The mean monthly rent for a one-bedroom apartment without a doorman in Manhattan is $2,651. Assume the standard deviation is $500. A real estate firm samples 100 apartments. What is the probability that the average rent of the sample is more than $2,708?

write only a number as your answer. Round to 4 decimal places.

To find the probability that the average rent of the sample is more than $2,708, we need to use the Z-score and the standard normal distribution.

The Z-score is calculated using the formula:
Z = (X - μ) / (σ / √n)

Where:
X = $2,708 (the value we want to find the probability for)
μ = $2,651 (mean monthly rent)
σ = $500 (standard deviation)
n = 100 (sample size)

Calculating the Z-score:
Z = (2,708 - 2,651) / (500 / √100)
Z = 57 / 50
Z = 1.14

Now, we need to find the area under the standard normal distribution curve to the right of the Z-score value of 1.14. We can use a Z-table or a calculator to find this value.

Using a Z-table or a calculator, the probability corresponding to a Z-score of 1.14 is approximately 0.8729.

Therefore, the probability that the average rent of the sample is more than $2,708 is approximately 0.8729, or 0.873 rounded to 3 decimal places.

Same process.