Uma indústria deseja avaliar se o detergente líquido neutro e o limão são igualmente preferidos pelos consumidores. De 250 consumidores entrevistados, 145 expressaram a sua preferência pelo detergente líquido neutro.

Passo 1

Primeiro, vamos entender o que a questão está pedindo. Temos uma indústria que quer saber se os consumidores preferem igualmente o detergente líquido neutro e o limão. Eles entrevistaram 250 pessoas, e 145 delas preferiram o detergente líquido neutro. Vamos usar estatística de proporções para ver se essa preferência é significativa ou se pode ser apenas por acaso.

Passo 2

Vamos estabelecer as hipóteses para o nosso teste:
  • Hipótese nula (H₀): A proporção de preferência pelo detergente é 50%, ou seja, os consumidores não têm preferência (p = 0,5).
  • Hipótese alternativa (H₁): A proporção de preferência pelo detergente é diferente de 50% (p ≠ 0,5).

Passo 3

Calculamos a proporção observada (p̂) de pessoas que preferiram o detergente:

Passo 4

Vamos calcular o erro padrão (EP) da proporção sob a hipótese nula:

Passo 5

Agora, calculamos o estatístico de teste (z):

Passo 6

Consultando uma tabela de distribuição normal ou usando uma calculadora, encontramos o valor-p para . Como o teste é bicaudal (estamos verificando se a proporção é diferente, tanto para cima quanto para baixo), multiplicamos a probabilidade de uma cauda por 2:

Passo 7

Comparando o valor-p com o nível de significância comum de 5% (0,05):
Como , rejeitamos a hipótese nula.

Resposta

Há evidências estatísticas para concluir que a preferência pelos consumidores entre detergente líquido neutro e limão não é igual, ou seja, o detergente líquido neutro é preferido significativamente.

Exercícios relacionados

Seja uma função definida pela equação . Calcule a derivada .
Seja . Calcule .
<html><head></head><body><div>(2.0pt) Utilizando os 5 passos dados em aula esboce o gráfico da seguinte função <span class="math-inline ">
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http://www.w3.org/2000/svg” width=“17.764ex” height=“2.452ex” role=“img” focusable=“false” viewBox=“0 -833.9 7851.8 1083.9”><title>h(x, y) = 4 x^2 + y^2</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“mi”><path data-c=“210E” d=“M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z”></path></g><g data-mml-node=“mo” transform=“translate(576,0)”><path data-c=“28” d=“M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 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<html><head></head><body><div>(2.0pt) Determine os valores máximos e mínimos locais e pontos de sela da função <span class="math-inline ">
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