Question

It has been determined that an agent of S.H.I.E.L.D. spends an average of 108 days per year identifying potential threats to human existence, with a standard deviation of 13.5 days. A random sample of 36 S.H.I.E.L.D. agents is taken.

a. What is the probability that the sample will have a mean of less than 105 days?

b. What is the probability that the sample will have a mean of more than 110 days?

c. What is the probability that the sample will have a mean between 109 and 114 days?

d. What is the probability that the sample will have a mean of no more than 104 days?

Answer 1

Population mean, = 108 days

Standard deviation, = 13.5 days

Sample size, n = 36

For sampling distribution of mean, = = 108 days

Standard error, =

=

= 2.25

a) P(sample will have a mean of less than 105 days) = P( < 105)

= P(Z < (105 - 108)/2.25)

= P(Z < -1.33)

= 0.0918

b) P(sample will have a mean of more than 110 days) = P( > 110)

= 1 - P( < 110)

= 1 - P(Z < (110 - 108)/2.25)

= 1 - P(Z < 0.89)

= 1 - 0.8133

= 0.1867

c) P(sample will have a mean between 109 and 114 days) = P(109 < < 114)

= P( < 114) - P( < 109)

= P(Z < (114 - 108)/2.25) - P(Z < (109 - 108)/2.25)

= P(Z < 2.67) - P(Z < 0.44)

= 0.9962 - 0.6700

= 0.3262

d) P(sample will have a mean of no more than 104 days) = P( < 104)

= P(Z < (104 - 108)/2.25)

= P(Z < -1.78)

= 0.0375