Question

A minority representation group accuses a major bank of racial discrimination in its recent hires for financial analysts. Exactly 16% of all applications were from minority members, and exactly 15% of the 2100 open positions were filled by members of the minority.

Required:
a. Find the mean of p, where p is the proportion of minority member applications in a random sample of 2100 that is drawn from all applications.
b. Find the standard deviation of p.

Answer 1

a) The mean is of
`\mu = 0.16`

b) The standard deviation is of
`s = 0.008`

Explanation:

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
`\mu = p` and standard deviation
`s = \sqrt{(p(1-p))/(n)}`

Question a:

Exactly 16% of all applications were from minority members

This means
`p = 0.16`, and thus, the mean is of
`\mu = p = 0.16`

b. Find the standard deviation of p.

2100 open positions, thus
`n = 2100`.

`s = \sqrt{(p(1-p))/(n)}`

`s = \sqrt{(0.16*0.84)/(2100)}`

`s = 0.008`

The standard deviation is of
`s = 0.008`