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do you think it would be unusual for an individual apartment to have a rent greater than $2781? assume the variable is normally distributed. the probability that an apartment has a rent greater than $2781 is what

To determine if it is unusual for an individual apartment to have a rent greater than $2781, we will use the normal distribution properties of the rent values. You provided the following key details:

- The mean monthly rent for a one-bedroom apartment in Manhattan is **$2643**.
- The standard deviation is **$502**.
- We need to find the probability that a single apartment has a rent greater than **$2781**.

### Steps:
1. **Calculate the z-score** for $2781 using the formula:
   \[
   z = \frac{X - \mu}{\sigma}
   \]
   where:
   - $ X = 2781 $ (the rent we're analyzing),
   - $ \mu = 2643 $ (the mean rent),
   - $ \sigma = 502 $ (the standard deviation).

2. Once we have the z-score, we can look up the corresponding probability from the standard normal distribution.

Let me compute that for you.The z-score for a rent of $2781 is approximately **0.275**.

Next, we will use the standard normal distribution to find the probability that a rent is less than $2781. Then, we can subtract that value from 1 to find the probability of a rent being greater than $2781.

Let me calculate that.The probability that an individual apartment has a rent greater than $2781 is approximately **0.3917**, or **39.17%**.

Since this probability is relatively high, it would **not** be considered unusual for an apartment to have a rent greater than $2781.

Do you have any questions or would you like further details on this process?

Here are 5 related questions you might consider:
1. How would the probability change if the rent threshold were higher, say $3000?
2. What is the probability that rent is between $2500 and $3000?
3. How does the probability change if the standard deviation is reduced?
4. What is the significance of a z-score of 1 in this context?
5. How does the sample size affect the probability in such problems?

**Tip:** Understanding z-scores can help you quickly assess whether a value is unusual or not, based on how far it is from the mean in terms of standard deviations.

Learn how to calculate the probability that an apartment rent exceeds $2781 in a normally distributed market. With a mean rent of $2643 and a standard deviation of $502, this problem involves calculating the z-score and using the standard normal distribution. Suitable for college-level statistics or advanced high school students.

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