The mean rent of a 3-bedroom apartment in Orlando is $1200. You randomly select 12 apartments around town. The rents are normally distributed with a standard deviation of $75. What is the probability that the mean rent is more than $1000?
A music industry researcher wants to estimate, with a 90% confidence level, the proportion of young urban people (ages 21 to 35 years) who go to at least 3 concerts a year. Previous studies show that 21% of those people (21 to 35 year olds) interviewed go to at least 3 concerts a year. The researcher wants to be accurate within 1% of the true proportion. Find the minimum sample size necessary.
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To find the probability that the mean rent is more than $1000, we can use the concepts of sampling distributions and the central limit theorem.
Step 1: Define the problem and the conditions:
- The mean rent of a 3-bedroom apartment in Orlando is $1200.
- You randomly select 12 apartments around town.
- The rents are normally distributed with a standard deviation of $75.
Step 2: Apply the central limit theorem:
According to the central limit theorem, for a large enough sample size (n ≥ 30), the distribution of the sample means will be approximately normally distributed regardless of the shape of the original population.
Since the sample size is 12, which is less than 30, we need to check if the population is normally distributed. However, since the problem statement states that the rents are normally distributed, we can proceed assuming that the sample means will also be normally distributed.
Step 3: Calculate the standard error of the mean:
The standard error (SE) of the mean is the standard deviation of the sample means and can be calculated using the formula:
SE = (Standard Deviation) / √(Sample Size)
= $75 / √12
Step 4: Calculate the z-score:
The z-score measures how many standard errors the sample mean is from the population mean. It can be calculated using the formula:
z = (Sample Mean - Population Mean) / SE
= ($1000 - $1200) / SE
Step 5: Find the probability:
Finally, we can use a standard normal distribution table or calculator to find the probability associated with the calculated z-score.
Alternatively, we can use statistical software like Excel or Python to find this probability. Specifically, using the "normsf" or "normsdist" functions. In Python, you can use the scipy.stats.norm library.
For example, using Python:
```python
import scipy.stats as stats
mean = 1200
sample_mean = 1000
standard_deviation = 75
sample_size = 12
standard_error = standard_deviation / (sample_size ** 0.5)
z_score = (sample_mean - mean) / standard_error
probability = 1 - stats.norm.cdf(z_score)
print("The probability that the mean rent is more than $1000 is", probability)
```
Using this approach, you can find the probability that the mean rent is more than $1000.