MATH 1110 - Section 001
Spring 2012
Quiz 08 Name:
March 27, 2012
Directions: Write legibly, the use of documents is forbidden, giving or receiving aid on this assignement is forbidden. You will get zero if any of these happen.
1. Quick questions: • Find the medians of the following set of data: 72, 71, 68, 72. • The best picture to visualize the quartiles is the histogram, the box plot, or the pie/bar chart ? • Given the following table, determine the relative frequencies. 4 points
Number of heads 0 1 2
Frequency 3 23 14
Answer • The middle two scores are 71 and 72, so the median is their average: • The box plot is the best picture to visualize the quartiles. • The relative frequency for this given table is:
3 40
23 40
14 40
71+72 2
= 71.5
2. The lawyers at a law firm are either associates or partners. At the end of each year, 30% of the associates leave the firm, 20% are promoted to partner and 50% remain associates. Also, 10% of the partners leave the firm at the end of each year. Assume that a lawyer who leaves the firm does not return. • Set up an absorbing stochastic matrix for the Markov process, in its standard form. • Find the stable matrix. 6 points
1 0.3 0.1 Answer The stochastic matrix A = 0 0.5 0 in which we can identify: 0 0.2 0.9 0.5 0 0.5 0 S = 0.3 0.1 and R = . Then I − R = 0.2 0.9 −0.2 0.1 The determinant of I −R is ∆ = 0.5×0.1−0×(−0.2), so the inverse of (I −R) is (I −R)−1 = 2 0 −1 0.3 0.1 Then S(I − R) = = 1 1 4 10 1 1 1 Therefore the stable matrix is: 0 0 0 0 0 0
2 4
0 10
MATH 1110 - Section 001
Spring 2012
Quiz 08 Name:
March 27, 2012
Directions: Write legibly, the use of documents is forbidden, giving or receiving aid on this assignement is forbidden. You will get zero if any of these happen.
1. Quick questions: • Find the medians of the following set of data: 72, 71, 68, 72. • The best picture to visualize the quartiles is the histogram, the box plot, or the pie/bar chart ? • Given the following table, determine the relative frequencies. 4 points
Number of heads 0 1 2
Frequency 3 23 14
Answer • The middle two scores are 71 and 72, so the median is their average: • The box plot is the best picture to visualize the quartiles. • The relative frequency for this given table is:
3 40
23 40
14 40
71+72 2
= 71.5
2. The lawyers at a law firm are either associates or partners. At the end of each year, 30% of the associates leave the firm, 20% are promoted to partner and 50% remain associates. Also, 10% of the partners leave the firm at the end of each year. Assume that a lawyer who leaves the firm does not return. • Set up an absorbing stochastic matrix for the Markov process, in its standard form. • Find the stable matrix. 6 points
1 0.3 0.1 Answer The stochastic matrix A = 0 0.5 0 in which we can identify: 0 0.2 0.9 0.5 0 0.5 0 S = 0.3 0.1 and R = . Then I − R = 0.2 0.9 −0.2 0.1 The determinant of I −R is ∆ = 0.5×0.1−0×(−0.2), so the inverse of (I −R) is (I −R)−1 = 2 0 −1 0.3 0.1 Then S(I − R) = = 1 1 4 10 1 1 1 Therefore the stable matrix is: 0 0 0 0 0 0
2 4
0 10
