Question

Let Y = e^x where X is normally distributed with μ = 1.2 and σ = 0.5. Compute the following values. [You may find it useful to reference the z table.]

a. Compute P(Y ≤ 9.6). (Round your intermediate calculations to at least 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)


b. Compute P(8.4 < Y < 9.7). (Leave no cells blank - be certain to enter "0" wherever required. Round your intermediate calculations to at least 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)


c. Compute the 70th percentile of Y. (Round your intermediate calculations to at least 4 decimal places, “z” value to 3 decimal places, and final answer to the nearest whole number.)

Answer 1

a]       P(Y ≤ 9.6) now taking log to the base e on both sides , we get

                    = Pr( X ≤ ln(9.6) ) = Pr( X ≤ 2.2618 ) = Pr( Z ≤ 2.12 ) = 0.9831

        Where Z = = = 2.12

b]

P(8.4 < Y < 9.7)   now taking log to the base e on both sides , we get

                         = P( ln(8.4) < X < ln(9.7)

                         = P( 2.1282 < X < 2.2721 )

                        = P( X < 2.2721) - P( X < 2.1282)

                        = P ( Z < 2.14) - P( Z < 1.86) = 0.9840 - 0.9683 = 0.0157

Where Z = = = 2.14 and Z = = = 1.86

c]

70th percentile of Y : Pr( Y < P_70 ) = 0.7

                                   Pr( X < ln(P_70) ) = 0.7

                        ==>     = 0.524 ......Z value

                       ==>       ln( P_70) = 1.2 + 0.5*0.524 = 1.4622

                      ==>        P_70 = exp(1.4622) = 4.315 nearest whole number is 4