Question

The local Housing Authority is studying the cost of apartments in our community. Based on a previous study, it was determined that monthly rent is normally distributed with a mean of $800 and a standard deviation of $200. Define the random variable, x, as the monthly rent of an apartment in our community. I asked a random sample of 30 apartment renters how much their rent is. I calculated the mean of the sample,

x⁻ a. What is the standard error of the mean? (Round to 2 decimal places.) b. State the probability distribution (sampling distribution) of
x⁻ c. What is the probability x⁻ will fall below $772 ? d. What is the probability x⁻
will fall between $790−$820 ? e. How much would that sample mean, x⁻, need to be in order to be in the top 5% of all sample means from samples of size 30 ?

Answer 1

The probability distribution of the sample mean, x⁻, can be approximated by a normal distribution when the sample size is large enough. The mean of the sampling distribution of x⁻ is equal to the mean of the population. The standard deviation of the sampling distribution of x⁻ is calculated using the formula: Standard Error of the Mean = Standard Deviation / sqrt(n).

Step-by-step explanation:

b. State the probability distribution (sampling distribution) of x⁻

The probability distribution of the sample mean, x⁻, can be approximated by a normal distribution when the sample size is large enough. This is known as the sampling distribution of the sample mean. The mean of the sampling distribution of x⁻ is equal to the mean of the population, which in this case is $800. The standard deviation of the sampling distribution of x⁻, also known as the standard error of the mean, can be calculated using the formula:

Standard Error of the Mean = Standard Deviation / sqrt(n)

where n is the sample size. In this case, we have a sample size of 30, so the standard error of the mean is:

Standard Error of the Mean = $200 / sqrt(30) ≈ $36.52