11The wind speed in San Francisco is normally distributed wi

11.The wind speed in San Francisco is normally distributed with a mean of 12.8 miles per hour and a standard deviation of 5.6 mph. a. Find the probability that the wind speed is between 5 mph and 15 mph. b. If eleven days are randomly selected, find the probability that the average wind speed exceeds 13 mph Find the wind speed needed for a day to be in the top 13%.

Solution

11. a.

First covert 5 and 15 to z scores:

z = (x - mean)/sd

(5 - 12.8)/5.6 = -1.39

(15 - 12.8)/5.6 = 0.39

P(5 < x < 12.8) = P(-1.39 < z < 0.39) = P(Z < 0.39) - P(z < -1.39) = .6517 - .0823 = 0.5694 (from standard normal table)

answer: .5694

11. b.

first convert x-bar of 13 to z score:

z = (x-bar - mean)/(sd/square root n)

(13 - 12.8)/(5.6/sqrt 11) = .12

P(Z > .12) = 1 - P(Z < .12) = 1 - .5478 = .4522 (from standard normal table)

answer: .4522

Find the wind speed needed for a day to be in the top 13%.

Top 13% is the 87th percentile. Looking up .87 on the standard normal table, give a z score of 1.13

convert z score to an x:

x = z*sd + mean = 1.13*5.6 + 12.8 = 19.128

answer: 19.128