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
4.9
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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