Question

In: Statistics and Probability

A large corporation collected data on the reasons both middle managers and senior managers leave the...

A large corporation collected data on the reasons both middle managers and senior managers leave the company. Some managers eventually retire, but others leave the company prior to retirement for personal reasons, including more attractive positions with other firms. Assume that the following matrix of one-year transition probabilities applies with the four states of the Markov process being retirement, leaves prior to retirement for personal reasons, stays as a middle manager, and stays as a senior manager. Leaves- Middle Senior Retirement Personal Manager Manager Retirement 1.00 0.00 0.00 0.00 Leaves-Personal 0.00 1.00 0.00 0.00 Middle Manager 0.05 0.07 0.79 0.09 Senior Manager 0.09 0.02 0.05 0.84 What states are considered absorbing states? Retirement and Leaves-Personal Why? The input in the box below will not be graded, but may be reviewed and considered by your instructor. Interpret the transition probabilities for the middle managers. Probability of retirement = Probability of leaving (personal) = Probability of staying middle manager = Probability of promotion to senior manager = Interpret the transition probabilities for the senior managers. Probability of retirement = Probability of leaving (personal) = Probability of staying middle manager = Probability of promotion to senior manager = What percentage of the current middle managers will eventually retire from the company? What percentage will leave the company for personal reasons? If required, round your answers to one decimal place. Retire from the company = % Leave the company for personal reasons = % The company currently has 940 managers: 670 middle managers and 270 senior managers. How many of these managers will eventually retire from the company? How many will leave the company for personal reasons? If required, round your answers to one decimal place. Retire from the company = % Leave the company for personal reasons = %

Solutions

Expert Solution

The transition probability matrix is,

Retirement Leaves- Personal Middle Manager Senior Manager

Retirement 1.00 0.00 0.00 0.00

Leaves-Personal 0.00 1.00 0.00 0.00

Middle Manager 0.05 0.07 0.79 0.09

Senior Manager 0.09 0.02 0.05 0.84

The absorbing states are Retirement and Leaves-Personal because the transition probabilities from Retirement/Leaves-Personal to any other state is 0.

For the middle managers. Probability of retirement = 0.05

Probability of leaving (personal) = 0.07

Probability of staying middle manager = 0.79

Probability of promotion to senior manager = 0.09

For the senior managers. Probability of retirement = 0.09

Probability of leaving (personal) = 0.02

Probability of staying middle manager = 0.05

Probability of promotion to senior manager = 0.84

The transition probability matrix in canonical form is,

Middle Manager Senior Manager Retirement Leaves- Personal  

Middle Manager 0.79 0.09 0.05 0.07

Senior Manager 0.05 0.84 0.09 0.02

Retirement 0.00 0.00 1.00 0.00

Leaves-Personal 0.00 0.00 0.00 1.00

From the above matrix,

Fundamental Matrix N = (I - Q)-1 is, (Where I is 2x2 Unit matrix)

Absorption Probabilities matrix B is,

B = NR

Thus,

Percentage of the current middle managers will eventually retire from the company = 55.3%

Percentage of the current middle managers will eventually leave from the company = 44.7%

Probability of the current middle managers will eventually retire from the company = 0.553

Probability of the current senior managers will eventually retire from the company = 0.735

Expected managers will eventually retire from the company = 670 * 0.553 + 270 * 0.735 = 569.0

Percentage = 569 / 970 = 58.7%

Probability of the current middle managers will eventually leave from the company = 0.447

Probability of the current senior managers will eventually leave from the company = 0.265

Expected managers will eventually leave from the company = 670 * 0.447 + 270 * 0.265 = 371.0

Percentage = 371 / 970 = 38.2%

orchestra answered 2 years ago
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