An airline company models the weight of each passengers luggage, Xi, as independent and identical random variables according to a normal distribution with mean/expectation μ = 35 pounds and standard deviation of σ = 5 pounds.

a. What is the probability a random passenger’s luggage is equal to 35 pounds? What about being between 30 and 40 pounds? Write out the integral form.

Now say the airline relaxes the normal distribution specification. Thus each luggage weights Xi are independent and identically distributed from some unknown distribution with mean 35 pounds and standard deviation of 5 pounds.

b. Say a sample of size n (where n is large) is taken from this distribution. What approximate n distribution does the sample mean, , X = ΑΣΧ, follow? State the type, expectation, and variance. What theorem are you using?

c. The airline is more interested in the total weight of all the luggages, rather than the mean/average n of them. What approximate distribution does follow? State the type, expectation, and variance.