Statistics

In the judicial case of United States vs. City of Chicago, discrimination was charged in a qualifying exam for the position of Fire Captain. In the table below Group A is a minority group and Group B is a Majority Group.

Passed Failed
Group A 10 14
Group B 417 145

A) If one of the test subjects is randomly selected, find the probability of getting someone who passed the exam.

B) Find the probability of randomly selecting one of the test subjects and getting someone who is in Group B or passed.

C) Find the probability of randomly selecting two different test subjects and finding that they are both in Group A.

D) Find the probability of randomly selecting one of the test subjects and getting someone who is in Group A and passed the exam.

E) Find the probability of getting someone who passed, given that the selected person is in Group A.

Based on the results above, can we make a probability argument that discrimination is present based on p = 0.05? Why or why not. I am not interested in theory here, only the impact of the probabilities above!

Suppose that we know that the average income in Malvern, PA is $30,000 and that the standard deviation is $2,000. Assume that household incomes in Malvern are normally distributed. Using the Empirical Rule, please provide the range of incomes that we can expect 68% of the data to lie within. What about 95% and 99.73%?

Assume that we have just learned that the population of Malvern is bimodal and not normally distributed. Using Chebychev's Theorem, what percent of our population can we expect to lie within plus or minus two standard deviations? What percent of our population can we expect to lie within plus or minus three standard deviations?

What kind of differences do we see in the above distributions when we compare the Empirical Rule to Chebychev's Theorem. Which approach provides more precision? Why can't we use plus or minus one standard deviation with the Chebychev analysis?