The average square footage in an apartment in a town is 1,800 square feet with a standard deviation of 120 square feet.

The square footage is normally distributed.
You randomly select 10 apartments in the town.
What is the probability that the mean will be more than 1900 square feet?

To find the probability that the mean will be more than 1900 square feet, we can use the Central Limit Theorem.

The Central Limit Theorem states that when a sample size is large enough, the distribution of sample means will be approximately normally distributed, regardless of the shape of the population distribution.

In this case, the sample size is 10, which is relatively small. However, we can still use the Central Limit Theorem because the population distribution is assumed to be normal.

To calculate the probability, we need to convert the sample mean into a z-score. The z-score measures the number of standard deviations a particular value is from the mean.

First, let's calculate the standard error of the mean (SEM), which is the standard deviation of the sample means:

SEM = standard deviation / √(sample size)
= 120 / √10
≈ 37.95

Next, we calculate the z-score using the formula:

z = (sample mean - population mean) / SEM
= (1900 - 1800) / 37.95
≈ 2.64

Now, we need to find the probability of having a z-score greater than 2.64. We can look up this probability using a standard normal distribution table or use a calculator.

The probability can be calculated as follows:

P(z > 2.64) ≈ 0.0040

Therefore, the probability that the mean square footage will be more than 1900 square feet is approximately 0.0040, or 0.40%.

To find the probability that the mean square footage will be more than 1900 square feet, we need to use the concept of the sampling distribution of the sample mean.

The mean of the sampling distribution, also known as the expected value of the sample mean, is equal to the population mean, which is given as 1800 square feet.

The standard deviation of the sampling distribution, also known as the standard error of the sample mean, is calculated by dividing the population standard deviation by the square root of the sample size. In this case, the population standard deviation is 120 square feet, and the square root of the sample size (10) is √10 ≈ 3.16.

Now, we can use these values to calculate the z-score, which allows us to find the probability of observing a sample mean greater than 1900 square feet.

The z-score formula is:
z = (x - μ) / σ,
where x is the sample mean, μ is the population mean, and σ is the standard deviation of the sampling distribution.

Plugging in the values, we get:
z = (1900 - 1800) / (120 / √10) ≈ 1.581,

Next, we need to find the probability associated with this z-score using a standard normal distribution table or a calculator. The probability that the mean square footage will be more than 1900 square feet can be found by looking up the z-score of 1.581 and finding the corresponding area in the right tail of the standard normal distribution.

Using a standard normal distribution table or calculator, we find that the area to the right of the z-score 1.581 is approximately 0.0569.

Therefore, the probability that the mean square footage will be more than 1900 square feet is approximately 0.0569, or 5.69%.