Question

10,000 people were asked about their taste for hot and iced coffee. Of the 10 thousand people, 4 thousand were women, of which 3 thousand like iced coffee, the rest like hot coffee. Of the men, 4 thousand like iced coffee, the rest like hot coffee. Use a 95% confidence level to establish a confidence interval for the proportion of the population of women who like hot coffee.

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Answer to a math question 10,000 people were asked about their taste for hot and iced coffee. Of the 10 thousand people, 4 thousand were women, of which 3 thousand like iced coffee, the rest like hot coffee. Of the men, 4 thousand like iced coffee, the rest like hot coffee. Use a 95% confidence level to establish a confidence interval for the proportion of the population of women who like hot coffee.

Expert avatar
Seamus
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94 Answers
Para establecer un intervalo de confianza para la proporción de mujeres que prefieren café caliente, podemos utilizar la fórmula del intervalo de confianza para una proporción:

\hat{p} \pm Z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

Donde:

- \hat{p} es la proporción muestral de mujeres que gustan de café caliente.
- Z es el valor crítico de la distribución normal estándar para el nivel de confianza del 95%, que es aproximadamente 1.96.
- n es el tamaño de la muestra, que en este caso es el total de mujeres encuestadas, es decir, 4000.

Primero, calculemos \hat{p} :
\hat{p} = \frac{\text{número de mujeres que gustan de café caliente}}{\text{total de mujeres encuestadas}} = \frac{1000}{4000} = 0.25

Ahora sustituimos los valores en la fórmula del intervalo de confianza:
\hat{p} \pm Z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = 0.25 \pm 1.96 \sqrt{\frac{0.25(0.75)}{4000}}

Calculamos el margen de error:
ME=1.96\sqrt{\frac{0.25(0.75)}{4000}}\approx0.0134

Por lo tanto, el intervalo de confianza del 95% para la proporción de mujeres que gustan de café caliente es:
0.25\pm0.0134

\textbf{Intervalo de Confianza:}(0.237,0.263)

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