Question

A bank is reviewing its risk management policies with regards to mortgages. To minimize the risk of lending, the bank wants to compare the typical mortgage owed by their clients against other homebuyers. The average mortgage owed by Americans is $306,500, with a standard deviation of $24,500. Suppose a random sample of 150 Americans is selected. Identify each of the following, rounding your answers to the nearest cent when appropriate:

1. $mu=?
2. $sigma=?
3. $=n=$
4. $mu_{overlinex}=$x=?
5. $sigma_{overlinex}=$x=?

Answer 1

Explanation:

We are given population mean μ=$306,500 and population standard deviation σ=$24,500, and want to find the mean and standard error of the sampling distribution, μx¯ and σx¯ for samples of size n=150.

By the Central Limit Theorem, the means of the two distributions are the same:

μx¯=μ=$306,500

To find the Standard Deviation of the sampling distribution, we divide the population standard deviation by the square root of the sample size:

σx¯=σn‾√=$24,500150‾‾‾‾√=$2,000.42

Answer 2

1.
`$ \mu = \$306,500 $`

2.
`\sigma = \$24,500`

3.
`n = 150`

4.
`$ \mu_(x)= \mu = \$306,500 $`

5.
`\sigma_x = \$ 2,000 \\\\`

Explanation:

The average mortgage owed by Americans is $306,500, with a standard deviation of $24,500.

From the above information, we know that,

The population mean is

`$ \mu = \$306,500 $`

The population standard deviation is

`\sigma = \$24,500`

Suppose a random sample of 150 Americans is selected

`n = 150`

Since the sample size is quite large then according to the central limit theorem, the sample mean is approximately normally distributed.

The sample mean would be the same as the population mean that is

`$ \mu_(x)= \mu = \$306,500 $`

The sample standard deviation is given by

`\sigma_x = (\sigma)/(√(n) )`

Where
`\sigma` is the population standard deviation and n is the sample size.

`\sigma_x = (24,500)/(√(150) ) \\\\\sigma_x = \$ 2,000 \\\\`

Therefore, the required parameters are:

1.
`$ \mu = \$306,500 $`

2.
`\sigma = \$24,500`

3.
`n = 150`

4.
`$ \mu_(x)= \mu = \$306,500 $`

5.
`\sigma_x = \$ 2,000 \\\\`