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.

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.

c. Compute an approximation for P ( p leq 0.15), which is the probability that there will be 15% or fewer minority member applications in a random sample of 2100 drawn from all applications. Round your answer to four decimal places.

Answer 1

a) 0.16

b) 0.0518

c)
`P(p \leq 0.15) = 0.4247`

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

For a proportion p in a sample of size n, we have that the mean is
`\mu = p` and the standard deviation is
`\sigma = \sqrt{(p(1-p))/(n)}`

In this problem, we have that:

`p = 0.16, n = 50`

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.

The mean of p is 0.16.

b. Find the standard deviation of p.

`\sigma = \sqrt{(0.16*0.84)/(50)} = 0.0518`

c. Compute an approximation for P ( p leq 0.15), which is the probability that there will be 15% or fewer minority member applications in a random sample of 2100 drawn from all applications. Round your answer to four decimal places.

This is the pvalue of Z when X = 0.15. So

`Z = (X - \mu)/(\sigma)`

`Z = (0.15 - 0.16)/(0.0518)`

`Z = -0.19`

`Z = -0.19` has a pvalue of 0.4247

`P(p \leq 0.15) = 0.4247`