Question

One reason that Normal distribution models show up so often is because they have some special and useful properties, many of which were covered in class. Here is another:

Mathematical Fact: If the variables X and Y are both normally distributed and independent, then new variables X + Y (the sum) and X – Y (the difference) are also normally distributed.

* The mean and standard deviation of the sum or difference are calculated using the properties of random variables from CH 2 of OpenIntro Statistics 3e textbook, sections 2.4.3 and 2.4.4.

Problem: A university administrator is interested in whether a new building can be planned and built on campus within a four-year time frame. He considers the process in two phases.

Phase I: Phase I involves lobbying the state legislature and governor for permission and funds, issuing bonds to obtain funds, and obtaining all the appropriate legal documents. Past experience indicates that the time required to complete phase I is approximately normally distributed with a mean of 16 months and standard deviation of 4 months.

If X = phase I time, then X ~ N(μ = 16 months, σ = 4 months).

Phase II: Phase II involves creation of blue prints, obtaining building permits, hiring contractors, and, finally, the actual construction of the building. Past data indicates that the time required to complete these tasks is approximately normally distributed with a mean of 18 months and a standard deviation of 12 months.

If Y = phase II time, then Y ~ N(μ = 18 months, σ = 12 months).
a) A new random variable, T = total time for completing the entire project, is defined as T = X + Y. What is the probability

distribution of T? (Give both the name of the distribution and its parameters.)
b) Find the probability that the total time for the project is less than four years. (In symbols, calculate P(T < 48 months).) c) Find the 95th percentile of the distribution of T.

Answer 1

If X = phase I time, then X ~ N(μ = 16 months, σ = 4 months), that is X is normally distributed with mean   and standard deviation

f Y = phase II time, then Y ~ N(μ = 18 months, σ = 12 months), , that is Y is normally distributed with mean   and standard deviation

a) T = total time for completing the entire project, is defined as T = X + Y.

the mean of T is

The variance of T is

The standard deviation of T is

Using the fact that If the variables X and Y are both normally distributed and independent, then new variables T=X + Y (the sum) is also normally distributed, we can say that

ans:

That is T is normally distributed with a mean of 34 months and a standard deviation of 12.6491 months

b) the probability that the total time for the project is less than four years (that is 4*12=48 months) is

ans: the probability that the total time for the project is less than four years is 0.8665

c) Let q be the value of the 95th percentile of the distribution of T. That is the probability that the total times for the project is less than q is 0.95

First we find the Z score for which

P(Z<z) = 0.95.

Using the standard normal tables we get that for z=1.645, P(Z<1.645)=0.95

Equating 1.645 to the z score of q, we get

ans: The 95th percentile of the distribution of T is 54.81 months