Question

at a small coffee shop, the distribution of the number of seconds it takes for a cashier to process an order is approximately normal with mean 276 seconds and standard deviation 38 seconds. which of the following is closest to the proportion of orders that are processed in less than 240 seconds?
a 0.17
b 0.25
c 0.36
d 0.83
e 0.95

Explanation:

Step1: Calculate the z - score

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x = 240$ (the value we are interested in), $\mu = 276$ (the mean), and $\sigma=38$ (the standard deviation). So, $z=\frac{240 - 276}{38}=\frac{- 36}{38}\approx - 0.95$.

Step2: Find the proportion using the standard normal table

We want to find $P(X < 240)$, which is equivalent to $P(Z<-0.95)$ when $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$ and $Z=\frac{X - \mu}{\sigma}$. Looking up $z=-0.95$ in the standard - normal table, we find that $P(Z < - 0.95)\approx0.17$.

Answer:

A. 0.17