5. The average rent for a studio apartment in NYC is $\$ 1680$ per month with a standard deviation of $\$ 200$. If a random sample of 40 people renting a studio apartment is taken, what is the probability that the mean rent of sample is less than $\$ 1500$ ?

Step 1 ：We are given a problem of probability involving normal distribution. The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger, regardless of the shape of the population distribution. This theorem applies here as we have a large sample size (n=40).

Step 2 ：We are given the population mean (μ = $1680), the population standard deviation (σ = $200), the sample size (n = 40), and we are asked to find the probability that the sample mean (x̄) is less than $1500.

Step 3 ：The first step is to find the standard deviation of the sample mean, also known as the standard error (SE), which is given by σ/√n. So, SE = \( \frac{200}{\sqrt{40}} = 31.622776601683793 \)

Step 4 ：Next, we calculate the z-score, which measures how many standard deviations an element is from the mean. The z-score for the sample mean is given by (x̄ - μ) / SE. So, z = \( \frac{1500 - 1680}{31.622776601683793} = -5.692099788303083 \)

Step 5 ：Finally, we use the z-score to find the probability. The probability that the sample mean is less than a certain value is the same as the probability that the z-score is less than the z-score of that value. We can find this probability from the standard normal distribution table. The probability that the sample mean is less than $1500 is extremely small, approximately 6.27e-09. This means that it is highly unlikely for the average rent of a random sample of 40 people to be less than $1500.

Step 6 ：Final Answer: The probability that the mean rent of the sample is less than $1500 is approximately \(\boxed{6.27 \times 10^{-9}}\).