Question

1. The general manager of the Hilton Hotel in Sydney is evaluating an employment screening test for the front office clerical staff. During this evaluation all new clerical employees are given the test. 70% pass the test; the rest fail. At a later time, after the new clerical employees have been working for a while, their performance is evaluated as being satisfactory or unsatisfactory. Historically, 80% of all clerical employees have been found to be satisfactory, and 75% of the satisfactory clerical employees in the evaluation of the employment screening test have passed the screening test. (a) From the given information, determine the probabilities of the following events: (i) pass- ing the test, (ii) having satisfactory performance and (iii) passing the test given satisfactory performance. [3 marks] (b) Using your answers to part (a), determine the probability of a clerical employee passing the test and having satisfactory performance. [1 mark] (c) Using your answer in parts (a) and (b), determine the probabilities of the a clerical employee: (i) failing the test and having satisfactory performance, (ii) failing the test and having unsat- isfactory performance, (iii) passing the test and having unsatisfactory performance, and (iv) having unsatisfactory performance. [4 marks) (d) Using your answers in part (c), determine the probabilities of a clerical employee: (i) failing the test given they are found to have unsatisfactory performance, (ii) failing the test given they are found to have satisfactory performance, (iii) passing the test given they are found to have unsatisfactory performance, (iv) unsatisfactory performance given they failed the test, (v) satisfactory performance given they passed the test, (vi) unsatisfactory performance given they passed the test, and (vii) satisfactory performance given they failed the test. [7 marks] (e) Using your answers in part (d), determine the following percentages: (i) clerical employees who failed the test and prove to be unsatisfactory, and (ii) clerical employees who passed the test and who prove to be satisfactory. [2 marks] (f) Government guidelines require screening tests to achieve at least 20% for part (i) in (e) and at least 60% for part (ii) in (e). Does this test meet those government requirements? Explain. [2 marks]

Answer 1

ANSWER::

1Q)

Let the events be defined as follows:
A: Clerical employee passing the test
B: Clerical employee being satisfactory

Given

• 70% pass the test, \implies P(A) = 0.7
• 80% employees are found to be satisfactory, \implies P(B) = 0.8
• 70% of the satisfactory clerical employees have passes, \implies P(A\ |\ B) = 0.75

(a) Probabilities of

(i) Passing the test = P(A) = 0.7
(ii) having satisfactory performance = P(B) = 0.8
(iii) passing the test given satisfactory performance = P(A\ |\ B) = 0.75

(b) Probability of passing the test AND having satisfactory performance =

(ANSWER)

P(ANB) = P(B)*P( A B) = 0.8 *0.75 = 0.6

NOTE:

P(AUB) = P(A) + P(B) - P(ANB) = 0.7 +0.8 -0.6 = 0.9

(c) Probabilities of

  (i) Failing the test and having satisfactory performance =

P(ANB) = P(B) - P(ANB) = 0.8 -0.6 = 0.2


(ii)   Failing the test and having unsatisfactory performance =

P( A B ) = P(AUB) = 1- P(AUB) = 1 -0.9 = 0.1

(iii) Passing the test and having unsatisfactory performance =

P(ANB) = P(A) - P(ANB) = 0.7 -0.6 = 0.1

(iv) having unsatisfactory performance =

P(B) = 1-PB) = 1 -0.8 = 0.2

(d) Probabilities of

(i) Failing the test given they are found to have unsatisfactory performance =

A B 0.1 P( A B ) = P(BC) = 0 P(BC) 0.2 = 0.5

(ii) Failing the test given they are found to have satisfactory performance =

P( A B ) = P( A B) P(B) 0.2 _ 025 0.8

(iii) Passing the test given they are found to have unsatisfactory performance =

   

PAB) 0.1 P(AB) = ? P(BC -03 = 0.5

(iv)   unsatisfactory performance given they failed the test =

P( B A ) 0.1 P(B° | 4) = P(4) 02 = 0.33

(v) satisfactory performance given they passed the test =

  

P(BA) = P(BNA) 0.6 PA) 07=0.857

  (vi) unsatisfactory performance given they passed the test =

P(BºA) P(BºA) = 2 0.1 P(A) = 07 = 0.143

  (vii) satisfactory performance given they failed the test =

P(B | AC) = P(BNA) 0.2 PLAC = 0.3 = 0.

(e) The percentage of

(i) clerical employees who failed the test and prove to be unsatisfactory =

100 * PLAN B)% = 100 * 0.1 % = 10 %

(ii) clerical employees who passed the test and who prove to be satisfactory. =

      

100 * P(ANB)% = 100*0.6 % = 60 %

f) Government guidelines require screening tests to achieve at least 20% for part (i) in (e) and at least 60% for part (ii) in (e). Does this test meet those government requirements?

Since the percentage in part (i) of (e) = 10 % which is LESS than 20 %, so NO this DOES NOT meet the government requirements.

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