An express mail service charges a special rate for any packa

An express mail service charges a special rate for any package that weighs less than 1 lb. Let x denote the weight of a randomly selected parcel that qualifies for this special rate. The probability distribution of x is specified by the density curve shown below. Use the fact that area of a trapezoid = (base)(average of two side lengths) to answer each of the following questions:

(a) What is the probability that a randomly selected package of this type weighs at most 0.5 lb? between 0.15 and 0.5 lb? at least 0.55 lb?
P (x ? 0.5) =  
P (0.15 ? x ? 0.5) =  
P (x ? 0.55) =  

(b) It can be shown that ?_x = 7/12 and ?²_x = 11/144. What is the probability that the value of x is more than 1 standard deviation from the mean value? (Round the answer to four decimal places.)
P (x is more than one standard deviation from the mean value) =

An express mail service charges a special rate for any package that weighs less than 1 lb. Let x denote the weight of a randomly selected parcel that qualifies for this special rate. The probability distribution of x is specified by the density curve shown below. Use the fact that area of a trapezoid = (base)(average of two side lengths) to answer each of the following questions: (a) What is the probability that a randomly selected package of this type weighs at most 0.5 lb? between 0.15 and 0.5 lb? at least 0.55 lb? P (x ? 0.5) = P (0.15 ? x ? 0.5) = P (x ? 0.55) = (b) It can be shown that x = 7/12 and x^2 = 11/144. What is the probability that the value of x is more than 1 standard deviation from the mean value? (Round the answer to four decimal places.) P (x is more than one standard deviation from the mean value) =

Solution

a)

P(X<0.5) = 0.5 * (0.5 + 1)/2 = 0.375

P(0.15 < x < 0.5) = P(X<0.5) - P(X<0.15)

= 0.375 - 0.5 * (0.5 + 0.65)/2

= 0.0875

P(X>0.55) = 1 - P(X<0.55)

= 1 - 0.5 * (0.5 + 0.55)/2

= 0.7375

b)

P(X > 7/12 + 11/144) = P(X>95/144)

= 1 - P(X<95/144)

= 1 -0.5 * (0.5 + 95/144)/2

= 0.7101