6) A sample was selected of 506 workers who currently receive vacation per year.

Question

6) A sample was selected of 506 workers who currently receive vacation per year. These workers were asked if they were willing to accept a small pay cut to get an additional week of paid vacation a year. The following table shows the two wee responses of these workers no response no 140 119 es Man Women 104 32 34 249 257 a)lf one person is selected at random from these 506 workers, find the following probabilities and name each one them appropriately: i) P(yes I woman) i) P(yes) ii) P(woman and no) iv) PC no response or man) b)Are the events "woman" and "yes" independent? Are they mutually exclusive? Explain why or why not.

Explanation / Answer

toatl no of workers = 506

a)

i)

n(yes) = n(yes and man) + n( yes and woman) = 77+104 = 181

P(yes) = n(yes)/total = 181/506 = 0.3577

ii)

P(yes | woman) = P(yes and woman)/P(woman)

P(yes and woman) = n(yes and woman)/total = 104/506

p(woman) = n(woman)/total = 257/506

P(yes | woman) = (104/506)/(257/506) = 104/257 = 0.4047

c)

P(woman and no) = n(woman and no)/total = 119/506 = 0.2352

d)

P(no reponse or man) = n(no reponse or man)/total

n(no reponse or man) = n(no response) +n(man) -n(no reponse and man)

= 66+249-32 = 283

P(no reponse or man) = 283/506 = 0.5593

e)

if woman and yes are mutulaly independent then P(yes/woman) should be equal to P(yes)

But by above caluclation in part(a) P(yes) = 0.3577 and P(yes | woman) = 0.4047 which are not equal so they are not independent

if two events are mutulally exclusive then the probablity of second event ocuuring given first event is zero i.e if woman and yes are mutulaly exclusive then P(yes | woman) = 0

But by above caluclation in part(a) P(yes | woman) = 0.4047 which is a non zero number .so both the events are not mutulally exclusive

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