Question

(1) Police response time to an emergency call is the difference between the time the call is first received by the dispatcher and the time a patrol car radios that it has arrived at the scene. Over a long period of time, it has been determined that the police response time has a normal distribution with a mean of 10.8 minutes and a standard deviation of 1.9 minutes. For a randomly received emergency call, find the following probabilities. (Round your answers to four decimal places.) (a) the response time is between 7 and 12 minutes (b) the response time is less than 7 minutes (c) the response time is more than 12 minutes (2) Sociologists say that 95% of married women claim that their husband's mother is the biggest bone of contention in their marriages (sex and money are lower-rated areas of contention). Suppose that seven married women are having coffee together one morning. Find the following probabilities. (Round your answers to three decimal places.) (a) All of them dislike their mother-in-law. (b) None of them dislike their mother-in-law. (c) At least five of them dislike their mother-in-law. (d) No more than four of them dislike their mother-in-law.

Answer 1

1) a) P(7 < X < 12)

   = P((7 - )/ < (X - )/ < (12 - )/)

   = P((7 - 10.8)/1.9 < Z < (12 - 10.8)/1.9)

   = P(-2 < Z < 0.63)

   = P(Z < 0.63) - P(Z < -2)

   = 0.7357 - 0.0228

   = 0.7129

b) P(X < 7)

= P((X - )/ < (7 - )/)

= P(Z < (7 - 10.8)/1.9)

= P(Z < -2)

= 0.0228

c) P(X > 12)

= P((X - )/ > (12 - )/)

= P(Z > (12 - 10.8)/1.9)

= P(Z > 0.63)

= 1 - P(Z < 0.63)

= 1 - 0.7357

= 0.2643

2) n = 7

    p = 0.95

It is a binomial distribution.

P(X = x) = nCx * p^x * (1 - p)^{n - x}

a) P(X = 7) = 7C7 * (0.95)^7 * (0.05)^0 = 0.6983

b) P(X = 0) = 7C0 * (0.95)^0 * (0.05)^7 = 0.0000

c) P(X > 5) = P(X = 5) + P(X = 6) + P(X = 7)

                   = 7C5 * (0.95)^5 * (0.05)^2 + 7C6 * (0.95)^6 * (0.05)^1 + 7C7 * (0.95)^7 * (0.05)^0

                   = 0.9962

d) P(X < 4) = 1 - P(X > 4)

                  = 1 - (P(X = 5) + P(X = 6) + P(X = 7))

                  = 1 - (7C5 * (0.95)^5 * (0.05)^2 + 7C6 * (0.95)^6 * (0.05)^1 + 7C7 * (0.95)^7 * (0.05)^0)

                  = 1 - 0.9962

                  = 0.0038