MONT 105N – Analyzing Environmental Data Discussion – Normal Probabilities using Tabulated Values March 15, 2019

Background

Probabilities for a standard normal random variable (i.e. normal distribution with μ = 0, σ = 1) are given in the table on the back of this sheet.

Important Fact: If Y is normal with mean μ and standard deviation σ, then

Z = \frac{Y − μ}{σ}

is standard normal, and the table can be applied to Z. In today’s discussion, you will practice using the table to answer questions about normally distributed quantities.

Discussion Questions

A) Let Z be a standard normal.

1) Find P (.23 < Z < .59)

2) Find P (−2.13 < Z < −0.56)

3) Find c such that P (Z > c) = .05 (Note: there’s no such c exactly appearing in the table, but you can estimate c with the information you have there.)

4) Find d such that P (−d < Z < d) = .98 (The same comment as in part 3) applies here.)

B) Y is normally distributed with mean μ = 6 and σ = 2. Find

1) P (6 < Y < 7)

2) P (4 < Y < 5)

3) P (4.5 < Y < 7.5)

C) SlimMints (yum!) are sold in two-packs with a stated label weight of 20.4 grams. The actual weights of the packages are normally distributed with mean μ = 21.37 and SD σ = .4. Let Y be the weight of a single package selected at random from the production line.

1) What is the probability P (Y > 20.4) (that is, greater than the stated label weight)?

2) If the company lowered the actual weights of the packages so that μ = 20.4 and σ = .4, what would the probability be of getting a package with weight Y < 19 (noticeably “light”)?

3) If n = 10 packages of mints are chosen at random from the production line, what is the probability that 6 of them will have weight < 19?