Aa Aa A study conducted by three law school professors found that asylum seekers

Question

Aa Aa A study conducted by three law school professors found that asylum seekers in the United States face broad disparities in the nation's immigration courts. The professors discovered that 54% of refugees who ask for asylum in the San Francisco immigration court win asylum, but only 12% are granted asylum in the Atlanta immigration court. [Source: Julia Preston, "Wide Disparities Found in Judging of Asylum Cases," The New York Times, May 31, 2007.] Select the appropriate distribution in the Distributions tool to help answer the questions that follow. Select a Distribution Distributions 0 1 23 You randomly select 30 refugees who are asking for asylum in the San Francisco immigration court. Let x denote the number of asylum seekers who win their cases. The probability that exactly 14 asylum seekers are granted asylum is The probability that at least 11 asylum seekers are granted asylum is The expected value of x is , and the standard deviation of x is

Explanation / Answer

Solution:

We are given

n = 30,

p = 54% = 0.54,

q = 1 – p = 1 – 0.54 = 0.46

Here,

n*p = 30*0.54 = 16.2 and

n*q = 30*0.46 = 13.8

n*p and n*q > 5,

so we can use normal approximation to binomial distribution.

Mean = n*p = 16.2

SD = sqrt(n*p*q) = sqrt(30*0.54*0.46) = 2.72983516

First we have to find P(X=14)

By using approximation rule for continuity correction factor 0.5, we have to find P(13.5<X<14.5)

P(13.5<X<14.5) = P(X<14.5) – P(X<13.5)

Z = (X – mean)/SD

Z = (14.5 – 16.2)/2.73 = -0.62271

P(Z< -0.62271) = 0.266737

P(X<14.5) = 0.266737

Now, for X = 13.5

Z = (13.5 – 16.2)/2.73 = -0.98901

P(Z< -0.98901) = 0.161329

P(X<13.5) = 0.161329

P(13.5<X<14.5) = P(X<14.5) – P(X<13.5)

P(13.5<X<14.5) = 0.266737 - 0.161329

P(13.5<X<14.5) = 0.105408

Required probability = 0.105408

The probability that exactly 14 asylum seekers are granted asylum is 0.1054.

Now, we have to find the probability that at least 11 asylum seekers are granted asylum.

We have to find P(X11)

Subtracting continuity correction factor 0.5, we have to find P(X10.5)

P(X10.5) = 1 – P(X<10.5)

Z = (10.5 – 16.2)/2.73 = -2.087912088

P(Z< -2.087912088) = 0.018402883

P(X<10.5) = 0.018402883

P(X10.5) = 1 – P(X<10.5)

P(X10.5) = 1 – 0.018402883

P(X10.5) = 0.981597117

Required probability = 0.981597117

The probability that at least 11 asylum seekers are granted asylum is 0.9816.

Expected value of x = mean of X = E(X) = n*p = 30*0.54 = 16.2

Standard deviation = sqrt(n*p*q) = sqrt(30*0.54*0.46) = 2.72983516

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