Question

a set of sofa prices are normally distributed with a mean of 2500 dollars and a standard deviation of 200 dollars. a leather sofa has a price of 2410 dollars. what proportion of sofa prices are higher than the price of the leather sofa? you may round your answer to four decimal places.

Explanation:

Step1: Calculate the z - score

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x = 2410$ (price of the leather sofa), $\mu = 2500$ (mean price), and $\sigma=200$ (standard deviation).
$z=\frac{2410 - 2500}{200}=\frac{- 90}{200}=-0.45$

Step2: Find the proportion of values above the z - score

We know that the total area under the normal - distribution curve is 1. We first find the cumulative distribution function value for $z=-0.45$ using a standard normal table or a calculator with a normal - distribution function. The cumulative distribution function $\Phi(-0.45)$ gives the proportion of values below $z = - 0.45$. Using a standard normal table or a calculator (e.g., in Python: scipy.stats.norm.cdf(-0.45)), we find that $\Phi(-0.45)\approx0.3264$.
The proportion of values above $z=-0.45$ is $1-\Phi(-0.45)$.
$1 - 0.3264 = 0.6736$

Answer:

$0.6736$