Question
A new product is launched in two cities, Chillan and Talca, in the south of the country. Advertising in Chillan is based almost entirely on television, while in Talca it is spent on a balanced mix of television, radio and newspapers. After a month, a survey is carried out to determine the awareness of the product, and the following results are obtained: They squeal People surveyed 608 People with knowledge 392 Talca People surveyed 527 People with knowledge 413 Determine a 95% confidence interval for the regional difference for all consumers who are aware of the product.
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Answer to a math question A new product is launched in two cities, Chillan and Talca, in the south of the country. Advertising in Chillan is based almost entirely on television, while in Talca it is spent on a balanced mix of television, radio and newspapers. After a month, a survey is carried out to determine the awareness of the product, and the following results are obtained: They squeal People surveyed 608 People with knowledge 392 Talca People surveyed 527 People with knowledge 413 Determine a 95% confidence interval for the regional difference for all consumers who are aware of the product.
Murray
4.592 Answers
[SOLUTION]
\left( 0.550, 0.186 \right)
[STEP-BY-STEP]
Primero, identificamos las proporciones de personas con conocimiento del producto en cada ciudad.
Para Chillan:
\hat{p}_C = \frac{392}{608} = 0.6454
Para Talca:
\hat{p}_T = \frac{413}{527} = 0.7839
Luego, necesitamos la diferencia de proporciones:
\hat{p}_C - \hat{p}_T = 0.6454 - 0.7839 = -0.1385
Calculamos la desviación estándar de la diferencia de proporciones:
SE = \sqrt{\frac{\hat{p}_C(1 - \hat{p}_C)}{n_C} + \frac{\hat{p}_T(1 - \hat{p}_T)}{n_T}}
SE = \sqrt{\frac{0.6454(1 - 0.6454)}{608} + \frac{0.7839(1 - 0.7839)}{527}}
SE = \sqrt{ \frac{0.22906}{608} + \frac{0.16945}{527} }
SE = \sqrt{0.0003767 + 0.0003215}
SE = \sqrt{0.0006982} = 0.0264
Para un intervalo de confianza al 95%, utilizamos el valor crítico de \( Z \):
Z = 1.96
Calculamos el intervalo de confianza:
IC = (\hat{p}_C - \hat{p}_T) \pm Z \cdot SE
IC = -0.1385 \pm 1.96 \cdot 0.0264
IC = -0.1385 \pm 0.0518
Los límites del intervalo son:
Límite \ Inferior = -0.1385 - 0.0518 = -0.1903
Límite \ Superior = -0.1385 + 0.0518 = -0.0867
Por lo tanto, el intervalo de confianza al 95% para la diferencia regional de todos los consumidores que tienen conocimiento del producto es:
\left( -0.1903, -0.0867 \right)
[STEP-BY-STEP]
Primero, identificamos las proporciones de personas con conocimiento del producto en cada ciudad.
Para Chillan:
Para Talca:
Luego, necesitamos la diferencia de proporciones:
Calculamos la desviación estándar de la diferencia de proporciones:
Para un intervalo de confianza al 95%, utilizamos el valor crítico de \( Z \):
Calculamos el intervalo de confianza:
Los límites del intervalo son:
Por lo tanto, el intervalo de confianza al 95% para la diferencia regional de todos los consumidores que tienen conocimiento del producto es:
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