Math Snap

STEP 1

1. The sample size is 86 apartments.
2. The population mean ($\mu$) is $2557.
3. The population standard deviation ($\sigma$) is $486.
4. The sample mean ($\bar{x}$) we are comparing to is $2627.
5. The distribution of sample means is approximately normal due to the Central Limit Theorem.

STEP 2

1. Calculate the standard error of the mean.
2. Calculate the z-score for the sample mean of $2627.
3. Use Excel to find the probability corresponding to the z-score.
4. Interpret the result to find the probability that the sample mean is greater than $2627.

STEP 3

Calculate the standard error of the mean (SEM) using the formula:
$$\text{SEM} = \frac{\sigma}{\sqrt{n}}$$ where $\sigma = 486$ and $n = 86$.
$$\text{SEM} = \frac{486}{\sqrt{86}} \approx 52.42$$

STEP 4

Calculate the z-score using the formula:
$$z = \frac{\bar{x} - \mu}{\text{SEM}}$$ where $\bar{x} = 2627$, $\mu = 2557$, and $\text{SEM} \approx 52.42$.
$$z = \frac{2627 - 2557}{52.42} \approx 1.336$$

STEP 5

Use Excel to find the probability corresponding to the z-score. In Excel, use the formula:
$$\text{=1 - NORM.S.DIST(1.336, TRUE)}$$ This gives the probability that the sample mean is greater than $2627.

Interpret the Excel result. The probability that the sample mean rent is greater than $2627 is approximately:
$$\boxed{0.0909}$$