7. [12 points] A particular telephone number is used to receive both voice calls

Question

7. [12 points] A particular telephone number is used to receive both voice calls and fax messages Suppose that 25% of the incoming calls involve fax messages. Let X-the number of fax messages among a random sample of 15 calls so that X Bin(15, 0.25) (a) Use the table of cumulative Binomial probabilities to calculate the chance of each of the following events. [2 points each (i) At most 3 of the calls involve fax messages (ii) Between 4 and 8 of the calls (inclusive of the endpoints) involve fax messages (iii) Exactly 4 of the calls involve fax messages (iv) P(2 s X

Explanation / Answer

(a) As we have to use the table of cumulaative binomial probabilities

(i) Pr (X <=3; 15, 0.25) = 0.4613

(ii)  Pr(4 <= X <= 8 ; 15; 0.25) = Pr (X <=8; 15, 0.25) -  Pr (X <=3; 15, 0.25) = 0.9958 - 0.4613 = 0.5345

(iii) Pr (X = 4; 15 ; 0.25) = 0.2252

(iv) Pr (2 <= X < 7 ; 15; 0.25) = Pr (X <7; 15, 0.25) -  Pr (X <=1; 15, 0.25) = 0.9438 - 0.0802 = 0.8636

(b) Mean of the number of call, out of 15 = np = 15 * 0.25 = 3.75 calls

Standard deviation of the numer of calls = sqrt(np(1-p) = sqrt [15 * 0.25 * 0.75] = 1.677 calls

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