Section 3.4 Problems A particular telephone number is used to receive both voice

Question

Section 3.4 Problems A particular telephone number is used to receive both voice calls and fax messages. Suppose that 25% of the incoming calls involve fax messages, and consider a sample of 25 incoming calls. What is the probability that 3- a At most 6 of the calls involhve a fax message? b. At least 6 of the calls involve a fax message? . Exactly 6 of the calls involve a fax message? d. More than 6 of the calls involve a fax message? e. What is the expected and the standard deviation of the number among the 25 calls that involve a fax message?

Explanation / Answer

3) n = 25

p = 0.25

a) P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)

                = 25C0 * (0.25)^0 * (0.75)^25 + 25C1 * (0.25)^1 * (0.75)^24 + 25C2 * (0.25)^2 * (0.75)^23 + 25C3 * (0.25)^3 * (0.75)^22 + 25C4 * (0.25)^4 * (0.75)^21 + 25C5 * (0.25)^5 * (0.75)^20 + 25C6 * (0.25)^6 * (0.75)^19 = 0.561

b) P(X > 6) = 1 - P(X < 6)

                = 1 - ( P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5))

                =,1 - ( 25C0 * (0.25)^0 * (0.75)^25 + 25C1 * (0.25)^1 * (0.75)^24 + 25C2 * (0.25)^2 * (0.75)^23 + 25C3 * (0.25)^3 * (0.75)^22 + 25C4 * (0.25)^4 * (0.75)^21 + 25C5 * (0.25)^5 * (0.75)^20)

              = 1 - 0.3783 = 0.6217

c) P(X = 6) = 25C6 * (0.25)^6 * (0.75)^19 = 0.1828

d) P(X > 6) = 1 - P(X < 6)

                = 1 - (25C0 * (0.25)^0 * (0.75)^25 + 25C1 * (0.25)^1 * (0.75)^24 + 25C2 * (0.25)^2 * (0.75)^23 + 25C3 * (0.25)^3 * (0.75)^22 + 25C4 * (0.25)^4 * (0.75)^21 + 25C5 * (0.25)^5 * (0.75)^20 + 25C6 * (0.25)^6 * (0.75)^19)

    = 1 - 0.561 = 0.439

e) mean = n * p = 25 * 0.25 = 6.25

sd = sqrt(n * p * (1 - p))

      = sqrt(25 * 0.25 * 0.75) = 2.165

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