## Final Exam Statistics & Probability

1. Question

The individual score of Statistics & Probability Class was reported as below

24, 58, 37, 55, 14, 4, 26, 18, 23, 9, 49, 58, 16, 48, 34, 23, 42, 48, 36, 90, 42, 40, 62, 20, 70, 80, 33, 35, 70, 0

1. Organize the data using stem-leaf display (4 points)
2. Calculate the mean and standard deviation (4 points)
3. Create the box plot of the data (show the formula and how the quartiles are calculated ! not only the answers). (4 points)
4. Calculate the skewness of the data using Pearson’s method, and name the shape of the frequency polygons based on the skewness. (4 points)
5. Define the range of values which is categorized into outliers. (4 points)

2. mean: 38.8  stdev: 22.05

3. Q1: 22.25  Q2: 36.5 Q3: 55.75  Min: 0 Max:90

4.  skewness: 0.313

5. outlier are defined for score > 106  or < -27.75  (max score is 100, minimum score is 0. Thus such score do not exist).

2. Question

The position of chief of police in the city of Corry, Pennsylvania, is vacant. A search committee of Corry residents is charge with the responsibility of recommending a new chief to the city council. There are 15 applicants, 5 of which are either female or members of minority. The search committee decides to interview all of the applicants. To begin, they randomly select three applicants to be interviewed on the first day, and none of the three is female or a member of a minority. The local news paper, the Corry Press, suggests discrimination in an editorial. What is the likelihood of this occurrence ? (20 points) $\frac {C^{10}_{3} C^{5}_{0}}{C^{15}_{3}} = 0.264$

3. Question:

At least one-half of an airplane’s engines are required to function in order for it to operate. If each engine independently functions with probability p, for what values is p is a 4-engine plane more likely to operate than a 2-engine plane ?  (20 points)

4-engine plane will work if at least 2 engines function. Thus  P(success) for 4 engines = P(x=2)+P(x=3)+P(x=4)

2-engine plane will work if at least 1 engine functions. Thus  P(success) for 2 engines = P(x=1)+P(x=2)

We have to calculate the value of p such as P(sucess) for 4 engines > P(success) for 2 engines.  To simplify the calculation, instead of calculating P(success), I calculate the P(failure) for 4 engines and P(failure) for 2 engines as follows

P(failure) for 4 engines = P(x=0)+P(x=1) = Combination(4;0)  p^0 (1-p)^4 + Combination(4;1)  p^1 (1-p)^3

P(failure) for 2 engines = P(x=0) = Combination(2;0)  p^0 (1-p)^2

Thus, the problem can be redefined as finding the value of p such as

P(failure) for 4 engines < than P(failure) for 2 engines

Hence,

Combination(4;0)  p^0 (1-p)^4 + Combination(4;1) p^1 (1-p)^3 < Combination(2;0)  p^0 (1-p)^2

Thus  (1-p)^4 + 4p(1-p)^3 < (1-p)^2

Since p is probabilty, thus (1-p) is always zero or positive, allowing us to divide the both sides by (1-p)^2.

(1-p)^2 + 4p(1-p) < 1

1 – 2p + p^2 + 4p – 4p^2 -1 < 0

-3 p^2 + 2p < 0

Both sides are divided by p, and the answer is obvious.

-3p +2 < 0 Hence, p > 0.67

4. Question

A laboratory blood test is 99% effective in detecting a certain disease when it is, in fact, present. However, the test also yields a “false positive” result for 1 percent of the healthy persons tested. (That is, if a healthy person is tested, then, with probability 0.01, the test result will imply he or she has the disease). If 0.5% of the population actually has the disease, what is the probability a person has the disease given that his test result is positive ? (20 points) Using Bayes rule, we obtain

P(has the disease|positive) = P(positive|has the disease) P(has the disease) / P(positive)

P(positive)= P(positive|has the disease) P(has the disease) + P(positive|healthy) P(healthy) = (0.99 x 0.005 ) + (0.01x 0.995)

P(has the disease|positive) = (0.99 x 0.005) / ( 0.99 x 0.005  + 0.01 x 0.995) = 0.3322

5. Question

A large manufacturing firm tests job applicants who recently graduated from college. The test scores are normally distributed with a mean of 500 and a standard deviation of 50. Management is considering placing a new hire in an upper level management position if the person scores in the upper 10 percent of the distribution. What is the lowest score a college graduate must earn to qualify for a responsible position? (20 points) 