Question

the distribution of durations for which apartments remain empty after the resident moves out for one property management company over the past 10 years was approximately normal with mean $mu = 85$ days and standard deviation $sigma = 29$ days. the property management company intends to update the kitchen appliances in the apartments that were empty for top 10% of durations. what is the minimum duration for which an apartment remained empty for the company to update the kitchen appliances? round to the nearest whole number. choose 1 answer:

Explanation:

Step1: Find the z - score

We want the top 10% of the normal distribution. The area to the left of the z - score we want is $1 - 0.10=0.90$. Looking up in the standard normal distribution table (z - table), the z - score corresponding to an area of 0.90 is approximately $z = 1.28$.

Step2: Use the z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the original normal distribution, $\mu$ is the mean, and $\sigma$ is the standard deviation. We know $\mu = 85$, $\sigma = 29$, and $z = 1.28$. Rearranging the formula for $x$ gives $x=\mu+z\sigma$.
Substitute the values: $x = 85+1.28\times29$.
First, calculate $1.28\times29 = 1.28\times29=37.12$.
Then, $x=85 + 37.12=122.12$.

Step3: Round the result

Rounding $122.12$ to the nearest whole number, we get $x = 122$.

Answer:

122