wsc">variância amostral. Então: </div><div class="t m0 x9 hd y14 ff4 fs3 fc0 sc0 ls14 ws7">\ue737 = <span class="ws5 v9">(\ue74a<span class="_2 blank"> </span>\u2212 1)</span></div><div class="t m0 xa he y15 ff4 fs3 fc0 sc0 ls9">\ue7ea<span class="fs5 lsa va">\ueb36</span><span class="ls4 vb">\ue735</span><span class="fs5 ls5 vc">\ueb36</span><span class="ls14 ws7 vb">\u223c \ue7ef<span class="_6 blank"></span><span class="fs5 ws8 vd">\uebe1\ueb3f\ueb35</span></span></div><div class="t m0 xb hc y16 ff4 fs5 fc0 sc0 lsb">\ueb36<span class="ff2 fs3 ls14 wsc ve"> </span></div><div class="t m0 x1 h6 y17 ff2 fs3 fc1 sc0 ls14 ws9">Observação<span class="fc0 wsc">: A população d<span class="_6 blank"></span>eve ter distribuição <span class="_6 blank"></span>normal para o resultado <span class="_6 blank"></span>ser verdadeiro. </span></div><div class="t m0 x1 h6 y18 ff2 fs3 fc0 sc0 ls14 wsc"> </div><div class="t m0 x1 h7 y19 ff3 fs4 fc1 sc0 ls14 wsc">Distribuição amostral da proporção amostral<span class="_0 blank"> </span> </div><div class="t m0 x1 h8 y1a ff2 fs3 fc0 sc0 ls14 wsc">Seja <span class="_6 blank"></span><span class="ff4 lsc">\ue74c<span class="ff2 ls14"> <span class="_7 blank"></span>a <span class="_7 blank"></span>proporção d<span class="_6 blank"></span>e <span class="_7 blank"></span>elementos <span class="_6 blank"></span>numa <span class="_7 blank"></span>população <span class="_6 blank"></span>com <span class="_7 blank"></span>uma <span class="_7 blank"></span>determinada <span class="_6 blank"></span>característica </span></span></div><div class="t m0 x1 h6 y1b ff2 fs3 fc0 sc0 ls14 wsc">de interesse. Define-se: </div><div class="t m0 xc hf y1c ff4 fs3 fc0 sc0 ls14 ws7">\ue73a<span class="_1 blank"> </span>= \ued5c<span class="wsc vf">1 <span class="ff2">, se o elemento tem a característica de interesse </span></span></div><div class="t m0 xd h8 y1d ff4 fs3 fc0 sc0 ls14 wsc">0 <span class="ff2">, caso contrário <span class="_8 blank"> </span><span class="v10"> </span></span></div><div class="t m0 x1 h8 y1e ff2 fs3 fc0 sc0 ls14 wsc">Sabe-se que <span class="ff4 wsa">\ue73a<span class="_1 blank"> </span>\u223c<span class="_2 blank"> </span>\ue724<span class="_6 blank"></span>\ue741\ue74e (\ue74c)<span class="ff2 wsc">. </span></span></div><div class="t m0 x1 h8 y1f ff2 fs3 fc0 sc0 ls14 wsc">Seja <span class="ff4">\ue73a</span></div><div class="t m0 x3 h9 y20 ff4 fs5 fc0 sc0 ls0">\ueb35<span class="fs3 ls14 ws0 v1">, \u2026 , \ue73a</span></div><div class="t m0 xe h9 y20 ff4 fs5 fc0 sc0 lsd">\uebe1<span class="ff2 fs3 ls14 wsc v1"> a.a de <span class="ff4 lse">\ue73a</span>. Define-se também: </span></div><div class="t m0 xf h8 y21 ff4 fs3 fc0 sc0 ls14 wsb">\ue73b<span class="_1 blank"> </span>=<span class="_2 blank"> </span>\uedcd \ue73a</div><div class="t m0 x10 hc y22 ff4 fs5 fc0 sc0 ls14">\uebdc</div><div class="t m0 x11 hc y23 ff4 fs5 fc0 sc0 ls14">\uebe1</div><div class="t m0 x12 hc y24 ff4 fs5 fc0 sc0 ls14 ws6">\uebdc \ueb40\ueb35</div><div class="t m0 x13 h6 y21 ff2 fs3 fc0 sc0 ls14 wsc"> </div><div class="t m0 x1 h8 y25 ff2 fs3 fc0 sc0 ls14 wsc">Como <span class="ff4">\ue73a</span></div><div class="t m0 x14 h9 y26 ff4 fs5 fc0 sc0 ls0">\ueb35<span class="fs3 ls14 ws0 v1">, \u2026 , \ue73a</span></div><div class="t m0 x15 hc y26 ff4 fs5 fc0 sc0 ls1">\uebe1<span class="ff2 fs3 ls14 wsc v1"> formam uma a.a (são independentes), então sabemos que <span class="_9 blank"> </span> </span></div><div class="t m0 x1 h10 y27 ff4 fs3 fc0 sc0 ls14 ws7">\ue73b<span class="_1 blank"> </span>\u223c \ue724\ue745<span class="_6 blank"></span>\ue74a<span class="_0 blank"></span><span class="ws2 v11">(</span><span class="ws0">\ue74a, \ue74c<span class="_0 blank"> </span><span class="ws2 v11">)</span><span class="ff2 wsc">. <span class="_4 blank"> </span></span><span class="lsf">\ue73b</span><span class="ff2 wsc"> <span class="_4 blank"> </span>é <span class="_4 blank"> </span>a <span class="_a blank"> </span>contagem <span class="_4 blank"> </span>de <span class="_4 blank"> </span>elementos <span class="_4 blank"> </span>na <span class="_4 blank"> </span>amostra <span class="_4 blank"> </span>que <span class="_4 blank"> </span>têm <span class="_4 blank"> </span>a <span class="_4 blank"> </span>c<span class="_0 blank"> </span>aracterística <span class="_4 blank"> </span>de </span></span></div><div class="t m0 x1 h6 y28 ff2 fs3 fc0 sc0 ls14 wsc">interesse. </div><div class="t m0 x1 h8 y29 ff2 fs3 fc0 sc0 ls14 wsc">Em seguida, definimos: <span class="_b blank"> </span><span class="ff4 ls10"> <span class="ls14 v12">\ue732</span></span></div><div class="t m0 x16 h11 y2a ff4 fs3 fc0 sc0 ls11">\uede0<span class="ls12 v13">=</span><span class="ls14 v4">\ue73b</span></div><div class="t m0 x17 h12 y2b ff4 fs3 fc0 sc0 ls13">\ue74a<span class="ls14 wsc vb"> </span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/b2963c8c-cf34-4aa4-947d-879773bcb628/bg2.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls14 wsc"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls14 wsc">2 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls14 wsc"> </div></div><div class="t m0 x1 h6 y2c ff2 fs3 fc0 sc0 ls14 wsc">Como sendo a proporção amostral. </div><div class="t m0 x18 h8 y2d ff5 fs3 fc0 sc0 ls14 wsd">\uf0b7<span class="ff6 ls15 wsc"> </span><span class="ff4 wse">\ue727 \ued63\ue732</span></div><div class="t m0 x19 h8 y2e ff4 fs3 fc0 sc0 ls16">\uede0<span class="ls14 wsf v13">\ued67<span class="_2 blank"> </span>=<span class="_2 blank"> </span>\ue727 \uf242<span class="fs5 v2">\uebd2</span></span></div><div class="t m0 x1a h13 y2f ff4 fs5 fc0 sc0 ls17">\uebe1<span class="fs3 ls14 ws10 v14">\uf243 =<span class="_1 blank"> </span></span><span class="ls14 v15">\ueb35</span></div><div class="t m0 x1b h13 y2f ff4 fs5 fc0 sc0 ls18">\uebe1<span class="fs3 ls19 v14">\ue727<span class="ls14 ws2 v11">[</span><span class="ls1a">\ue73b<span class="ls1b v11">]</span><span class="ls12">=</span></span></span><span class="ls14 ws8 v15">\uebe1\uebe3</span></div><div class="t m0 x1c h14 y2f ff4 fs5 fc0 sc0 ls1c">\uebe1<span class="fs3 ls14 ws7 v14">= \ue74c<span class="ff2 wsc">. </span></span></div><div class="t m0 x18 h8 y30 ff5 fs3 fc0 sc0 ls14 wsd">\uf0b7<span class="ff6 ls15 wsc"> </span><span class="ff4 wsa">\ue738\ue73d\ue74e \ued6b\ue732</span></div><div class="t m0 xe h8 y31 ff4 fs3 fc0 sc0 ls16">\uede0<span class="ls14 ws11 v13">\ued6f<span class="_2 blank"> </span>=<span class="_2 blank"> </span>\ue738\ue73d\ue74e \uf240<span class="fs5 v2">\uebd2</span></span></div><div class="t m0 x1d h13 y32 ff4 fs5 fc0 sc0 ls17">\uebe1<span class="fs3 ls14 ws10 v14">\uf241 =<span class="_c blank"> </span></span><span class="ls14 v15">\ueb35</span></div><div class="t m0 x1e h15 y32 ff4 fs5 fc0 sc0 ls17">\uebe1<span class="fs6 ls1d v1">\uec2e</span><span class="fs3 ls14 wsa v14">\ue738\ue73d\ue74e <span class="ws2 v11">(</span><span class="ls1a">\ue73b<span class="ls1b v11">)</span><span class="ls12">=</span></span></span><span class="ls14 wsc v15">\uebe1 \uebe3 </span><span class="ls1e v16">(</span><span class="ls14 ws8 v15">\ueb35\ueb3f\uebe3<span class="v11">)</span></span></div><div class="t m0 x1f h15 y32 ff4 fs5 fc0 sc0 ls17">\uebe1<span class="fs6 ls1f v1">\uec2e</span><span class="fs3 ls12 v14">=</span><span class="ls14 wsc v15">\uebe3 </span><span class="ls1e v16">(</span><span class="ls14 ws8 v15">\ueb35\ueb3f\uebe3<span class="v11">)</span></span></div><div class="t m0 x20 h16 y32 ff4 fs5 fc0 sc0 ls20">\uebe1<span class="ff2 fs3 ls14 wsc v14">. </span></div><div class="t m0 x1 h6 y33 ff2 fs3 fc1 sc0 ls14 wsc">Aplicação do TCL </div><div class="t m0 x1 h6 y34 ff2 fs3 fc0 sc0 ls14 wsc">Perceba que a proporção amostral é uma média amostral: </div><div class="t m0 x21 h8 y35 ff4 fs3 fc0 sc0 ls14">\ue732</div><div class="t m0 x22 h11 y36 ff4 fs3 fc0 sc0 ls11">\uede0<span class="ls12 v13">=</span><span class="ls14 v4">\ue73b</span></div><div class="t m0 x23 h17 y37 ff4 fs3 fc0 sc0 ls21">\ue74a<span class="ls22 vb">=</span><span class="ls14 v17">1</span></div><div class="t m0 x24 he y37 ff4 fs3 fc0 sc0 ls23">\ue74a<span class="ls14 wsb vb">\uedcd \ue73a</span></div><div class="t m0 x13 hc y38 ff4 fs5 fc0 sc0 ls14">\uebdc</div><div class="t m0 x25 hc y39 ff4 fs5 fc0 sc0 ls14">\uebe1</div><div class="t m0 x11 h18 y3a ff4 fs5 fc0 sc0 ls14 ws6">\uebdc \ueb40\ueb35<span class="_d blank"> </span><span class="fs3 ws7 v18">= \ue73a</span></div><div class="t m0 x26 h8 y36 ff4 fs3 fc0 sc0 ls8">\ued24<span class="ff2 ls14 wsc v13"> </span></div><div class="t m0 x1 h8 y3b ff2 fs3 fc0 sc0 ls14 wsc">Portanto, se <span class="ff4 ws12">\ue74a<span class="_2 blank"> </span>\u2192 \u221e</span> podemo<span class="_0 blank"> </span>s aplicar o TCL, e então: </div><div class="t m0 x27 hd y3c ff4 fs3 fc0 sc0 ls14 ws13">\ue73c =<span class="_e blank"> </span><span class="v9">\ue732</span></div><div class="t m0 xf h8 y3d ff4 fs3 fc0 sc0 ls24">\uede0<span class="ls14 ws5 v13">\u2212 \ue74c</span></div><div class="t m0 x1c hb y3e ff4 fs3 fc0 sc0 ls14 ws2">\ueda7<span class="ls25 v5">\ued6c</span><span class="ws14 v4">\ue74c(1 \u2212 \ue74c<span class="_0 blank"> </span>)</span></div><div class="t m0 x16 h19 y3f ff4 fs3 fc0 sc0 ls26">\ue74a<span class="ls27 v14">\ued70</span><span class="ls14 ws1 v19">\u2248<span class="_2 blank"> </span>\ue730 (0,1)<span class="ff2 wsc"> </span></span></div><div class="t m0 x1 h6 y40 ff2 fs3 fc0 sc0 ls14 wsc"> </div><div class="t m0 x28 h6 y41 ff2 fs3 fc0 sc0 ls14 wsc"> </div><div class="t m0 x1 h6 y42 ff2 fs3 fc0 sc0 ls14 wsc"> </div><div class="t m0 x1 h1a y43 ff2 fs0 fc0 sc0 ls14 wsc"> </div><div class="t m0 x1 h1a y44 ff2 fs0 fc0 sc0 ls14 wsc"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
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