Question

a store manager records the number of customers at the store each week for several years. the data are approximately normally distributed with a mean of 1245 and a standard deviation of 35. the manager wants to predict future customer totals. using these data, what percentage of weeks will have between 1280 and 1315 customers? %

Explanation:

Step1: Calcular los valores z

El valor z se calcula con la fórmula \( z=\frac{x - \mu}{\sigma} \), donde \( \mu = 1245 \), \( \sigma=35 \).
Para \( x = 1280 \): \( z_1=\frac{1280 - 1245}{35}=\frac{35}{35}=1 \)
Para \( x = 1315 \): \( z_2=\frac{1315 - 1245}{35}=\frac{70}{35}=2 \)

Step2: Usar la propiedad de la distribución normal

En una distribución normal, el porcentaje de datos entre \( z = 1 \) y \( z = 2 \) se calcula como \( P(1<Z<2)=P(Z < 2)-P(Z < 1) \)
Sabemos que \( P(Z < 1)=0.8413 \) y \( P(Z < 2)=0.9772 \)
Entonces \( P(1<Z<2)=0.9772 - 0.8413=0.1359 \)

Answer:

13.59