Question

An electrical company manufactures batteries that have a duration that is distributed approximately normally, with a mean of 700 hours and a standard deviation of 40 hours. Find the probability that a randomly selected battery has an average life of less than 810 hours.

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Answer to a math question An electrical company manufactures batteries that have a duration that is distributed approximately normally, with a mean of 700 hours and a standard deviation of 40 hours. Find the probability that a randomly selected battery has an average life of less than 810 hours.

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Birdie
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Para encontrar la probabilidad de que una batería seleccionada al azar tenga una vida útil promedio de menos de 810 horas, usaremos las propiedades de la distribución normal. Dado: Media (\( \mu \)) = 700 horas Desviación estándar (\( \sigma \)) = 40 horas Queremos encontrar \( P(X < 810) \), donde \( X \) representa la vida útil promedio de una batería. Primero, estandaricemos el valor 810 usando la fórmula de puntuación z: \[ Z = \frac \] Dónde: - \( X \) es el valor que nos interesa (810 horas). - \( \mu \) es la media (700 horas). - \( \sigma \) es la desviación estándar (40 horas). \[ Z = \frac \] \[ Z = \frac \] \[ Z = 2,75 \] Ahora que tenemos el puntaje z, usaremos una tabla de distribución normal estándar o una calculadora para encontrar la probabilidad correspondiente a este puntaje z. Según la tabla de distribución normal estándar o la calculadora, la probabilidad de que \( Z < 2,75 \) sea aproximadamente 0,9970. Por lo tanto, la probabilidad de que una batería seleccionada al azar tenga una vida promedio de menos de 810 horas es aproximadamente 0,9970 o 99,70%.

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