Question

a set of bicycle prices are normally distributed with a mean of 300 dollars and a standard deviation of 50 dollars. a sports bicycle has a price of 380 dollars. what proportion of bicycle prices are lower than the price of the sports bicycle? 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 $x$ is the value from the data set, $\mu$ is the mean, and $\sigma$ is the standard deviation. Here, $\mu = 300$, $\sigma=50$, and $x = 380$. So, $z=\frac{380 - 300}{50}=\frac{80}{50}=1.6$.

Step2: Find the proportion using the standard normal distribution

We want to find $P(X<380)$, which is equivalent to $P(Z < 1.6)$ when $X$ is normally distributed with mean $\mu = 300$ and standard deviation $\sigma = 50$. Looking up the value of $P(Z < 1.6)$ in the standard - normal table (or using a calculator with a normal - distribution function), we get $P(Z<1.6)=0.9452$.

Answer:

$0.9452$